Coherent and Chaotic Modifications of Fluctuation Theorems: Theory, Semiclassical Structure, and Molecular-Scale Applications
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Coherent and Chaotic Modifications of Fluctuation Theorems: Theory, Semiclassical Structure, and Molecular-Scale Applications Andrei T. Patrascu FAST Foundation, Destin FL, 32541, USA email: andrei.patr[email protected] Classical fluctuation theorems impose an exponential bias against entropy–reducing trajectories, rendering access to low–entropy states statistically rare. In this work we develop a comprehensive quantum and semiclassical framework showing how coherence, phase structure, and chaotic dynamics reshape these thermodynamic statistics. We begin by formulating trajectory–level entropy production within a precisely defined operational measurement scheme: a coherence–preserving, ancilla–assisted protocol that yields a well-defined entropy–production operator and its spectral projectors. Using this construction, we derive a quantum fluctuation theorem in which the classical ratio P (+Σ) /P ( − Σ) = eΣ/kB is augmented by multiplicative interference factors originating from off–diagonal contributions in the transition amplitudes. We then establish the equivalence between this operator expression and a semiclassical formulation based on the Van Vleck–Gutzwiller propagator. The stationary–phase evaluation yields explicit classical actions, stability determinants, and Maslov indices, making the interference corrections fully transparent as coherent sums over distinct trajectory families. This demonstration provides a detailed and verifiable derivation of the modified ratio and its positivity and norm bounds. To illustrate the consequences of coherence on fluctuation statistics, we analyse two concrete settings. First, we construct solvable finite–dimensional quantum models in which the full entropy– production distribution, its diagonal (classical) limit, and its interference corrections can be computed exactly. Second, we study chaotic and mixed–phase–space systems where chaos–assisted tunnelling supplies multiple near-degenerate pathways that naturally amplify interference. In both cases we provide explicit numerical examples in which oscillatory deviations from the classical fluctuation theorem appear within experimentally relevant coherence windows. Finally, we show how purification and controlled ancilla phases form an operational resource that selectively aligns interfering trajectory contributions into the entropy–decreasing sector, thereby reducing the fluctuation–theorem ratio in a microreversible and thermodynamically consistent manner. We demonstrate how this mechanism can be incorporated into minimal open–system models of chemo–mechanical cycles, including molecular–scale machines with internal resonances or coherent subspaces. Overall, this work delivers a unified and fully detailed account of how quantum coherence, phase control, and classical chaos jointly modify nonequilibrium fluctuation relations, providing a transparent theoretical foundation and concrete model systems in which these effects can be verified. I. INTRODUCTION A. Background on fluctuation theorems Fluctuation theorems (FTs) are exact identities that constrain the statistics of irreversible processes far from equilibrium. They quantify, in a precise way, how likely it is to observe apparent “violations” of the second law when one looks at the dynamics of a small system over a finite observation time. The earliest such relations were discovered in nonequilibrium molecular dynamics simulations by Evans, Cohen and Morriss, and subsequently generalized by Evans and Searles to a broader class of driven steady states [ 1 , 2 ]. Gallavotti and Cohen then proved an asymptotic version for deterministic, chaotic dynamical systems in nonequilibrium steady states [ 3 ], while Lebowitz and Spohn provided a formulation tailored to Markovian stochastic dynamics [ 4 ]. Around the same time, Jarzynski and Crooks derived closely related relations for the fluctuating work performed during arbitrary finite-time protocols, now known as the Jarzynski equality and the Crooks fluctuation relation [ 5 , 6 ]. Modern accounts in stochastic thermodynamics and quantum thermodynamics place these results in a unified trajectory-level framework [7–10]. In the classical stochastic setting, a particularly transparent formulation of the steady-state FT involves the stochastic entropy production Σ[Γ] along a single realization (trajectory) Γof the dynamics. For a fixed observation time τ, the FT can be written schematically as P(+Σ) P(−Σ) = exp Σ/kB,(1) where P (Σ) is the probability density (or mass function) of the random variable Σover trajectories of length τ , and kB is Boltzmann’s constant. Equation (1) is the detailed form of the fluctuation theorem.
2 It states that trajectories with positive entropy production are exponentially more likely than those with the opposite negative entropy production of the same magnitude. The second law for the average entropy production, hΣi ≥ 0, then emerges as a simple corollary of this statistical bias. In the remainder of this subsection we recall a standard (and concrete) derivation of Eq. (1) for a continuous-time Markov jump process, following the stochastic thermodynamics formulation of Seifert [ 7 , 8 ]. This derivation will fix notation and make precise what we mean by entropy production Σas a functional of a trajectory, providing the baseline to which we will later compare quantum and semiclassical modifications. Markov jump processes and trajectory weights Consider a classical system with a finite set of states i = 1 , . . . , N evolving as a continuous-time Markov jump process. The dynamics is specified by a time-dependent rate matrix W ( t )with elements wij ( t ) ≥ 0 for i6 = j , where wij ( t )is the transition rate from state i to state j . The escape rate from state i at time tis ri(t)≡X j6=i wij(t).(2) Let pi(t)denote the probability to occupy state iat time t. The master equation reads d dtpi(t) = X j(6=i)wji(t)pj(t)−wij (t)pi(t).(3) A single trajectory over the time interval [0, τ]can be specified by Γ = i0, t1, i1, t2, . . . , tK, iK,(4) where ik is the state occupied between jump times tk and tk+1 , with 0 = t0< t1<··· < tK< tK+1 = τ , and K is the (random) number of jumps in [0 , τ ]. The path probability density for such a trajectory, given an initial probability distribution pi(0), is P[Γ] = pi0(0) "K−1 Y k=0 e−Rtk+1 tk rik(t)dt wikik+1 (tk+1)#e−Rτ tK riK(t)dt .(5) The exponential factors represent the probability of not leaving the current state during each waiting interval, while the wikik+1 factors represent the actual jumps. To define entropy production we also consider the time-reversed trajectory. Let the time-reversal of Γ be ˜ Γ = ˜ i0,˜ t1,˜ i1,...,˜ tK,˜ iK≡iK, τ −tK, iK−1, . . . , τ −t1, i0,(6) i.e. we traverse the same sequence of states in reverse order, with time-reversed jump times. The reverse dynamics is defined by a rate matrix ˜ W ( t )with elements ˜wij ( t )and initial distribution ˜pi (0). Its path weight for ˜ Γis ˜ P[˜ Γ] = ˜p˜ i0(0) "K−1 Y k=0 e−R˜ tk+1 ˜ tk ˜r˜ ik(t)dt ˜w˜ ik˜ ik+1 (˜ tk+1)#e−Rτ ˜ tK ˜r˜ iK(t)dt ,(7) with ˜ri(t) = Pj6=i˜wij(t). Stochastic entropy production as a path functional The central object of stochastic thermodynamics [ 7 , 8 ] is the total entropy production along a single trajectory, defined as Σ[Γ] ≡kBln P[Γ] ˜ P[˜ Γ] .(8)
3 Physically, Σ[Γ] quantifies how much more likely the observed trajectory is than its time-reversed counterpart. When the dynamics is microreversible and the choice of the reverse process is appropriate, this ratio reduces to entropy production in the system plus entropy production in the environment (medium). Let us consider the simplest and most relevant case: a time-homogeneous Markov process in a nonequilibrium steady state, driven by fixed thermodynamic forces (affinities) such as constant temperature and chemical-potential differences. In this setting the forward and reverse dynamics share the same rates, ˜wij =wij ,(9) and we take the initial distribution of the forward process to be the stationary distribution pss i , with ˜pi (0) = pss i as well. Because the escape rates and waiting-time factors depend only on the visited states and not on the direction of time, the exponential waiting-time factors in Eqs. (63) and (7) cancel in the ratio P[Γ]/˜ P[˜ Γ]. One finds P[Γ] ˜ P[˜ Γ] =pss i0 pss iK K−1 Y k=0 wikik+1 wik+1ik .(10) Taking the logarithm and multiplying by kB, we obtain Σ[Γ] = kBln pss i0 pss iK +kB K−1 X k=0 ln wikik+1 wik+1ik .(11) The first term is the change in stochastic system entropy ssys ( t ) = −kBln pi(t) ( t )between the initial and final times. In a stationary state, pi ( t ) = pss i is constant, hence its contribution vanishes on average. The second term is the entropy production in the medium (heat baths, chemostats, etc.), and is fully expressed in terms of rate ratios. Thermodynamic consistency is encoded through local detailed balance: whenever a jump i→j exchanges an amount qi→jof heat with a heat bath at inverse temperature β, the rates obey wij wji = exp βqi→j.(12) Substituting this into Eq. (11) and grouping the heat exchanges along the trajectory yields k−1 B Σ[Γ] = βQ [Γ]+∆ ssys [Γ], where Q [Γ] = Pkqik→ik+1 is the total heat into the bath and ∆ ssys the change in system entropy along the trajectory [ 7 , 8 ]. Thus Σ[Γ] is not an abstract symbol but a trajectory-level entropy production with direct physical meaning: system entropy change plus entropy flow to the environment. Detailed and integral fluctuation theorems Having defined Σ[Γ] through Eq. (8) , we can derive the detailed FT (1) in a few steps. The probability density of the random variable Σis P(Σ) = X Γ δΣ−Σ[Γ]P[Γ] ,(13) where the sum denotes a sum over all jump sequences and an integral over all jump times. Similarly, the distribution of −Σunder the same forward dynamics is P(−Σ) = X Γ δ−Σ−Σ[Γ]P[Γ] .(14) Using Σ[ ˜ Γ ] = − Σ[Γ] (which follows directly from Eq. (8) ) and the fact that the mapping Γ 7→ ˜ Γ is a one-to-one correspondence between forward and reversed paths, we may rewrite this as P(−Σ) = X Γ δΣ−Σ[˜ Γ]P[˜ Γ] .(15)
4 Inserting the path-ratio identity (8) in the form P[Γ] = ˜ P[˜ Γ] expΣ[Γ]/kB, we obtain P(Σ) = X Γ δΣ−Σ[Γ]P[Γ] = X Γ δΣ−Σ[Γ]˜ P[˜ Γ] eΣ[Γ]/kB =eΣ/kBX Γ δΣ−Σ[Γ]˜ P[˜ Γ] = eΣ/kB˜ P(−Σ) .(16) For a steady state where the forward and reversed processes coincide (P=˜ P), this simplifies to P(Σ) P(−Σ) = exp Σ/kB,(17) which is precisely Eq. (1) . This is the Evans–Searles/Gallavotti–Cohen detailed fluctuation theorem for entropy production in a classical Markovian steady state [1–4]. Integrating both sides of Eq. (17) over all Σ, weighted by e−Σ/kB, yields the integral FT [7] e−Σ/kB≡ZdΣP(Σ) e−Σ/kB= 1 .(18) By Jensen’s inequality, he−Σ/kBi ≥ exp−hΣi/kB, we immediately recover the second law in the form hΣi ≥ 0.(19) Thus, the conventional macroscopic irreversibility emerges from an exact identity relating the statistics of all fluctuations, including rare entropy-decreasing events. Work fluctuation relations: Jarzynski and Crooks The same structure underlies the work-based relations of Jarzynski and Crooks for systems driven out of equilibrium by a time-dependent protocol λt [ 5 , 6 , 10 ]. Consider a system initially prepared in canonical equilibrium at inverse temperature β , with control parameter λ0 , and then driven by λt from t = 0 to t = τ . For each realization of the protocol, a fluctuating work W [Γ] is performed on the system. Jarzynski’s equality states that e−βW =e−β∆F,(20) where ∆ F is the equilibrium free-energy difference between the initial and final parameter values. This is an integral FT for the work, from which the usual inequality hWi ≥ ∆ F (i.e. non-negative average dissipated work) follows. Crooks’ relation refines Eq. (20) into a detailed FT by comparing the work distributions for the forward protocol λt and the time-reversed protocol ˜ λt = λτ−t . Denoting by PF ( W )and PR ( W )the corresponding distributions, one finds [6] PF(W) PR(−W)= exp β(W−∆F).(21) This is again of the form (17) with Σ = β ( W− ∆ F ) kB , which can be interpreted as the entropy production associated with the work performed during the protocol. Summary and outlook In summary, classical fluctuation theorems express a remarkable symmetry of trajectory-level statistics: • A well-defined stochastic entropy production Σ[Γ] can be assigned to each trajectory via the log-ratio of forward and time-reversed path probabilities, Eq. (8). • In steady state, the distribution P (Σ) obeys the detailed FT (17) , which implies the integral FT (18) and the second law hΣi ≥ 0.
5 • For driven processes, analogous relations hold for fluctuating work, Eqs. (20) and (21) , which can be viewed as specific incarnations of the same underlying trajectory symmetry. Throughout this paper we will use Σto denote such a trajectory-level entropy production functional in the classical limit. The central aim of the following sections is to understand how quantum coherence and interference, together with underlying classical structures such as chaotic phase space, modify the probabilities P (+Σ) and P ( − Σ) entering Eq. (1) , and hence reshape the statistics of entropy-decreasing events in a way that remains fully consistent with thermodynamic principles. B. Quantum systems, coherence and measurement schemes In quantum systems, the formulation of fluctuation theorems requires an additional conceptual step beyond the classical setting reviewed in Sec. I A: one must specify how thermodynamic observables are defined and which measurement protocol is used to access their statistics. This is because, unlike in the classical Markovian picture where a trajectory Γhas a well-defined sequence of states at all times, a quantum system described by a density operator ρ ( t )lives in a superposition of energy or configuration eigenstates, and the statistics of work or entropy production depend crucially on the measurement backaction induced by the chosen scheme. These issues are central in quantum stochastic thermodynamics and have been analyzed extensively in Refs. [9, 11, 12]. Broadly speaking, three families of approaches can be distinguished: 1. Two-point measurement (TPM) schemes, where one performs projective measurements of an observable at the beginning and end of the process, and defines stochastic thermodynamic quantities from the resulting eigenvalue differences. 2. Interferometric or ancilla-based schemes, where one couples the system to an auxiliary degree of freedom (qubit, harmonic mode, etc.) and infers work or entropy-production statistics from phase shifts in the ancilla, without performing strong measurements on the system at the initial time. 3. Quantum-jump or continuous-measurement schemes, where the dynamics is unraveled into quantum trajectories consisting of non-unitary evolution segments punctuated by quantum jumps, each jump carrying a definite energy or particle exchange with the environment. While these approaches often yield the same fluctuation relations at the level of averaged quantities, they differ in their treatment of quantum coherence. TPM protocols, in particular, destroy all coherences in the measurement basis at the initial time, whereas interferometric schemes can preserve them and thus reveal interference effects in the fluctuation statistics. In this work we will focus on two precise schemes: • a TPM-based scheme, which provides a “classical-like” reference in which all initial coherences are removed; • a coherence-preserving ancilla-based scheme, in which interference between different quantum pathways appears explicitly as corrections to the classical fluctuation theorem. Before introducing these schemes in detail, we briefly review the structure of quantum dynamics and the role of coherence. Open quantum dynamics and the role of coherence Consider a quantum system S with Hilbert space HS and Hamiltonian HS ( t ), coupled to one or several environments (heat baths, particle reservoirs, chemical chemostats, . . . ). At the level of the combined system–environment state ρSE(t), the dynamics is unitary: ρSE(t) = U(t, 0) ρSE(0) U†(t, 0) ,(22) where U ( t, 0) is the time-ordered exponential of the total Hamiltonian (including interactions). The reduced state of the system is obtained by tracing out the environment, ρS ( t ) = TrE [ ρSE ( t )]. In many physically relevant regimes (weak coupling, Born–Markov approximation, negligible memory effects), the
6 reduced dynamics can be described by a quantum Markov semigroup generated by a Lindblad–Gorini– Kossakowski–Sudarshan (LGKS) master equation [13, 14, 142]: d dtρS(t) = LtρS(t)=−i ~HS(t), ρS(t)+X αLα(t)ρS(t)L† α(t)−1 2L† α(t)Lα(t), ρS(t),(23) where Lα ( t )are Lindblad jump operators and {·,·} denotes the anticommutator. This description is particularly convenient for connecting to stochastic thermodynamics, since each jump operator can often be associated with a specific energy or particle exchange channel, and hence with a heat or matter current into the environment. Quantum coherence refers to the presence of off-diagonal matrix elements of ρS ( t )in a distinguished basis, often taken to be the energy eigenbasis {|n ( t ) i} of HS ( t )or a pointer basis selected by decoherence [ 16 , 17 ]. In the resource-theoretic approach, one identifies coherence as a resource relative to a fixed “incoherent” basis, and quantifies it via functionals such as the `1 -norm of coherence or the relative entropy of coherence [ 18 ]. For our purposes, it suffices to note that if ρS (0) has off-diagonal elements ρmn (0) = hm|ρS (0) |ni in the basis in which thermodynamic quantities are defined (typically the energy basis), then coherent superposition of energy eigenstates can, in principle, affect the statistics of work, heat and entropy production. The key question is how to associate stochastic thermodynamic variables with individual realizations of a quantum process in a way that is both operationally meaningful and compatible with microreversibility. Different measurement schemes answer this question differently, and the treatment of coherence is where they diverge. Two-point projective measurement (TPM) scheme The TPM scheme was introduced in the early formulations of the quantum Jarzynski equality and Crooks relation by Kurchan and Tasaki [ 11 , 12 ] and developed systematically in Refs. [ 9 ]. It is now the standard “work measurement” scheme in quantum fluctuation relations. Let us briefly recall its structure in the simplest setting of a closed system driven by a time-dependent Hamiltonian HS ( t )over a protocol t∈[0, τ]. (i) At t = 0, perform a projective measurement of the system Hamiltonian HS (0), with spectral decomposition HS (0) = PnE0 n|n0ihn0| . The initial state collapses to one of the eigenstates |n0i with probability p0 n = hn0|ρS (0) |n0i , and any initial coherence between distinct energy eigenstates is destroyed. (ii) Evolve the system unitarily according to U ( τ, 0) = Texp{−i ~Rτ 0HS ( t ) dt} , where T denotes time-ordering. (iii) At t = τ , perform a second projective measurement of HS ( τ ) = PmEτ m|mτihmτ| . The conditional probability for obtaining outcome mgiven initial outcome nis pm|n=hmτ|U(τ, 0)|n0i2.(24) A stochastic work value is assigned to each pair (n, m)as Wmn =Eτ m−E0 n.(25) The probability distribution of work is PTPM(W) = X m,n δW−Wmnpm|np0 n.(26) If the initial state is the canonical equilibrium state ρS (0) = e−βHS(0)/Z0 , Jarzynski’s equality and Crooks’ relation follow straightforwardly from microscopic reversibility of the unitary dynamics [ 9 , 11 , 12 ], in direct analogy with the classical case reviewed in Sec. I A. The crucial feature of this scheme is that the first projective measurement forces the initial state to become diagonal in the energy basis. Any initial coherence in that basis is removed before the system evolves under the time-dependent Hamiltonian. As a consequence, the TPM scheme is effectively classical
7 with respect to the chosen measurement basis: the work statistics depend only on the classical probabilities p0 n and the transition matrix Tmn = pm|n . While this makes TPM conceptually clean and mathematically convenient, it also implies that any potential role of quantum coherence in work or entropy-production fluctuations is erased at the outset. Similar TPM constructions can be used to define entropy production in open systems by performing projective measurements of the total Hamiltonian of system plus bath, or of suitable commuting observables that encode energy and particle numbers, at the beginning and end of the protocol [ 9 ]. In all such cases, the first measurement destroys coherences between eigenstates of the measured observables, and the resulting fluctuation theorems resemble their classical counterparts with Σinterpreted as a log-ratio of classical trajectory probabilities, as in Eq. (1). Interferometric, ancilla-based schemes To probe the influence of quantum coherence on fluctuation statistics, one must resort to measurement schemes that do not impose an initial projective measurement on the system. A powerful approach is to couple the system to an auxiliary quantum degree of freedom (an “ancilla”) and store information about thermodynamic quantities in the ancilla’s phase. This philosophy underlies several interferometric schemes for measuring the characteristic function of work or entropy production in closed and open quantum systems [ 9 ], and has been implemented experimentally in platforms such as NMR and trapped ions (we will return to specific setups in later sections). The basic idea can be illustrated for work in a closed system. Instead of measuring HS (0) and HS ( τ ) projectively, one introduces an ancilla A with Hilbert space HA and an initial state | + iA (e.g. an eigenstate of σx for a qubit). One then engineers a sequence of controlled unitaries that correlate the ancilla with the system’s energy at the beginning and end of the protocol in a phase-sensitive way. Schematically, one implements a unitary of the form Utot =e−iuHS(τ)⊗IAU(τ, 0) ⊗IAe+iuHS(0) ⊗IA,(27) acting on ρS (0) ⊗| + ih + |A , where u is a real parameter conjugate to work. After tracing out the system, the reduced state of the ancilla contains the characteristic function of work, G(u) = TrSeiuHS(τ)U(τ, 0) e−iuHS(0) ρS(0) U†(τ, 0),(28) encoded as a phase shift in its off-diagonal density-matrix elements. By performing tomography on the ancilla as a function of u , one reconstructs G ( u )and obtains the full work distribution via Fourier transform, without ever measuring the system projectively at t = 0. Crucially, if ρS (0) contains coherences in the energy basis, they remain present during the evolution and directly affect G ( u )and hence the work distribution. The same strategy can be adapted to entropy production and to open-system settings by replacing HS (0) and HS ( τ )with suitable operators that encode the relevant thermodynamic quantities, or by embedding the LGKS dynamics (23) into a unitary dilation on S + E . What matters for the present work is that these ancilla-based protocols: • provide an operationally well-defined way to access the statistics of thermodynamic variables without performing an initial projective measurement on the system; • preserve initial coherences in the relevant basis, allowing interference between different quantum pathways to show up in the full distribution of work or entropy production; • can be designed so as to reproduce the TPM statistics when the initial state is diagonal, thereby ensuring consistency with the classical limit and with known fluctuation theorems. Quantum trajectories and jump-based entropy production A complementary route to quantum fluctuation theorems is provided by quantum trajectories: individual realizations of the LGKS master equation (23) obtained by continuous monitoring of the environment [ 9 , 142 ]. In a quantum-jump unraveling, for instance, the system undergoes piecewise deterministic evolution interrupted by stochastic quantum jumps associated with the Lindblad operators Lα . Each jump
8 Lα corresponds to a specific energy or particle exchange with a reservoir, and one can assign a stochastic entropy-production increment ∆Σ α to that jump based on local detailed balance at the trajectory level. Summing these increments along a quantum trajectory yields a stochastic entropy-production functional Σ[ γ ], where γ denotes a sequence of jump times, jump types and non-unitary evolution segments. Under suitable microreversibility conditions on the LGKS generator, one can then prove detailed and integral fluctuation theorems for the distribution of Σ[γ]that closely resemble their classical counterparts [9]. Quantum-jump approaches thus provide a natural quantum generalization of the classical Markov-chain framework of Sec. I A, and they will be important for connecting our semiclassical and chaos-based analysis to open-system models of molecular machines in later sections. They also illustrate again the central role of the measurement scheme: the definition of a quantum trajectory already presupposes a specific continuous monitoring of the environment. Measurement schemes used in this work The discussion above highlights a key lesson: in quantum thermodynamics, there is no unique, measurement-independent notion of “trajectory-level entropy production”. Instead, one must specify an operational scheme—TPM, ancilla-based interferometry, or continuous monitoring—that determines how stochastic thermodynamic variables are defined and accessed. In the remainder of this paper we will work with two specific schemes: 1. ATPM-based scheme that serves as our classical-like reference. In this scheme, an initial projective measurement removes all coherences in the relevant basis, and the resulting fluctuation relations for entropy production reduce to the familiar classical form of Eq. (1). 2. Acoherence-preserving ancilla-based scheme, inspired by the interferometric protocols sketched above. In this scheme, the full quantum state (including initial coherences) evolves unitarily or under a Markovian open-system dynamics, while an ancilla encodes the characteristic function of an entropy-production operator. The associated probability distribution exhibits interference-induced corrections to Eq. (1), which we will analyze in detail. By comparing these two schemes within a unified framework, we will be able to isolate and quantify the genuine effects of quantum coherence and interference on fluctuation statistics, and to connect them to underlying classical structures such as chaotic phase space and tunnelling pathways. C. Coherent and chaotic structures The discussion in Secs. I A and I B has emphasized two complementary aspects of nonequilibrium thermodynamics. On the one hand, classical fluctuation theorems such as Eq. (1) describe entropy production in terms of path probabilities of a stochastic process. On the other hand, quantum fluctuation relations require a choice of measurement scheme, and their statistics can depend on the presence or absence of initial coherences. To understand how quantum coherence and classical chaos together reshape fluctuation statistics, it is useful to recall two structural features of quantum dynamics in the semiclassical regime: 1. coherent superposition of many trajectories in Hilbert space, which leads to interference corrections to otherwise classical probabilities; 2. the proliferation of nearly equivalent pathways in systems whose classical counterparts exhibit chaotic dynamics and mixed phase-space structure, including the phenomenon of chaos-assisted tunnelling. In this subsection we sketch these ideas at a level sufficient to motivate our later, more technical development. We will see that chaos provides a natural source of many near-degenerate pathways, while coherence determines how they combine constructively or destructively in the fluctuation-theorem ratio P(+Σ)/P(−Σ).
9 Coherent superposition of semiclassical paths Consider a closed quantum system with configuration-space coordinates q and a unitary evolution operator U ( τ, 0). The transition amplitude from an initial wave packet |ψ0i to a final wave packet |φi after time τcan be written as A(φ←ψ0) = hφ|U(τ, 0)|ψ0i.(29) In the semiclassical limit, this amplitude admits an approximation in terms of classical trajectories connecting the supports of |ψ0i and |φi . For simplicity, think of |ψ0i and |φi as narrow Gaussian wave packets centered at phase-space points ( q0,p0 )and ( qf,pf )respectively. Then, under suitable conditions, the propagator K ( qf,q0 ; τ ) = hqf|U ( τ, 0) |q0i can be approximated by the Van Vleck–Gutzwiller form [72, 73, 166] K(qf,q0;τ)≈(2πi~)−d/2X αDα(qf,q0;τ)1/2expi ~Sα(qf,q0;τ)−iµαπ 2,(30) where the sum runs over all classical trajectories α connecting q0 to qf in time τ , Sα is the classical action along trajectory α , Dα is a stability (Van Vleck) determinant encoding the local behaviour of nearby trajectories, µα is a Maslov index, and d is the number of degrees of freedom [ 72 , 166 ]. Convolution of this propagator with the initial and final packets produces an amplitude of the schematic form A(φ←ψ0)≈X α CαeiSα/~,(31) where the Cα are complex coefficients that depend smoothly on the stability determinants, wave-packet envelopes and Maslov indices. The corresponding transition probability is P(φ←ψ0) = A(φ←ψ0)2≈X α|Cα|2+X α6=β CαC∗ βexpi ~Sα−Sβ.(32) The first term, Pα|Cα|2 , is what one would obtain by adding the contributions of each trajectory incoherently, i.e. as if they were mutually exclusive alternatives. This term therefore plays the role of a “classical” probability, and in many contexts it can indeed be related to a classical phase-space density transported along the Liouville flow [ 22 , 72 ]. The second term captures interference between distinct trajectories and can substantially modify the total probability. To appreciate the possible magnitude of interference effects, consider a situation in which N trajectories contribute with comparable moduli, |Cα|≈|C| for α = 1 , . . . , N . The incoherent (diagonal) contribution then scales as Pdiag ≡ N X α=1 |Cα|2≈N|C|2.(33) If the phases of the CαeiSα/~ are effectively random, the off-diagonal sum in Eq. (32) averages out and P ( φ←ψ0 ) ≈Pdiag . By contrast, if the phases can be aligned—for instance by tuning external parameters or the initial state so that all Sα/~differ only by integer multiples of 2π—then A(φ←ψ0)≈ N X α=1 CαeiSα/~≈NC, (34) and the corresponding probability is P(φ←ψ0)≈N2|C|2=N Pdiag .(35) In other words, coherent addition of N trajectories can, in principle, enhance a transition probability by a factor of order Nrelative to its incoherent counterpart. More generally, the Cauchy–Schwarz inequality yields a universal bound on this enhancement. Writing zα=CαeiSα/~, we have P(φ←ψ0) = N X α=1 zα 2 ≤N X α=1 |zα|2 ≤N N X α=1 |zα|2=N Pdiag .(36)
16 Thus, if the forward process visits the states in the order i0→i1→ ··· → iK , the reversed process visits them in the reverse order iK→iK−1→ ··· → i0, with jump times reflected about τ/2. To define the path weight of ˜ Γ , we must specify a reversed dynamics, characterized by a rate matrix ˜ W ( t ) = [ ˜wij ( t )]. In principle, ˜ W ( t )can be any rate matrix; different choices correspond to different notions of time reversal. In the simplest case, relevant for steady-state fluctuation theorems, one takes the forward process to be time-homogeneous and defines the reversed process to have the same rates, ˜wij ( t ) = wij ( t ) = wij , with initial distribution given by the stationary distribution of the forward process [ 36 ]. In more general situations (e.g. non-stationary driving), the reversed process is chosen so that an appropriate detailed balance or local detailed-balance condition holds; these choices will be discussed in Sec. II B. For the moment, we simply assume that a reversed rate matrix ˜ W ( t )has been specified. The path weight ˜ P [ ˜ Γ ]of the reversed trajectory under the reversed dynamics is then given by the same construction as in Eq. (63), but with tildes on all quantities: ˜ P[˜ Γ] = ˜p0 ˜ i0"K−1 Y k=0 exp −Z˜ tk+1 ˜ tk ˜r˜ ik(u)du!˜w˜ ik˜ ik+1 (˜ tk+1)#exp−Zτ ˜ tK ˜r˜ iK(u)du = ˜p0 iK"K−1 Y k=0 exp −Zτ−tk τ−tk+1 ˜riK−k(u)du!˜wiK−kiK−k−1(τ−tk+1)#exp−Zτ−tK 0 ˜ri0(u)du,(65) where ˜ri(t) = Pj6=i˜wij(t)and ˜p0 iis the initial distribution for the reversed process. In later subsections, the ratio P [Γ] /˜ P [ ˜ Γ ]will be identified with an exponential of the stochastic entropy production associated with the trajectory, and different choices of ˜ W ( t )will correspond to different physical notions of time reversal (e.g. reversal of protocol, steady-state reversal, adjoint dynamics) as discussed in Refs. [ 7 , 36 ]. For now, the key point is that Eqs. (63) and (65) provide explicit and rigorous expressions for the probability densities of forward and reversed trajectories in a general continuous-time Markov jump process. B. Classical entropy production along a trajectory Having obtained explicit expressions for the path weights of a forward Markov jump process and its time-reversed counterpart in Sec. II A, we now define the classical stochastic entropy production along a single trajectory. Our treatment follows the path-space approach of Lebowitz and Spohn [ 4 ] and the stochastic thermodynamics framework developed by Seifert [7, 8], see also [36, 38]. Entropy production as a log-ratio of path probabilities Let P [Γ] denote the probability density of a trajectory Γof the forward process in the time interval [0 , τ ], and let ˜ P [ ˜ Γ ]denote the probability density of the time-reversed trajectory ˜ Γ under the reverse process, as given in Eqs. (63) and (7) , respectively. The stochastic total entropy production along the forward trajectory Γis defined as Σ[Γ] ≡kBln P[Γ] ˜ P[˜ Γ] .(66) This definition has a clear probabilistic meaning: Σ[Γ] measures how much more likely the observed trajectory is under the forward process than its time-reversed partner is under the reverse process. A large positive value of Σ[Γ] indicates a strongly time-asymmetric trajectory (typical of irreversible behavior), while Σ[Γ] < 0corresponds to a trajectory that is more typical of the reverse process than of the forward one. Equation (66) also implies that the average entropy production can be written as a Kullback–Leibler divergence (relative entropy) between the forward and reverse path measures: hΣi ≡ X Γ Σ[Γ] P[Γ] = kBX Γ P[Γ] ln P[Γ] ˜ P[˜ Γ] ,(67)
17 where the sum symbol denotes the sum/integral over all trajectories. This is just hΣi=kBDPk˜ P≥0,(68) with D ( Pk˜ P )the relative entropy. The non-negativity of h Σ i is thus an immediate consequence of the convexity of the logarithm and does not require any further approximations [ 4 , 38 ]. This fact underlies the second law of thermodynamics in stochastic form. Explicit expression for a Markov jump process We now evaluate Eq. (66) using the explicit path weights for a Markov jump process. For clarity we focus on the simplest and physically most relevant case: a time-homogeneous Markov process with rates wij (independent of t) and a unique stationary distribution pss = (pss 1, . . . , pss N)satisfying X jwijpss j−wjipss i= 0 for all i . (69) We take the forward process to start from the stationary distribution, so that pi (0) = pss i , and choose the reverse process to have the same rates and initial distribution, ˜wij =wij,˜pi(0) = pss i.(70) In this steady-state setting, the master equation preserves the stationary distribution, and the waiting-time statistics are stationary. For a trajectory Γ = {i0, t1, i1, . . . , tK, iK}as in Eq. (4), the forward path weight is P[Γ] = pss i0"K−1 Y k=0 e−rik(tk+1−tk)wikik+1 #e−riK(τ−tK),(71) where ri = Pj6=iwij is the (time-independent) escape rate from state i . The time-reversed trajectory ˜ Γ = {iK, τ −tK, iK−1, . . . , τ −t1, i0}has path weight ˜ P[˜ Γ] = pss iK"K−1 Y k=0 e−riK−k(˜ tk+1−˜ tk)wiK−kiK−k−1#e−ri0(τ−˜ tK),(72) with ˜ tk=τ−tK+1−k. Since riis time-independent, we have ˜ tk+1 −˜ tk= (τ−tK−k−1)−(τ−tK−k) = tK−k−tK−k−1,(73) and similarly τ−˜ tK = tK− 0. Comparing Eqs. (71) and (72) , we see that all the exponential survival factors involving escape rates cancel in the ratio P[Γ]/˜ P[˜ Γ]: P[Γ] ˜ P[˜ Γ] =pss i0 pss iK K−1 Y k=0 wikik+1 wik+1ik .(74) Taking the logarithm and multiplying by kByields the stochastic entropy production functional Σ[Γ] = kBln pss i0 pss iK +kB K−1 X k=0 ln wikik+1 wik+1ik .(75) The first term depends only on the initial and final states of the trajectory, while the second term is a sum over all jumps. Decomposition into system and medium contributions The total entropy production (75) can be decomposed into a system part and a medium (environment) part. To do so, one introduces the stochastic system entropy [7, 8] ssys(t)≡ −kBln pXt(t),(76)
18 where pi ( t )is the ensemble probability of finding the system in state i at time t under the forward dynamics. For a trajectory Γ = {i0, t1, . . . , tK, iK} the system entropy at the beginning and at the end of the observation interval is ssys(0) = −kBln pi0(0), ssys(τ) = −kBln piK(τ),(77) and the stochastic system entropy change along Γis ∆ssys[Γ] ≡ssys(τ)−ssys(0) = −kBln piK(τ) + kBln pi0(0).(78) In a stationary state, pi ( τ ) = pi (0) = pss i , and the system entropy change vanishes on average, though not necessarily on each trajectory. The remaining part of Eq. (75) is naturally interpreted as the entropy change in the environment (the medium): Σmed[Γ] ≡kB K−1 X k=0 ln wikik+1 wik+1ik .(79) In many physical situations the system is coupled to one or several heat baths and particle reservoirs such that the transition rates obey a local detailed-balance condition. For example, if all transitions exchange heat with a single thermal bath at inverse temperature β, one typically has [7, 8, 38] ln wikik+1 wik+1ik =β qik→ik+1 ,(80) where qik→ik+1 is the heat transferred to the bath during the jump ik→ik+1 . Inserting Eq. (80) into Eq. (79) gives Σmed[Γ] = β Q[Γ],(81) where Q [Γ] = Pkqik→ik+1 is the total heat dissipated into the bath along the trajectory. More general forms of local detailed balance can account for multiple reservoirs with different temperatures and chemical potentials, in which case Σ med becomes a sum over products of thermodynamic forces and associated currents [38]. Combining Eqs. (66), (78) and (79), we can write the total entropy production as Σ[Γ] = ∆ssys[Γ] + Σmed[Γ],(82) which is the central balance relation of stochastic thermodynamics [ 7 , 8 ]. The first term describes the change in Shannon entropy of the system, while the second term accounts for the entropy change in the environment. In a nonequilibrium steady state, the average system entropy change over long times vanishes and the mean entropy production is entirely due to the medium, 1 τhΣi −−−−→ τ→∞ 1 τhΣmedi ≥ 0.(83) Remarks and outlook The decomposition (82) admits several refinements. In slowly driven nonequilibrium steady states, the medium entropy production can be further decomposed into an excess part and a housekeeping part, as in the Hatano–Sasa formulation of steady-state thermodynamics [ 37 ]. In the rigorous mathematical framework of Ge and Qian, Σ[Γ] is related to the time derivative of a relative-entropy functional that plays the role of a generalized nonequilibrium free energy [ 38 ]. These refinements, however, are not needed for our present purposes. What will be crucial in later sections is that Σ[Γ] is a well-defined functional of the entire trajectory, constructed from the ratio of forward and reverse path weights via Eq. (66) , and that it admits the explicit representation (75) in terms of jump rates and stationary probabilities. This will allow us to define coarse-grained sectors of positive and negative entropy production, and to compare classical and quantum fluctuation relations in a precise and operationally meaningful way.
19 C. Coarse-grained entropy sectors and projectors (classical) In Sec. II B we introduced the trajectory-level entropy production Σ[Γ] for a continuous-time Markov jump process, defined via the log-ratio of forward and time-reversed path weights, Σ[Γ] ≡kBln P[Γ] ˜ P[˜ Γ] ,(84) and we obtained explicit expressions in terms of stationary probabilities and rate ratios, see Eq. (75) . We now use this functional to define coarse-grained sectors of the trajectory space, distinguished by the sign and magnitude of Σ[Γ]. In the classical setting, these sectors are described by characteristic functions on path space, which play the role of “projectors” onto sets of trajectories. This provides the precise classical template that we will later generalize to the quantum case, where these projectors become genuine orthogonal projectors on a Hilbert space. Entropy-production distribution and detailed FT We first recall the probability distribution of the random variable Σ[Γ]. The probability density P ( σ ) for observing a value σof the entropy production in the time interval [0, τ]is P(σ)≡X Γ δσ−Σ[Γ]P[Γ],(85) where the sum symbol denotes the sum over all possible sequences of states and the integral over all possible jump times (i.e. over all trajectories Γ), and δ is the Dirac delta. Likewise, we may define ˜ P ( σ ) as the entropy-production distribution for the reversed process, ˜ P(σ)≡X Γ δσ−Σ[˜ Γ]˜ P[˜ Γ].(86) Using the definition (84) , we can relate these two distributions. Starting from Eq. (85) , we insert P[Γ] = exp Σ[Γ]/kB˜ P[˜ Γ], obtaining P(σ) = X Γ δσ−Σ[Γ]eΣ[Γ]/kB˜ P[˜ Γ] =eσ/kBX Γ δσ−Σ[Γ]˜ P[˜ Γ],(87) where in the second step we used the property of the delta function to replace Σ[Γ] by σ inside the exponential. The mapping Γ 7→ ˜ Γ is a bijection between forward and reversed trajectories, and Σ[ ˜ Γ ] = − Σ[Γ] by construction. We may therefore rewrite the sum over Γas a sum over reversed trajectories and simultaneously change the argument of the delta, P(σ) = eσ/kBX ˜ Γ δσ+ Σ[˜ Γ]˜ P[˜ Γ] =eσ/kB˜ P(−σ).(88) In a steady state with the same dynamics used for the forward and reversed processes (Sec. II B), the path measures coincide, P=˜ P, and Eq. (88) reduces to the classical detailed fluctuation theorem P(σ) P(−σ)=eσ/kB.(89) This relation can also be obtained from a large-deviation analysis of Markov processes, where Σ /τ satisfies a symmetry of its rate function (see e.g. Ref. [39]).
20 Discretization and classical projectors on path space In practice, the entropy-production variable Σmay take a continuum of possible values. To define coarse-grained sectors and to prepare for a quantum analogue in terms of projectors, it is convenient to introduce a discretization (or coarse-graining) of the real line. Let {Ir}r∈R be a partition of R into disjoint intervals (bins) R=[ r∈R Ir, Ir∩Ir0=∅for r6=r0.(90) For each bin Ir, we introduce a classical projector on trajectory space, Π(cl) r(Γ) ≡(1,if Σ[Γ] ∈Ir, 0,otherwise.(91) Mathematically, Π (cl) r is simply the characteristic function of the subset of trajectories whose entropy production falls in Ir. These projectors satisfy the relations Π(cl) r(Γ) Π(cl) r0(Γ) = δrr0Π(cl) r(Γ),X r∈R Π(cl) r(Γ) = 1 for all Γ,(92) which closely resemble the algebraic properties of orthogonal projectors on a Hilbert space. The coarsegrained probability of bin Iris Pr≡X Γ Π(cl) r(Γ) P[Γ] = ZIr dσ P(σ),(93) where we used Eq. (85) to identify Prwith the integral of P(σ)over Ir. Of particular interest are the sign sectors, corresponding to positive and negative entropy production. We define the positive and negative bins I+= [0,∞), I−= (−∞,0),(94) and the corresponding projectors Π(cl) +(Γ) ≡(1,Σ[Γ] ≥0, 0,Σ[Γ] <0,Π(cl) −(Γ) ≡(1,Σ[Γ] <0, 0,Σ[Γ] ≥0.(95) These obey Π(cl) +(Γ) Π(cl) −(Γ) = 0,Π(cl) +(Γ) + Π(cl) −(Γ) = 1,(96) for all trajectories Γ. The coarse-grained probabilities of positive and negative entropy production are P(+) ≡X Γ Π(cl) +(Γ) P[Γ] = Z∞ 0 dσ P(σ), P(−)≡X Γ Π(cl) −(Γ) P[Γ] = Z0 −∞ dσ P(σ).(97) Coarse-grained detailed FT for sign sectors We now derive a relation between P (+) and P ( − )that follows from the exact detailed FT (89) . Starting from Eq. (97) and using the symmetry P(−σ) = e−σ/kBP(σ), we obtain P(−) = Z0 −∞ dσ P(σ) = Z∞ 0 dσ P(−σ) =Z∞ 0 dσ e−σ/kBP(σ).(98) Combining this with P(+) = Z∞ 0 dσ P(σ),(99)
21 we see that the ratio P(+)/P(−)can be written as P(+) P(−)=Z∞ 0 dσ P(σ) Z∞ 0 dσ e−σ/kBP(σ) =De−Σ/kBE−1 Σ>0,(100) where h···iΣ>0 denotes an average with respect to the conditional distribution of Σrestricted to the positive sector, hf(Σ)iΣ>0≡1 P(+) Z∞ 0 dσ f(σ)P(σ).(101) Equivalently, we can work directly at the trajectory level. Using the path-weight relation P [Γ] = eΣ[Γ]/kB˜ P[˜ Γ] in Eq. (97) and changing variables Γ7→ ˜ Γ, we find P(+) = X Γ Π(cl) +(Γ) P[Γ] =X Γ Π(cl) +(Γ) eΣ[Γ]/kB˜ P[˜ Γ] =X Γ Π(cl) +(˜ Γ) eΣ[˜ Γ]/kB˜ P[Γ].(102) Since Σ[˜ Γ] = −Σ[Γ] and Π(cl) +(˜ Γ) = Π(cl) −(Γ), this becomes P(+) = X Γ Π(cl) −(Γ) e−Σ[Γ]/kB˜ P[Γ].(103) In a steady state where ˜ P=P, we obtain P(+) = DΠ(cl) −(Γ) e−Σ[Γ]/kBE, P(−) = DΠ(cl) −(Γ)E,(104) where h·i denotes an average with respect to the forward path measure P[Γ]. Therefore, P(+) P(−)=DΠ(cl) −(Γ) e−Σ[Γ]/kBE DΠ(cl) −(Γ)E=De−Σ[Γ]/kBEΣ<0,(105) where now h·iΣ<0denotes an average over trajectories conditioned on Σ[Γ] <0. The relations (100) and (105) are the natural coarse-grained versions of the detailed FT. They express the ratio of the total probabilities for positive and negative entropy production in terms of an average of e−Σ/kB over the appropriate sector. In particular, since e−Σ/kB> 1whenever Σ < 0, we immediately obtain P(+) P(−)=De−Σ/kBEΣ<0>1,(106) i.e. at any finite time, positive entropy-production trajectories are more probable in total than negative ones, in accordance with the second law. Summary and outlook In summary, we have: • Defined a discretization of entropy-production values and introduced classical “projectors” Π (cl) r (Γ) onto path sectors with Σ[Γ] ∈Ir, Eq. (91). • Shown that the probabilities Pr of these sectors can be written both as sums over trajectories and as integrals of P(σ)over the bins Ir, Eq. (93).
22 • Derived coarse-grained fluctuation relations, such as Eqs. (100) and (105) , relating the total probabilities of positive and negative entropy production to sector averages of e−Σ/kB. In the classical setting, these projectors are simply characteristic functions on path space. In the quantum setting, we will introduce operator-valued projectors Π ± acting on a Hilbert space and define the probabilities of positive and negative entropy production as expectation values Tr [Π ±ρ ]for a suitable evolved state ρ . The classical construction developed here therefore provides a precise template for the quantum generalization to be discussed in Sec. III. III. QUANTUM ENTROPY PRODUCTION AND MEASUREMENT SCHEMES A. Quantum dynamics and microreversibility In the classical framework of Sec. II, entropy production was defined on the space of trajectories of a Markov jump process, and the fluctuation theorem followed from the log-ratio of forward and reversed path weights. In the quantum case, the appropriate notion of “trajectory” is more subtle: it depends on how measurements are performed, and on how the system interacts with its environment. Nevertheless, at a fundamental level the dynamics of a quantum system S coupled to an environment E is unitary. This global unitarity, together with a suitable notion of time reversal, encodes the microreversibility that underlies quantum fluctuation relations [9, 28]. In this subsection we introduce the basic ingredients of this description: global unitary evolution of S + E , the time-reversal operation and its action on the dynamics, and the reduced dynamics of S as a completely positive trace-preserving (CPTP) map with a microreversible dilation, following standard treatments of open quantum systems [13, 14, 142] and quantum fluctuation theorems [9, 11, 12, 40]. Global unitary dynamics of system and environment Let HS and HE be the Hilbert spaces of the system and environment, respectively, and let HSE = HS⊗HEdenote the tensor product. The total Hamiltonian is taken to be of the form HSE(t) = HS(λt)⊗IE+IS⊗HE+Hint,(107) where: •HS ( λt )is the (possibly time-dependent) system Hamiltonian, controlled by a protocol λt (e.g. an external field or parameter varied in time); •HE is the environment Hamiltonian, assumed time-independent and large enough to act as a thermal or chemical reservoir; •Hint is the interaction Hamiltonian between system and environment. The global state ρSE(t)evolves unitarily according to the von Neumann equation d dtρSE(t) = −i ~HSE(t), ρSE(t),(108) with formal solution ρSE(t) = U(t, 0) ρSE(0) U†(t, 0), U(t, 0) = Texp−i ~Zt 0 HSE(u)du,(109) where Tdenotes time ordering. We assume that at the initial time the system and environment are uncorrelated, ρSE(0) = ρS(0) ⊗ρref E,(110) where ρS (0) is the initial state of the system and ρref E is a fixed reference state of the environment. For thermodynamic applications, ρref E is typically taken to be an equilibrium (Gibbs) state at inverse temperature βand possibly fixed chemical potentials: ρref E=1 ZE exp −β(HE−X α µαNα),(111)
23 with ZE the partition function and Nα conserved particle-number operators. This choice encodes the thermodynamic parameters (temperature, chemical potentials) of the environment and will be crucial for relating microscopic reversibility to macroscopic entropy production. The reduced state of the system at time tis obtained by tracing over the environment: ρS(t) = TrEρSE(t)= TrEU(t, 0) ρS(0) ⊗ρref EU†(t, 0).(112) This defines a linear map Et,0on the space of system density operators, ρS(t) = Et,0[ρS(0)],Et,0[ρ]≡TrEU(t, 0) ρ⊗ρref EU†(t, 0).(113) It is a standard result that Et,0 is completely positive and trace preserving (CPTP) [ 142 ]. The complete positivity follows from the Stinespring dilation: any map of the form (113) is a physical quantum channel. Trace preservation is guaranteed by unitarity and the cyclicity of the partial trace: TrSEt,0[ρ]= TrSEU(t, 0) ρ⊗ρref EU†(t, 0)= TrSEρ⊗ρref E= TrS[ρ],(114) for any system state ρ. If the environment is large and the system–environment coupling satisfies standard weak-coupling and Markovian assumptions, the family of maps {Et,0}t≥0 forms a quantum Markov semigroup generated by a Lindblad–Gorini–Kossakowski–Sudarshan (LGKS) master equation [ 13 , 14 , 142 ], as already recalled in Eq. (23). In that case, Et,0=etLwith Lthe LGKS generator. Time reversal and microreversibility To formulate fluctuation relations, we must consider not only the forward process (driven by the protocol λt for t∈ [0 , τ ]), but also a suitably defined time-reversed process. At the microscopic level, time reversal is represented by an antiunitary operator Θon HSE [ 28 , 142 ]. For systems with well-defined time-reversal properties (e.g. spinless particles without magnetic fields), one may take Θ = K , complex conjugation in a suitable basis. More generally, Θcan be written as Θ = UΘK , where UΘ is a unitary and Kdenotes complex conjugation. The time-reversal operation acts on operators Aand states ρas ΘAΘ−1,ΘρΘ−1,(115) and reverses the sign of odd quantities (such as momenta) while leaving even quantities (such as positions) invariant. For the total Hamiltonian (107), we assume that there exists a decomposition of Θas Θ = ΘS⊗ΘE,(116) with antiunitary Θ S and Θ E acting on system and environment separately, such that the following microreversibility condition holds: ΘHSE(λt) Θ−1=HSE(˜ λτ−t),(117) where ˜ λτ−t is the time-reversed driving protocol. For example, if the control parameter is a magnetic field B ( t )that changes sign under time reversal, then ˜ λτ−t involves −B ( τ−t ); for fields that are even under time reversal, ˜ λτ−t=λτ−t. Let UF ( τ, 0) denote the unitary evolution operator generated by HSE ( λt ), as in Eq. (109) , and let UR ( τ, 0) denote the unitary generated by the time-reversed Hamiltonian HSE ( ˜ λτ−t )when the protocol is run from t= 0 to t=τ. Under the microreversibility condition (117), one can show [28, 40] that UR(τ, 0) = Θ U† F(τ, 0) Θ−1.(118) This relation expresses the fact that the time-reversed dynamics is obtained by conjugating the inverse of the forward unitary by the time-reversal operator. It is the quantum analogue of microscopic reversibility in classical Hamiltonian dynamics (Liouville’s theorem) and is the central dynamical ingredient behind quantum fluctuation relations [9, 11, 12, 28].
24 Reduced dynamics and microreversible dilations In thermodynamic fluctuation theorems, one compares probabilities of events under the forward and reversed processes. At the level of the reduced system dynamics, the forward process is described by the CPTP map EF≡ E(λ) τ,0defined in Eq. (113) with the forward protocol λt: EF[ρ] = TrEUF(τ, 0) ρ⊗ρref EU† F(τ, 0).(119) Similarly, the reversed process is described by a CPTP map ER≡ E(˜ λ) τ,0 built from the reversed unitary UR(τ, 0) and (typically) the same environmental reference state ρref E: ER[ρ] = TrEUR(τ, 0) ρ⊗ρref EU† R(τ, 0).(120) The pair ( EF,ER ), together with the choice of initial system states for the forward and reversed processes, constitutes the quantum analogue of the forward and backward Markov processes in the classical stochastic picture. To make the microreversibility relation (118) more explicit at the level of the reduced dynamics, it is useful to write EFin Kraus form. Let ρref E=X α pα|eαiheα|(121) be a spectral decomposition of the environment state. Inserting an orthonormal basis {|eαi} of HE on both sides of UFin Eq. (119), we obtain EF[ρ] = X α,β pαheβ|UF(τ, 0)|eαiρheα|U† F(τ, 0)|eβi =X α,β Mβα ρ M† βα,(122) where the Kraus operators Mβα on HSare defined by Mβα ≡√pαheβ|UF(τ, 0)|eαi.(123) Trace preservation of EFimplies the constraint X α,β M† βαMβα =IS.(124) Using the microreversibility relation (118) , and assuming that ρref E is invariant under time reversal (Θ Eρref E Θ −1 E = ρref E ), one can show that the Kraus operators of the reversed map ER are related to those of EFby a generalized time-reversal operation [9, 40]. Schematically, one finds ˜ Mαβ ∝ΘSM† βαΘ−1 S,(125) up to weights that encode the populations pα of the environment eigenstates and, in thermal settings, the Boltzmann factors associated with energy exchanges. The precise proportionality factors depend on the choice of forward and backward initial states for S and on whether one considers two-point projective measurements or coherence-preserving schemes [9, 28, 40]. The key point is that the pair of maps ( EF,ER )admits a microreversible dilation at the level of S + E : both arise from the same environment reference state ρref E and from unitaries related by Eq. (118) . This ensures that forward and backward processes are related by a quantum generalization of microscopic reversibility, and it is precisely this structure that will underlie the operator-level definition of entropy production and the quantum fluctuation theorem developed in the following subsections. In particular, when HSE describes weak coupling to a thermal reservoir at inverse temperature β , the microreversibility condition implies a Kubo–Martin–Schwinger (KMS) condition for reservoir correlation functions and a corresponding quantum detailed-balance condition for the LGKS generator [ 28 , 142 ]. This guarantees the existence of a stationary Gibbs state ρeq S∝e−βHS and provides the thermodynamic meaning of the entropy production operator constructed later: its eigenvalues quantify the log-ratio of forward and backward transition probabilities in a way that directly generalizes the classical expression of Sec. II.
25 B. TPM definition of entropy production A widely used way to define fluctuating thermodynamic quantities in quantum systems is the two-point projective measurement (TPM) scheme. In its standard form, the TPM protocol associates a stochastic work (and, in suitable contexts, a stochastic entropy production) with each realization of a quantum process by performing projective measurements at the beginning and at the end of the driving protocol. This scheme was first used in the derivations of the quantum Jarzynski equality and Crooks fluctuation relation by Kurchan and Tasaki [ 11 , 12 ], and has since become the standard reference framework for quantum fluctuation theorems [9, 27, 28, 41]. In this subsection we recall the TPM scheme in detail, first for the work performed on a closed system, and then interpret the resulting fluctuating quantity as an entropy-production random variable in an isothermal setup. We also emphasize that the first projective measurement removes all initial coherences in the measurement basis, rendering the subsequent dynamics effectively classical with respect to that basis. TPM scheme for work in a closed system We consider a finite-dimensional system S undergoing a unitary evolution generated by a time-dependent Hamiltonian HS ( t )over a protocol t∈ [0 , τ ]. We assume that the protocol is externally controlled by a parameter λt, so that HS(t) = HS(λt), and that the initial state ρS(0) is arbitrary for the moment. The TPM protocol proceeds in three steps: (i) Initial projective measurement. At time t= 0, one measures the system Hamiltonian HS(0) = X n E0 n|n0ihn0|,(126) obtaining outcome E0 nwith probability p0 n= TrSΠ0 nρS(0),Π0 n≡ |n0ihn0|.(127) Conditioned on outcome n, the post-measurement state is ρS(0+|n) = Π0 nρS(0)Π0 n p0 n =|n0ihn0|.(128) Averaging over outcomes, the unconditional post-measurement state is ρS(0+) = X n Π0 nρS(0)Π0 n,(129) i.e. the dephased version of ρS (0) in the energy basis at t = 0. All initial coherences hn0|ρS (0) |m0i with n6=mare thus destroyed by the first measurement. (ii) Unitary evolution. For 0 < t < τ , the system evolves unitarily according to the Schrödinger equation with Hamiltonian HS(t), yielding the propagator UF(τ, 0) = Texp−i ~Zτ 0 HS(t)dt.(130) Conditioned on initial outcome n, the state just before the final measurement is ρS(τ−|n) = UF(τ, 0) |n0ihn0|U† F(τ, 0).(131) (iii) Final projective measurement. At time t=τ, one measures the final Hamiltonian HS(τ) = X m Eτ m|mτihmτ|,(132) obtaining outcome Eτ mwith conditional probability pF m|n= TrSΠτ mρS(τ−|n)=hmτ|UF(τ, 0)|n0i2,Πτ m≡ |mτihmτ|.(133) The joint probability of outcomes (n, m)in the forward process is then pF mn =p0 npF m|n= TrSΠτ mUF(τ, 0) Π0 nρS(0) Π0 nU† F(τ, 0)Πτ m.(134)
32 The unitary eiuˆ Σ=X s eiuσs|sihs|(181) generates phase factors eiuσsassociated with each entropy eigenvalue. To access the characteristic function GΣ ( u )in Eq. (175) , we perform a controlled operation between the ancilla and the SE degrees of freedom at time t=τ: ˆ V(u)≡ |0iAh0|⊗ISE +|1iAh1|⊗eiuˆ Σ.(182) This is a Ramsey-type gate: if the ancilla is in | 0 iA , nothing happens to SE ; if it is in | 1 iA , the unitary eiuˆ Σ is applied. Such a gate can be implemented by an interaction Hamiltonian of the form Hint ( t ) = g ( t ) ˆ Σ⊗| 1 iAh 1 | switched on for an appropriate duration, in direct analogy with the schemes of Refs. [46, 47]. Applying ˆ V(u)to the forward state ρF SEA(τ)yields ρ(u) SEA ≡ˆ V(u)ρF SEA(τ)ˆ V†(u) = ˆ V(u)ρF SE(τ)⊗|+iAh+|ˆ V†(u).(183) Ancilla readout and characteristic function of ˆ Σ The reduced state of the ancilla after the controlled operation is ρA(u)≡TrSEρ(u) SEA.(184) A straightforward calculation (following the same algebra as in Refs. [46, 47]) gives ρA(u) = 1 21χΣ(u)∗ χΣ(u) 1 , χΣ(u)≡TrSEeiuˆ ΣρF SE(τ).(185) In other words, the off-diagonal element of ρA(u)encodes the characteristic function GΣ(u)≡TrSEeiuˆ ΣρF SE(τ)=χΣ(u).(186) The expectation values of the ancilla Pauli operators are hˆσ(A) xiu= TrAˆσ(A) xρA(u)= Re χΣ(u),(187) hˆσ(A) yiu= TrAˆσ(A) yρA(u)=−Im χΣ(u).(188) Combining Eqs. (187) and (188), we obtain GΣ(u) = χΣ(u) = hˆσ(A) xiu−ihˆσ(A) yiu.(189) Thus, by repeating the protocol for different values of the counting field u , and measuring the ancilla’s Bloch vector components hˆσ(A) xiu and hˆσ(A) yiu , one can reconstruct the full characteristic function GΣ ( u ) and hence the entropy-production distribution Pqm(σ)via Eq. (176). Importantly, the only projective measurement in this protocol is the final readout of the ancilla. At no point is a projective measurement performed directly on S (or E ), in sharp contrast with the TPM scheme of Sec. III B. Preservation of initial coherence Because the system and environment evolve unitarily (or under a Markovian CPTP dynamics obtained by tracing out further degrees of freedom) from ρS (0) ⊗ρref E to ρF SE ( τ ), and because the ancilla couples via the unitary ˆ V ( u )in Eq. (182) , the overall evolution from t = 0 to t = τ is completely coherent. The initial state ρS (0) is never projected onto an energy eigenbasis (or any other measurement basis) at t = 0; instead, its coherences play a direct role in determining ρF SE(τ)and hence GΣ(u).
33 To see this more explicitly, write the final SE state in the eigenbasis {|si} of ˆ Σ, ρF SE(τ) = X s,s0 ρF ss0|sihs0|.(190) Then GΣ(u) = TrSEeiuˆ ΣρF SE(τ)=X s eiuσsρF ss.(191) The diagonal elements ρF ss generally contain interference contributions from different microscopic pathways of the unitary UF ( τ, 0) leading to the same entropy eigenvalue σs . In this sense, the eigenprojectors Π s = |sihs| collect coherent contributions from many underlying trajectories, and the statistics of ˆ Σ retain the effect of initial coherence and quantum interference in the dynamics. By contrast, in the TPM scheme the first projective measurement at t = 0 replaces ρS (0) by a classical mixture Pnp0 n|n0ihn0| , completely removing coherence between energy eigenstates. The resulting trajectory probabilities and entropy production Σ TPM ( n, m )are therefore insensitive to initial coherence. The ancilla-based protocol presented here avoids this dephasing and provides a genuine coherencepreserving measurement of entropy production. Recovery of TPM statistics for diagonal initial states When the initial state ρS (0) is diagonal in the initial Hamiltonian eigenbasis (e.g., a canonical Gibbs state), the ancilla-based scheme reduces to the TPM statistics. This mirrors the analysis of work distributions in interferometric schemes [46, 47]. To make this connection explicit, consider the TPM entropy production Σ TPM ( n, m )defined in Eq. (146) , associated with pairs of energy eigenvalues ( E0 n, Eτ m )of HS (0) and HS ( τ ). As discussed in Sec. III C, we can represent the TPM entropy production as the spectrum of an operator ˆ ΣTPM =X n,m ΣTPM(n, m)|n, mihn, m|,(192) acting on an abstract “history” Hilbert space with basis {|n, mi} corresponding to pairs of energy outcomes. In this representation, it is convenient to encode the TPM forward and backward joint probabilities into diagonal density operators on the same history space: χF≡X n,m pF(n, m)|n, mihn, m|, χR≡X n,m pR(m, n)|m, nihm, n|,(193) where pF ( n, m )and pR ( m, n )are the forward and backward joint TPM probabilities introduced in Sec. III B. The TPM entropy-production operator is then obtained as the operator-valued log-likelihood ratio between the forward and backward diagonal states: ˆ ΣTPM ≡kBln χF−ln χR,(194) where the logarithms are well-defined on the support of χF and χR . Acting on the history basis {|n, mi} , this reproduces the scalar TPM entropy-production values Σ TPM ( n, m )via ˆ ΣTPM|n, mi = ΣTPM(n, m)|n, mi. The TPM joint probabilities pF ( n, m )and pR ( m, n )are encoded in diagonal states χF and χR as in Eq. (193) , and the operator ˆ ΣTPM is precisely the log-likelihood ratio operator of Eq. (194) for this scheme. In the Ramsey protocol, if we choose ˆ Σ = ˆ ΣTPM and prepare ρS (0) diagonal in the initial energy basis (for instance, a Gibbs state), then the characteristic function GΣ ( u )generated by the ancilla coincides with the characteristic function of TPM entropy production, GΣ(u) = X n,m eiuΣTPM(n,m)pF(n, m),(195)
34 and the Fourier transform (176) reproduces the TPM distribution PTPM (Σ). This is the entropyproduction analogue of the result established in Refs. [ 46 , 47 ] for work distributions: for diagonal initial states, the Ramsey interferometric protocol yields exactly the same statistics as the TPM definition. In summary, the ancilla-based measurement protocol presented here extends the TPM framework in a precise way: • It provides a fully unitary, coherence-preserving realization of the entropy-production operator ˆ Σ and its characteristic function GΣ(u). • It involves no projective measurement on the system at t = 0 (or at intermediate times), thereby avoiding the destruction of initial coherence. • It reduces to the TPM entropy-production statistics when the initial state is diagonal in the energy basis, ensuring consistency with the standard quantum fluctuation theorems. In the following sections, we will use this protocol as the operational basis for our analysis of how quantum coherence and classically chaotic structures modify the fluctuation relations for entropy production. IV. GENERAL QUANTUM FT: DECOMPOSITION INTO DIAGONAL AND INTERFERENCE PARTS A. Probabilities for positive/negative sectors We now apply the operator framework of Secs. III C and III D to define, in full generality, the quantum probabilities associated with positive and negative entropy-production sectors. The goal of this subsection is purely structural: to express these probabilities in a basis that diagonalizes the entropy-production operator ˆ Σ and to separate contributions coming from diagonal density-matrix elements (“classical” populations) and from off-diagonal elements (“coherences”). In Sec. IV B we will use this structure to derive the decomposition of the fluctuation-theorem ratio into a classical factor and an interference factor. Forward evolution and entropy sectors Let H denote the Hilbert space on which the entropy-production operator ˆ Σ acts. This may be the Hilbert space of the system S alone, or that of an extended system (e.g. S plus an environment degrees of freedom included via a dilation, or S plus a trajectory register), as discussed in Sec. III A. We assume that the forward process from t= 0 to t=τis described, at this level, by a unitary U(τ)on H, so that ρ(τ) = U(τ)ρ0U†(τ),(196) where ρ0 is the initial state on H . This covers both closed-system dynamics (where U ( τ )is the physical time-evolution operator) and open-system dynamics described by a CPTP map E , thanks to the Stinespring dilation theorem [ 50 , 101 ]: any CPTP map can be realized as a unitary on an enlarged space followed by a partial trace. The entropy-production operator ˆ Σhas spectral decomposition ˆ Σ = X s σs|sihs|, σs∈R,(197) with orthonormal eigenbasis {|si} of H . The spectral projectors onto positive and negative entropy sectors were defined in Sec. III C as Π+≡X σs>0|sihs|,Π−≡X σs<0|sihs|.(198) (If needed, one may also introduce Π0=Pσs=0 |sihs|.) The quantum probabilities of observing positive and negative entropy production in the forward protocol, when the scheme of Sec. III D is used to measure ˆ Σ, are then Pqm(±)≡Tr[Π±ρ(τ)] = TrΠ±U(τ)ρ0U†(τ).(199) These expressions are the quantum analogue of the classical coarse-grained probabilities P ( ± )defined in Eq. (97) : instead of summing over trajectories with Σ[Γ] ≷ 0, we take the expectation values of the spectral projectors Π±in the final quantum state.
35 Expansion in the eigenbasis of ˆ Σ To connect Eq. (A29) with classical-like and interference contributions, we expand all operators in the eigenbasis {|si} of ˆ Σ. Writing ρ0=X m,n ρ0,mn |mihn|, ρ0,mn ≡ hm|ρ0|ni,(200) and defining the Hermitian operators X±≡U†(τ) Π±U(τ),(201) with matrix elements X±,mn ≡ hm|X±|ni=hm|U†(τ) Π±U(τ)|ni,(202) we can rewrite Eq. (A29) as Pqm(±) = TrΠ±U(τ)ρ0U†(τ)= TrU†(τ) Π±U(τ)ρ0 =X m,n ρ0,mn X±,nm.(203) This is precisely the expansion indicated in the outline: Pqm(±) = X m,n ρ0,mn hn|U†(τ) Π±U(τ)|mi.(204) Because both ρ0 and X± are Hermitian, we have ρ∗ 0,mn = ρ0,nm and X∗ ±,mn = X±,nm , so that (203) is real: Pqm(±) = X m ρ0,mm X±,mm +X m6=n ρ0,mn X±,nm =X m ρ0,mm X±,mm + 2 Re"X m<n ρ0,mn X±,nm#.(205) The first term depends only on the diagonal populations ρ0,mm in the ˆ Σ -eigenbasis and is therefore “classical” in character. The second term depends on the off-diagonal elements ρ0,mn (with m6 = n ) and encodes interference between different ˆ Σ-eigenstates. Diagonal (classical) reference probabilities Motivated by Eq. (205), we define the diagonal (or dephased) version of the initial state, ∆[ρ0]≡X m ρ0,mm |mihm|,(206) which is obtained by applying the dephasing (or pinching) channel in the ˆ Σ -eigenbasis [ 50 , 101 ]. This channel kills all off-diagonal elements in that basis and represents the state one would obtain after an ideal projective measurement of ˆ Σat t= 0, followed by discarding the measurement outcome. We then define the classical reference probabilities for positive and negative entropy production as Pdiag(±)≡TrΠ±U(τ) ∆[ρ0]U†(τ)=X m ρ0,mm X±,mm.(207) These are exactly the diagonal contributions in Eq. (205) . Physically, Pdiag ( ± )are the probabilities one would obtain if the initial state were first projectively measured in the eigenbasis of ˆ Σ (or, operationally, in the eigenbasis of the observable whose outcomes are used to construct ˆ Σ ), and then evolved under U ( τ ). In other words, Pdiag ( ± )correspond to a classical mixture of initial eigenstates |mi with probabilities pm = ρ0,mm , each of which contributes independently to the positive or negative sector after the unitary.
36 The difference between the full quantum probability and the diagonal reference probability defines the interference contribution, ∆Pint(±)≡Pqm(±)−Pdiag(±) = 2 Re"X m<n ρ0,mn X±,nm#.(208) By construction, Pqm(±) = Pdiag(±)+∆Pint(±).(209) When the initial state is diagonal in the ˆ Σ -eigenbasis, i.e. ρ0 = ∆[ ρ0 ], all off-diagonal elements vanish, ρ0,mn = 0 for m6=n, and hence ∆Pint(±) = 0. In that limit Pqm(±) = Pdiag(±),(210) so the quantum probabilities reduce to their classical mixture counterparts, in direct analogy with the TPM scheme of Sec. III B. From the perspective of stochastic thermodynamics, Pdiag ( ± )provide the natural “classical baseline” for the positive and negative entropy sectors, obtained by erasing initial coherence in the measurement basis. The quantity ∆ Pint ( ± )then measures the net effect of coherence and interference on the occupation of the positive and negative sectors. In Sec. IV B we will rewrite the fluctuation-theorem ratio in terms of Pdiag ( ± )and ∆ Pint ( ± )and derive bounds on the relative magnitude of the interference contributions, thereby making precise the idea that the quantum FT can be factored into a classical and an interference part. B. Splitting into diagonal and off-diagonal contributions In Sec. IV A we expressed the quantum probabilities of positive and negative entropy production in the forward protocol as Pqm(±)≡TrΠ±U(τ)ρ0U†(τ),(211) where ρ0 is the initial state on the Hilbert space H on which the entropy-production operator ˆ Σ acts, U ( τ )is the global forward unitary, and Π ± are the spectral projectors onto the positive and negative eigen-subspaces of ˆ Σ, ˆ Σ = X s σs|sihs|,Π+=X σs>0|sihs|,Π−=X σs<0|sihs|.(212) We now use this operator expression to split Pqm ( ± )into contributions coming from diagonal populations and from off-diagonal coherences of the initial state, in the eigenbasis {|si} of ˆ Σ . This will provide the “classical + interference” structure that underlies our later decomposition of the fluctuation-theorem ratio. Algebraic splitting in the eigenbasis of ˆ Σ It is convenient to switch to the Heisenberg picture and move the unitary U ( τ )onto the projectors. Define K±≡U†(τ) Π±U(τ),(213) so that Eq. (211) becomes Pqm(±) = Trρ0K±.(214) The operators K± are orthogonal projectors (unitary conjugates of Π ± ), and satisfy 0 ≤K±≤I , with K++K−+K0=Iif a zero-entropy subspace is included via a projector K0. We now expand ρ0and K±in the eigenbasis {|si} of ˆ Σ: ρ0=X m,n ρ0,mn |mihn|, ρ0,mn ≡ hm|ρ0|ni,(215)
37 K±=X m,n K±;mn |mihn|, K±;mn ≡ hm|K±|ni.(216) Substituting these into Eq. (214) and using the orthonormality of the basis states, Tr [ |mihn||n0ihm0| ] = δn,n0δm,m0, we obtain Pqm(±) = X m,n ρ0,mn K±;nm =X m ρ0,mm K±;mm +X m,n m6=n ρ0,mn K±;nm.(217) The first term involves only the diagonal elements ρ0,mm of the initial state in the ˆ Σ -eigenbasis; the second term involves only the off-diagonal matrix elements ρ0,mn , m6 = n , and thus vanishes whenever the initial state is diagonal in this basis. This is precisely the splitting anticipated in the outline: the terms with m = n correspond to a classical mixture over the eigenstates |mi of ˆ Σ , whereas the terms with m6 = n encode the contribution of quantum coherences between different eigenstates. Definition of Pdiag(±)and ∆Pint(±) We now introduce the diagonal (dephased) part of ρ0in the ˆ Σ-eigenbasis, ρdiag 0≡X m pm|mihm|, pm≡ρ0,mm,(218) and the purely off-diagonal (coherent) part ρcoh 0≡X m,n m6=n ρ0,mn |mihn|.(219) By construction, ρ0 = ρdiag 0 + ρcoh 0 and {pm} is a classical probability distribution, pm≥ 0and Pmpm = Tr[ρ0]=1. The decomposition (217) can now be written compactly as Pqm(±) = Trρdiag 0K±+ Trρcoh 0K± =X m pmK±;mm +X m,n m6=n ρ0,mn K±;nm.(220) This motivates the following definitions: Pdiag(±)≡Trρdiag 0K±=X m pmK±;mm,(221) ∆Pint(±)≡Trρcoh 0K±=X m,n m6=n ρ0,mn K±;nm.(222) With these definitions, we obtain the desired splitting Pqm(±) = Pdiag(±)+∆Pint(±).(223) The quantity Pdiag ( ± )is the probability one would obtain if the initial state were first dephased in the ˆ Σ -eigenbasis, i.e. if one applied the completely dephasing channel ∆[ ρ0 ] = ρdiag 0 (cf. resource-theoretic treatments of coherence [ 51 , 52 ]), and then evolved with the same dynamical map. In that sense, Pdiag ( ± ) defines a classical reference probability for the positive/negative sectors. The correction term ∆ Pint ( ± )vanishes whenever ρ0 is diagonal in the ˆ Σ basis (no initial coherence) or whenever K± is diagonal in that basis (no coherent mixing between different ˆ Σ -eigenstates under the Heisenberg evolution). In general, it quantifies the net effect of initial coherence and dynamical interference on the occupation of the positive and negative entropy sectors.
38 Properties and physical interpretation A few basic properties follow directly from the definitions. (i) Classical reference term Pdiag(±).Because K±is a projector, we have 0≤K±;mm =hm|K±|mi ≤ 1for all m. (224) Hence Pdiag(±)is a convex combination of numbers in [0,1], 0≤Pdiag(±) = X m pmK±;mm ≤1.(225) In the classical limit discussed in Sec. II, each label m can be identified with a coarse-grained classical state, and K±;mm plays the role of a classical indicator of whether a trajectory starting in m contributes to the positive or negative entropy sector. In that case Pdiag ( ± )reduces to the classical coarse-grained probability P(±)defined in Eq. (97). (ii) Interference term ∆Pint(±).The interference term involves only off-diagonal elements, ∆Pint(±) = X m6=n ρ0,mn K±;nm.(226) Using the Hermiticity of ρ0and K±, one finds ∆Pint(±) = X m<nρ0,mn K±;nm +ρ0,nm K±;mn∈R.(227) Thus ∆ Pint ( ± )is real, but it can be positive or negative; there is no simple sign constraint. A trivial bound follows from Cauchy–Schwarz: ∆Pint(±)≤X m6=n|ρ0,mn||K±;nm|,(228) and more refined bounds can be obtained by exploiting the structure of ρ0 and K± in particular models. Physically, ∆ Pint ( ± )quantifies the net contribution of quantum coherence to the probability of observing positive or negative entropy production. It depends jointly on: •the initial coherence of the state, encoded in the off-diagonal elements ρ0,mn in the ˆ Σbasis; and • the coherent mixing induced by the dynamics, encoded in the off-diagonal elements K±;nm of the Heisenberg-evolved coarse-graining operators. This separation into a diagonal (population) contribution and an off-diagonal (coherence) contribution is closely related to the decompositions used in the resource theory of coherence [ 52 ] and in analyses of thermodynamic constraints beyond free-energy relations [ 51 ], but here it is formulated directly at the level of the positive/negative entropy sectors. (iii) Classical limit. Whenever the initial state is diagonal in the ˆ Σ basis, ρ0 = ρdiag 0 , we have ∆Pint(±)=0and thus Pqm(±) = Pdiag(±).(229) If, in addition, K± is (approximately) diagonal in the same basis (for instance in a semiclassical limit where decoherence selects {|si} as pointer states), then K±;mn ≈ 0for m6 = n and the dynamics acts classically on the probabilities pm . In this regime both the dynamics and the statistics of entropy production can be faithfully described by a classical stochastic process, and the quantum fluctuation theorem reduces to its classical counterpart, as in Sec. II. By contrast, any deviation of Pqm ( ± )from Pdiag ( ± )is a purely quantum effect: it signals the presence of coherence in the initial state and/or coherent mixing under the dynamics. In the next subsection we will use the decomposition (223) to rewrite the fluctuation-theorem ratio Pqm (+) /Pqm ( − )as a product of a classical factor, built from Pdiag(±), and a well-defined interference correction built from ∆Pint(±).
39 C. Diagonal part and classical FT In Sec. IV B we decomposed the quantum probabilities of positive and negative entropy production into a diagonal (population) part and an interference part, Pqm(±) = Pdiag(±)+∆Pint(±),(230) with Pdiag(±) = Tr[∆Σ[ρ0]B±],∆Pint(±) = Tr[ρcoh B±],(231) where ∆ Σ [ ρ0 ]is the dephased (diagonal) part of the initial state in the eigenbasis of ˆ Σ and ρcoh its coherent part. In this subsection we show that, under standard microreversibility assumptions, the diagonal contribution Pdiag ( ± )obeys the usual classical fluctuation theorem (FT). Thus, the diagonal sector provides a bona fide classical baseline, and all deviations from classical FT arise from the interference corrections ∆Pint(±). Setup and assumptions We work in the operator framework of Secs. III and IV. The key ingredients are: •A Hilbert space Hon which the entropy-production operator ˆ Σacts, with spectral decomposition ˆ Σ = X s σs|sihs|, σs∈R.(232) The labels s index the operational outcomes of the forward and backward protocols (e.g. path records, energy-transfer histories, or ancilla outcomes), and the eigenvalues σs are the associated stochastic entropy-production values. • A forward global unitary UF ( τ )and a backward unitary UR ( τ )related by microreversibility as in Sec. III A, e.g. UR(τ)=ΘU† F(τ) Θ−1,(233) where Θis an antiunitary time-reversal operator acting on the total Hilbert space of system plus environment. • An environment initially prepared in a KMS (thermal equilibrium) state at inverse temperature β , ρref E=e−β(HE−PαµαNα) ZE ,(234) which is invariant under time reversal and satisfies the KMS condition [ 53 , 54 ]. This ensures that microscopic time reversal is compatible with macroscopic thermodynamic parameters. • An initial state of the extended system (system plus environment plus any ancilla used to encode the outcomes s) of the form ρF ext(0) = ρ0⊗ρref E,(235) for the forward protocol, and a corresponding initial state ρR ext (0) for the reversed protocol, chosen so that the stationary or equilibrium conditions appropriate for a steady-state FT hold [9, 28]. The crucial dynamical assumption is that microreversibility and the KMS condition imply a Crooks-type relation for the probabilities of the outcomes sin the forward and backward protocols: pF s pR s = expσs kB, σs∈R,(236) where pF s (resp. pR s ) is the probability of outcome s in the forward (resp. reversed) experiment. This is precisely the operator-level log-likelihood construction of Sec. III C written in component form; the eigenvalues σs are defined so that Eq. (236) holds. Equation (236) is the discrete-outcome analogue of the classical path-space relation P[Γ]/˜ P[˜ Γ] = eΣ[Γ]/kBin Sec. II.
40 Diagonal initial states and detailed FT for Pdiag(σ) We now specialise to initial states that are diagonal in the ˆ Σ-eigenbasis: ρ0= ∆Σ[ρ0] = X s p(0) s|sihs|, p(0) s≥0,X s p(0) s= 1.(237) In this case the coherent part ρcoh vanishes, and hence Pqm(±) = Pdiag(±),(238) i.e. the full quantum probabilities are entirely determined by the diagonal sector. Let us group the outcomes s according to their entropy-production value σs . For each real number σ in the spectrum of ˆ Σ, we define the sets Sσ≡ {s:σs=σ},S−σ≡ {s:σs=−σ},(239) and the corresponding forward and backward probability densities PF diag(σ)≡X s∈Sσ pF s,(240) PR diag(σ)≡X s∈Sσ pR s.(241) Because the eigenvalues σsare defined via the log-likelihood ratio (236), we have for each s, pF s=eσs/kBpR s.(242) Grouping terms with the same value of σ, we obtain PF diag(σ) = X s∈Sσ pF s=X s∈Sσ eσs/kBpR s=eσ/kBX s∈Sσ pR s.(243) Similarly, for the value −σ, PR diag(−σ) = X s∈S−σ pR s=X s0∈Sσ pR s0,(244) where in the last step we simply relabel the index, using the fact that if σs = −σ then there is a corresponding outcome with eigenvalue σ in the reversed protocol. Combining these expressions gives the detailed fluctuation theorem for the diagonal sector: PF diag(σ) = eσ/kBPR diag(−σ).(245) This is the direct quantum analogue of the classical relation P ( σ ) = eσ/kB˜ P ( −σ )in Sec. II C, but here restricted to the diagonal part of the initial state. In the steady-state situation—either equilibrium or a nonequilibrium steady state maintained by time-independent driving—the forward and backward protocols share the same stationary initial state, and the corresponding path-space measures coincide: PF diag =PR diag. Equation (245) then reduces to Pdiag(σ) Pdiag(−σ)=eσ/kB,(246) which is precisely the usual steady-state FT for the distribution of entropy-production values, but now at the level of the diagonal quantum sector. In other words, when the initial state is diagonal in the ˆ Σ -eigenbasis and microreversibility holds, the diagonal part of the dynamics obeys the classical detailed FT.
41 Sign-coarse-grained probabilities Pdiag(±) We now translate the detailed FT (246) into a relation between the coarse-grained sign probabilities Pdiag(±), defined in analogy with Eq. (97) as Pdiag(+) ≡X σs>0 pF s=Z∞ 0 dσ Pdiag(σ),(247) Pdiag(−)≡X σs<0 pF s=Z0 −∞ dσ Pdiag(σ).(248) Using the detailed FT (246), we can re-express Pdiag(−)as an integral over the positive axis: Pdiag(−) = Z0 −∞ dσ Pdiag(σ) = Z∞ 0 dσ Pdiag(−σ) =Z∞ 0 dσ e−σ/kBPdiag(σ).(249) Combining Eqs. (247) and (249) gives the coarse-grained FT for the diagonal sector: Pdiag(+) Pdiag(−)=Z∞ 0 dσ Pdiag(σ) Z∞ 0 dσ e−σ/kBPdiag(σ) =De−Σ/kBE−1 Σ>0,diag ,(250) where hf(Σ)iΣ>0,diag ≡Z∞ 0 dσ f(σ)Pdiag(σ) Z∞ 0 dσ Pdiag(σ) (251) denotes an average with respect to the diagonal distribution restricted to the positive sector. This is the exact analogue of the classical sign-coarse-grained FT derived in Sec. II C, now written explicitly for the diagonal quantum sector. In the special case where the entropy production takes only two symmetric values ± Σ 0 (e.g. a minimal two-outcome process), one has Pdiag(σ) = Pdiag(+) δ(σ−Σ0) + Pdiag(−)δ(σ+ Σ0),(252) and Eq. (246) reduces to Pdiag (+) /Pdiag ( − ) = eΣ0/kB , in the familiar steady-state FT form. Equation (250) is the corresponding generalisation for an arbitrary spectrum of entropy-production values. Summary To summarise: • When the initial state is diagonal in the eigenbasis of the entropy-production operator ˆ Σ and the dynamics is microreversible with a KMS environment, the diagonal sector obeys a detailed FT identical in form to the classical steady-state FT, Eq. (246). • The sign-coarse-grained diagonal probabilities Pdiag ( ± )obey the same coarse-grained FT as in the classical Markov case, Eq. (250). • In this regime, the full quantum probabilities coincide with the diagonal ones, Pqm ( ± ) = Pdiag ( ± ), and the quantum FT reduces exactly to the classical FT. Any deviation from the classical FT in our general framework is therefore entirely due to the interference corrections ∆ Pint ( ± )arising from coherences in the initial state and/or from coherent mixing under the dynamics.
48 Hilbert space and Hamiltonian The system Hilbert space is three-dimensional, HS= span|0i,|1i,|2i,(292) where the basis vectors can be interpreted, for example, as: • |0i: “empty” or unbound state, • |1i: substrateor fuel-bound state, • |2i: product-bound or advanced mechanical state. We take the system Hamiltonian to be HS(λt) = E0|0ih0|+E1|1ih1|+E2|2ih2|+~Ω(λt) 2|0ih1|+|1ih0|,(293) where: •E0, E1, E2are bare energies of the three states, and • Ω( λt )is a real-valued Rabi frequency controlled by an external protocol λt (for instance, an oscillating field or a driven tunnelling amplitude). The coherent coupling between | 0 i and | 1 i allows for the build-up of superpositions in that subspace. The state | 2 i is not coherently coupled in HS ( λt )and will instead be populated by incoherent transitions driven by the environment. For much of what follows, it is convenient to focus on a cyclic protocol of duration τ, HS(λ0) = HS(λτ),(294) so that the system Hamiltonian returns to its initial form at the end of the cycle. A simple case, which we will adopt in explicit calculations, is a piecewise constant drive where Ω( λt )is switched between two values during the cycle. System–bath coupling and Lindblad operators The system interacts weakly with a thermal reservoir at inverse temperature β and fixed chemical potentials. The environment drives incoherent transitions between the energy eigenstates, modelled in the Born–Markov and secular approximations by a Lindblad–Gorini–Kossakowski–Sudarshan (LGKS) master equation, d dtρS(t) = −i ~[HS(λt), ρS(t)] + D[ρS(t)],(295) where the dissipator Dis of the standard form D[ρ] = X αLαρL† α−1 2{L† αLα, ρ}.(296) To realise a “toy molecular machine” structure, we choose the dissipative channels as follows. We single out one transition, | 1 i ↔ | 2 i , as the chemically or mechanically driven step, and allow an additional thermal relaxation between |2iand |0i: L21 =√γ21 |2ih1|, L12 =√γ12 |1ih2|,(297) L20 =√γ20 |2ih0|, L02 =√γ02 |0ih2|.(298) Physically, the pair ( L21, L12 )describes an exchange of energy (and possibly chemical work) with a “fuel” reservoir that drives the 1 → 2transition, while ( L20, L02 )describes equilibration between | 2 i and | 0 i with a thermal bath. The rates obey local detailed balance, γ21 γ12 = exp −β(E2−E1)−∆µ,(299) γ20 γ02 = exp −β(E2−E0),(300)
49 where ∆ µ is an effective chemical affinity associated with the conversion of a fuel molecule (e.g. ATP to ADP). When ∆ µ > 0, the forward transition 1 → 2is favoured, and the cycle 0 ↔ 1 ↔ 2 ↔ 0can sustain a nonzero steady-state current, as in standard stochastic models of molecular motors [116]. The LGKS equation (295) generates a CPTP map Et,0 = e tL on density operators, where the Liouvillian L combines the coherent Hamiltonian evolution and the dissipative part D . For the cyclic protocol of duration τ, the net effect of one cycle on the system state can be written as ρS(τ) = EF[ρS(0)],EF≡ T expZτ 0 dt Lλt,(301) with T denoting time ordering. This map admits a Kraus representation EF [ ρ ] = PαMαρM† α obtained by Stinespring dilation, and a corresponding microreversible backward map ER as discussed in Sec. III A. Entropy-production operator for one cycle For this model, we define the entropy production associated with one driving cycle by combining the LGKS dynamics with the operator framework of Secs. III B–III C. We focus on an isothermal situation where the relevant thermodynamic quantities are heat exchanges with the bath and the chemical work performed by the fuel reservoir. As in Sec. III B, we can introduce a TPM-based entropy-production variable by performing projective measurements of the system Hamiltonian at the beginning and end of the cycle, assuming an initial Gibbs state ρeq S(0) = e−βHS(λ0) Z0 ,(302) and defining the stochastic work Wmn and TPM entropy production Σ TPM ( m, n ) = β ( Wmn − ∆ F ), with ∆ F the equilibrium free-energy change. For a strictly cyclic protocol HS ( λτ ) = HS ( λ0 )one has ∆ F = 0, and hence ΣTPM(m, n) = βEm−En,(303) where En and Em are eigenvalues of HS ( λ0 )associated with the initial and final energy eigenstates. Following Sec. III C, we may then define a TPM entropy-production operator on a history Hilbert space with basis {|n, mi}, ˆ ΣTPM =X n,m ΣTPM(m, n)|n, mihn, m|.(304) However, in order to preserve coherence between | 0 i and | 1 i during the cycle, we will instead work with the coherence-preserving, ancilla-based measurement protocol of Sec. III D. In that setting, the dynamics over one cycle EF is implemented as a unitary UF ( τ )on an enlarged Hilbert space including the bath and ancilla, and the entropy-production operator ˆ Σ is defined as the log-likelihood ratio operator for the forward and backward outcome distributions, cf. Sec. III C. Its spectral decomposition takes the general form ˆ Σ = X s σs|sihs|, σs=kBln pF s pR s ,(305) where s labels the outcomes of the ancilla-based protocol (e.g. effective “jump histories” of the three-level system over one cycle), and pF s , pR s are the corresponding forward and backward probabilities. The projectors onto positive and negative entropy-production sectors are Π+=X σs>0|sihs|,Π−=X σs<0|sihs|,(306) as in Sec. III C. In the following subsections we will compute the probabilities Pqm ( ± )and the interference corrections I± for this specific model, both for TPM and for the coherence-preserving ancilla scheme, and compare them with the classical reference dynamics obtained by dephasing in the energy basis.
50 B. Exact computation of P(Σ) with and without coherence We now use the three-level model of Sec. V A to compute exactly the entropy-production distribution P (Σ) associated with one driving cycle. Because the Hilbert space of the system is finite dimensional and the LGKS generator (295) is time piecewise constant, the one-cycle map EF in Eq. (301) can be diagonalised analytically in a finite-dimensional operator basis. This allows us to evaluate the characteristic function of entropy production and to obtain the full distribution by Fourier inversion, both for the TPM scheme and for the coherence-preserving ancilla protocol of Sec. III D. The corresponding distributions will be denoted by PTPM(Σ) and Pancilla(Σ), respectively. Spectral decomposition of the one-cycle channel We work in Liouville space, representing operators on HS as vectors and the one-cycle map EF as a linear superoperator EF : B ( HS ) → B ( HS ). Because dim HS = 3, the Liouville space is 9-dimensional, and EFcan be diagonalised in terms of right and left eigenoperators {Rα, Lα}8 α=0, EF[Rα]=ΛαRα,(307) E† F[Lα]=Λ∗ αLα,(308) with biorthogonality and completeness relations TrL† αRβ=δαβ,X α RαTrL† α·=IB(HS).(309) Here Λ α are the eigenvalues of EF ; for a CPTP map one of them is Λ 0 = 1, with R0 = ρss S the unique steady state (for a primitive channel) and L0=IS[142]. Any initial state ρS(0) can be expanded in the left eigenoperators as ρS(0) = X α cα(0) Rα, cα(0) = TrL† αρS(0),(310) and its state after one cycle is ρS(τ) = EF[ρS(0)] = X α Λαcα(0) Rα.(311) TPM distribution PTPM(Σ) We first consider the TPM scheme of Sec. III B, applied to a single driving cycle of the three-level system. Let {|εji}2 j=0 and {εj} denote the eigenvectors and eigenvalues of the initial and final Hamiltonian HS(λ0) = HS(λτ), with projectors Πj=|εjihεj|. We assume that the system starts in the Gibbs state ρeq S(0) = e−βHS(λ0) Z0 = 2 X j=0 peq jΠj, peq j=e−βεj Z0 .(312) The TPM protocol projects onto Π j at t = 0, evolves according to EF , and measures Π k at t = τ . The joint forward probability of outcomes (j, k)is pF(j, k) = peq jpF k|j, pF k|j= Tr[ΠkEF[Πj]] .(313) Using the spectral decomposition (311) with ρS(0) = Πj, we have EF[Πj] = X α ΛαTrL† αΠjRα,(314) and hence pF k|j=X α ΛαTrL† αΠjTr[ΠkRα].(315)
51 Because the protocol is cyclic, the free-energy change vanishes, ∆ F = 0, and the TPM entropy production associated with (j, k)is ΣTPM(k, j) = βεk−εj.(316) The forward TPM entropy-production distribution is then PTPM(Σ) = X j,k δΣ−ΣTPM(k, j)pF(j, k),(317) which in this three-level model is a sum of at most nine delta peaks at the values ΣTPM(k, j). It is often convenient to work with the characteristic function GTPM(u)≡eiuΣTPM =X j,k eiuΣTPM(k,j)pF(j, k),(318) which can be written compactly as GTPM ( u ) = Trheiuˆ ΣTPM χFi , where ˆ ΣTPM is the history operator in Eq. (304) and χF is the diagonal forward state defined in Eq. (193) . The distribution (317) is recovered by inverse Fourier transform, PTPM(Σ) = 1 2πZ+∞ −∞ du e−iuΣGTPM(u).(319) Because GTPM ( u )is a finite sum of exponentials (318) with frequencies Σ TPM ( k, j ), the integral (319) can be evaluated exactly, yielding (317). By construction, initial coherences in the energy basis are destroyed by the first projective measurement, and the subsequent dynamics of the TPM trajectories is effectively classical. The distribution PTPM (Σ) thus provides the “classical” baseline against which we will compare the coherence-preserving ancilla protocol. Coherence-preserving Pancilla(Σ) We now turn to the ancilla-based Ramsey protocol of Sec. III D. The dynamics over one cycle is implemented unitarily on an extended Hilbert space Hext = HS⊗HE⊗HA by UF ( τ ), and the entropyproduction operator ˆ Σfor the protocol has spectral decomposition ˆ Σ = X s σs|sihs|, σs=kBln pF s pR s ,(320) as in Eq. (305) . We assume that the initial system state ρS (0) is arbitrary (not necessarily diagonal in the energy basis), while the environment and ancilla are prepared as in Sec. III D. After one cycle and the controlled operation ˆ V(u), the reduced ancilla state ρA(u)encodes the characteristic function Gancilla(u)≡eiuˆ ΣF= Trextheiuˆ ΣρF ext(τ)i,(321) which is obtained operationally from ancilla expectation values via Eq. (189) . In the eigenbasis {|si} of ˆ Σ , ρF ext(τ) = X s,s0 ρF ss0|sihs0|,(322) such that Gancilla(u) = X s eiuσspF s, pF s=ρF ss0s0=s.(323) The corresponding entropy-production distribution for the ancilla protocol is Pancilla(Σ) = X s δ(Σ −σs)pF s=1 2πZ+∞ −∞ du e−iuΣGancilla(u),(324)
52 which again is a finite sum of delta peaks at the eigenvalues σsof ˆ Σ. To make contact with the diagonal/interference decomposition of Sec. IV, we split the initial system state into its diagonal and coherent parts in the ˆ Σ -eigenbasis, ρS (0) = ρdiag S + ρcoh S , and propagate both through the extended unitary UF ( τ ). This induces a corresponding splitting of the final extended state and of the probabilities pF sinto diagonal and interference contributions, pF s=pF s,diag + ∆pF s,int,(325) which yields Pancilla(Σ) = Pdiag(Σ) + ∆Pint(Σ),(326) with Pdiag(Σ) = X s δ(Σ −σs)pF s,diag,(327) ∆Pint(Σ) = X s δ(Σ −σs) ∆pF s,int.(328) Upon integrating over positive and negative Σ, these reduce to the sign-resolved probabilities Pdiag ( ± ) and ∆Pint(±)of Sec. IV B. The corresponding interference quotients I±= ∆Pint(±)/Pdiag(±)enter the factorised FT ratio (263). When the initial state ρS (0) is diagonal in the eigenbasis of HS ( λ0 )(or, more generally, in the eigenbasis of ˆ Σ ), the coherent part ρcoh S vanishes, and Pancilla (Σ) reduces to Pdiag (Σ). In the limit where ˆ Σ coincides with the TPM entropy-production operator ˆ ΣTPM and the ancilla protocol is implemented in the energy basis, Pancilla (Σ) further reduces to the TPM distribution PTPM (Σ) defined in Eq. (317) . Deviations of Pancilla (Σ) from PTPM (Σ) therefore quantify the effect of initial coherence and coherent dynamics on the statistics of entropy production in this finite-dimensional model. In the next subsection we will evaluate PTPM (Σ) and Pancilla (Σ) explicitly for representative parameter choices, and we will compare the resulting FT ratios with the general predictions of Sec. IV C. Explicit I±and modified ratio Having obtained the entropy-production distributions PTPM (Σ) and Pancilla (Σ) in Sec. V B, we now evaluate explicitly the diagonal and interference contributions for the three-level model, and compute the interference quotients I±and the modified fluctuation-theorem ratio. Sign-resolved probabilities and diagonal/interference split For any entropy-production distribution P(Σ) we define the sign-resolved probabilities P(+) = Z∞ 0 dΣP(Σ), P(−) = Z0 −∞ dΣP(Σ),(329) in analogy with Eq. (97) . In the TPM scheme, the distribution PTPM (Σ) is given by Eq. (317) . Since it is a sum of delta peaks, the integrals in (329) reduce to finite sums, PTPM(+) = X j,k: ΣTPM(k,j)>0 pF(j, k),(330) PTPM(−) = X j,k: ΣTPM(k,j)<0 pF(j, k),(331) with ΣTPM(k, j)and pF(j, k)given by Eqs. (316) and (313). For the ancilla protocol, the distribution Pancilla(Σ) is given by Eq. (324), Pancilla(Σ) = X s δ(Σ −σs)pF s,(332)
53 so that Pancilla(+) = X σs>0 pF s, Pancilla(−) = X σs<0 pF s.(333) These can be written in the operator form of Sec. IV A as Pancilla(±) = TrΠ±ρF ext(τ),(334) where ρF ext ( τ )is the final SE +ancilla state and Π ± are the spectral projectors of ˆ Σ , see Eqs. (198) and (320). To isolate the diagonal and interference contributions, we decompose the initial system state in the eigenbasis of ˆ Σ (which, in this finite-dimensional example, can be obtained numerically together with the eigenvalues σs), ρS(0) = ρdiag S+ρcoh S, ρdiag S= ∆Σ[ρS(0)], ρcoh S=ρS(0) −ρdiag S,(335) with ∆ Σ the dephasing map defined in Eq. (206) . Propagating both parts through the extended unitary and the ancilla protocol, we obtain a decomposition of the final extended state and of the probabilities pF s , pF s=pF s,diag + ∆pF s,int,(336) which induces the diagonal and interference parts of the sign-resolved probabilities, Pdiag(±)≡X σs≷0 pF s,diag,(337) ∆Pint(±)≡X σs≷0 ∆pF s,int.(338) By construction, Pancilla(±) = Pdiag(±)+∆Pint(±),(339) in agreement with the operator identity (223). Explicit interference quotients I± The interference quotients I±≡∆Pint(±) Pdiag(±)(340) can now be computed explicitly for any chosen initial state ρS (0) of the three-level system. To connect with the physical interpretation of the model, we consider initial states with coherence in the {| 0 i,| 1 i} subspace, ρS(0) = p0c0 c∗p10 0 0 p2 in the basis {|0i,|1i,|2i}, p0+p1+p2= 1,(341) where c quantifies the coherence between the “empty” and “fuel-bound” states. In the absence of this coherence (c= 0), the initial state is diagonal and one has ∆Pint(±)=0and I±= 0. In practice, Pdiag ( ± )and ∆ Pint ( ± )can be obtained by running the ancilla protocol twice: once with the initial state ρS (0) and once with its dephased version ∆ Σ [ ρS (0)], using the same dynamics and readout. The difference between the two runs yields ∆ Pint ( ± ), and division by Pdiag ( ± )gives I± . In the three-level model, these quantities can also be computed analytically by inserting the explicit matrix elements of ρS (0) and of the Heisenberg-evolved operators K± = U† ( τ )Π ±U ( τ )in the general expressions of Sec. IV B, Pdiag(±) = X m ρ0,mm K±;mm,(342) ∆Pint(±) = X m6=n ρ0,mn K±;nm,(343)
54 with indices m, n now running over the eigenstates of ˆ Σ . For small coherence |c| 1, this yields an approximately linear dependence |I±|∝|c|, consistent with the bounds derived in Sec. IV E: |I±| ≤ C`1(ρS(0)) Pdiag(±),(344) where C`1is the `1-coherence in the ˆ Σ-basis. Modified FT ratio for TPM and ancilla schemes The fluctuation-theorem ratio for the TPM scheme is defined as RTPM(Σ) ≡PTPM(+Σ) PTPM(−Σ),(345) where PTPM ( ± Σ) denote the restrictions of PTPM (Σ) to positive and negative values, respectively. For the cyclic three-level model with an initial Gibbs state, the TPM framework satisfies the standard detailed FT, and hence RTPM(Σ) = expΣ kB,(346) For the ancilla protocol, the sign-coarse-grained probabilities Pancilla ( ± )obey the factorised relation of Sec. IV D, Pancilla(+) Pancilla(−)= expΣcl kB1 + I+ 1 + I− ,(347) where Σ cl is the coarse-grained classical entropy-production scale defined in Eq. (261) . In the three-level model, the spectrum of ˆ Σ is narrow enough that Σ cl ≈ Σin the sense that it tracks the dominant positive values of Σcontributing to Pdiag(Σ). Equation (347) explains the oscillatory deviations of the ancilla curves from the classical line: as Σ varies, the interference quotients I± modulate the ratio exp (Σ /kB )through the factor (1 + I+ ) / (1 + I− ). In particular, when I+> I−> 0the quantum ratio is enhanced above the classical prediction; when −1< I+< I−<0it is suppressed. The three-level model thus provides an explicit realisation of the abstract decomposition developed in Sec. IV. The TPM statistics PTPM (Σ) realise the classical baseline dictated by the diagonal sector, whereas the ancilla-based statistics Pancilla (Σ) reveal the quantitative impact of coherence and interference on the fluctuation theorem via the interference quotients I±. To make the abstract expressions more concrete, we now show the full phase dependence of the coarse-grained FT ratio in the finite-dimensional example. For each value of the ancilla phase φ we evolve the coherent initial state ρ0 ( φ ) = |ψ ( φ ) ihψ ( φ ) | and its dephased counterpart ∆[ ρ0 ], extract the sector probabilities Pqm(±;φ)and Pdiag(±), and compute Rqm(φ) = ln Pqm(+; φ) Pqm(−;φ), Rdiag(φ) = ln Pdiag(+) Pdiag(−).(348) As expected from the operator analysis of Sec. IV, Rdiag ( φ )is independent of φ (it only depends on populations), while Rqm ( φ )acquires a nontrivial oscillatory dependence via the interference quotients I±(φ). D. Interpretation The three-level model allows us to interpret the interference quotients I± and the modified fluctuationtheorem ratios in microscopic terms. In particular, we can see explicitly how initial phase differences between coherent components of the state control both the magnitude and the sign of I± , and hence the oscillatory deviations from the classical FT.
55 0123456 ancilla phase 0.6 0.4 0.2 0.0 0.2 R ( ) = ln[ P (+)/ P ( )] R diag( ) R qm( ) Figure 1. Coarse-grained FT log-ratio R ( φ ) = ln [ P (+; φ ) /P ( − ; φ )] for the three-level finite-dimensional model as a function of the ancilla phase φ . The dashed black curve shows the classical baseline Rdiag ( φ )obtained from the dephased initial state, which is independent of φ because it depends only on populations. The solid curve shows the full quantum result Rqm ( φ )for the coherent initial state |ψ ( φ ) i = ( | 1 i + eiφ| 2 i ) /√2 . The clear oscillatory dependence of Rqm ( φ )on φ illustrates explicitly how coherence and phase control modify the FT ratio through the interference factors I±(φ), while the diagonal FT structure remains unchanged. Phase control of the interference contribution Recall from Sec. IV B that the interference contribution to the sign-resolved probabilities can be written as ∆Pint(±) = X m6=n ρ0,mn K±;nm,(349) where ρ0,mn = hm|ρ0|ni are the matrix elements of the initial state in the eigenbasis {|mi} of ˆ Σ and K±;nm = hn|K±|mi are the matrix elements of the Heisenberg-evolved projectors K± = U† ( τ )Π ±U ( τ ). Writing these complex numbers in polar form, ρ0,mn =|ρ0,mn|eiϕmn , K±;nm =|K±;nm|eiθ±;nm ,(350) we obtain ∆Pint(±)=2X m<n |ρ0,mn||K±;nm|cosϕmn +θ±;nm,(351) which makes the role of phases completely explicit. The magnitudes |ρ0,mn| and |K±;nm| fix the scale of possible interference effects, while the phase sums ϕmn + θ±;nm determine their sign and effective value. The interference quotients I±=∆Pint(±) Pdiag(±)(352) are therefore controlled by the same cosines, subject to the normalisation by the diagonal baseline Pdiag(±). A particularly transparent situation arises when only a single pair ( m, n )contributes significantly to the off-diagonal sum in Eq. (351). In that case ∆Pint(±)≈2|ρ0,mn||K±;nm|cosϕmn +θ±;nm,(353)
56 and the sign of the interference correction is simply the sign of the cosine. Changing the initial relative phase ϕmn by π flips the sign of ∆ Pint ( ± ), while leaving the diagonal sector Pdiag ( ± )unchanged. This directly shows that one can invert the deviation of the quantum FT ratio from the classical line by appropriate phase engineering of the initial coherent superposition. More generally, one can implement phase control by a unitary that is diagonal in the ˆ Σ-eigenbasis, V(α) = X m eiαm|mihm|,(354) applied to the initial state: ρ07→ ρ0 0 = V ( α ) ρ0V† ( α ). This transformation preserves all diagonal elements ρ0,mm and hence leaves Pdiag(±)unchanged, but shifts the phases of coherences, ρ0 0,mn = ei(αm−αn)ρ0,mn, ϕ0 mn =ϕmn +αm−αn.(355) Substituting into Eq. (351), one finds ∆P0 int(±;α)=2X m<n |ρ0,mn||K±;nm|cosϕmn +αm−αn+θ±;nm,(356) so that the set of phases {αm} can be used as “knobs” to tune the interference corrections between their extremal values permitted by the bounds of Sec. IV E. In the three-level model, this corresponds to simple phase shifts on the coherent superposition between | 0 i and | 1 i , which can in principle be implemented by suitable local control pulses. Three-level example and connection to the figures In the three-level molecular-machine model of Sec. V A, we considered initial states with coherence only between the “empty” and “fuel-bound” states, ρS(0) = p0|c|eiφ 0 |c|e−iφ p10 0 0 p2 , p0+p1+p2= 1.(357) In this case, the dominant contribution to ∆ Pint ( ± )comes from the coherence between | 0 i and | 1 i and takes the form ∆Pint(±)≈2|c||K±;10|cosφ+θ±;10,(358) where θ±;10 encodes dynamical phases accumulated during the cycle and depends on the driving protocol and dissipative rates. The interference quotients are then approximately I±≈2|c||K±;10|cosφ+θ±;10 Pdiag(±).(359) Thus |c| sets the overall strength of coherence-induced corrections, while the initial phase φ determines whether interference is constructive or destructive in each sign sector. In particular: • For phases such that cos ( φ + θ+;10 ) > 0and cos ( φ + θ−;10 ) < 0, the positive-entropy sector is enhanced and the negative-entropy sector is suppressed relative to the diagonal baseline, yielding I+>0> I−. • For phases shifted by π , the signs are reversed, I+< 0 < I− , leading to suppression of positiveentropy events and enhancement of negative-entropy events. At the same time, the decoherence analysis of Sec. IV E ensures that increasing dephasing in the entropy-production basis reduces |c| and hence drives I±→ 0, causing the quantum and classical curves to merge. The three-level model thus provides a clear illustration of how phase coherence can be used to steer fluctuation properties in a controlled way, while remaining fully consistent with the general fluctuation-theorem framework.
57 0123456 ancilla phase 0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 diag( ) qm( ) ( ) Figure 2. Mean entropy production per protocol h Σ i ( φ )for the three-level finite-dimensional model as a function of the ancilla phase φ . The dotted curve shows the diagonal (classical) value h Σ idiag ( φ )computed from the dephased initial state; it is independent of φ . The solid curve shows the full quantum mean entropy production h Σ iqm ( φ )obtained from the coherent initial state. The dashed curve is the excess entropy production ∆ h Σ i ( φ ) = h Σ iqm ( φ ) −h Σ idiag ( φ ). The fact that ∆ h Σ i ( φ )changes sign while h Σ iqm ( φ ) ≥ 0for all φ illustrates that coherence can either suppress or enhance dissipation relative to the classical baseline, in full agreement with the modified FT structure and the bounds on the interference quotients. The same mechanism can be visualised at the level of mean entropy production. Using the jump-based definition of medium entropy production for the three-level Lindblad model, we compute for each phase φthe quantum and diagonal averages, hΣiqm(φ),hΣidiag(φ),(360) and their difference ∆hΣi(φ) := hΣiqm(φ)−hΣidiag(φ).(361) Figure 2 shows that (i) both mean entropy productions are strictly non-negative, as required by the second law, but (ii) the quantum value can be either larger or smaller than the classical one depending on φ . In other words, coherence and phase control can either enhance or reduce dissipation relative to the dephased baseline, without ever driving the net entropy production negative. VI. SEMICLASSICAL / VAN VLECK–GUTZWILLER FORMULATION A. Van Vleck–Gutzwiller propagator and trajectory sums In the previous sections we have developed an operator-level description of entropy production and fluctuation relations, and we have analysed in detail a finite-dimensional model. We now turn to the semiclassical regime, where coherent sums over large numbers of classical trajectories provide a natural route to understanding how interference corrections arise and how they are modulated by classical chaos. The central tool is the Van Vleck–Gutzwiller (VVG) semiclassical propagator [ 68 , 70 , 166 ], which approximates the quantum time-evolution kernel in terms of a sum over classical paths weighted by their actions, stability determinants, and Maslov indices.
64 By contrast, the interference term (400) involves pairs of distinct trajectories ( α, β )within the same sector, weighted by products C(±) αC(±)∗ β and phase factors whose arguments are the action differences S(±) α−S(±) β . Geometrically, this phase is proportional to the symplectic area enclosed by the closed loop formed by following trajectory α forward and trajectory β backward in time [ 73 ]. When this area is large compared to ~ , the phase oscillates rapidly under small perturbations of energy, time or external parameters, and the corresponding interference term averages out. Significant contributions to ∆ P(SC) int ( ± ) thus come from pairs of trajectories whose action differences remain small on the scale of ~ , e.g. nearly identical trajectories, time-reversed partners, or correlated orbit pairs in chaotic dynamics [74]. Semiclassical interference quotients To connect with the general structure of Sec. IV D, we introduce the semiclassical interference quotients I(SC) ±≡∆P(SC) int (±) P(SC) diag (±),(404) whenever P(SC) diag (±)>0. With this definition, Eq. (401) becomes Pqm(±) = P(SC) diag (±)1 + I(SC) ±,(405) and the semiclassical fluctuation-theorem ratio can be written as Pqm(+) Pqm(−)=P(SC) diag (+) P(SC) diag (−) 1 + I(SC) + 1 + I(SC) − ,(406) in direct analogy with Eq. (263) . In the limit where interference terms vanish (for instance, under strong dephasing or when off-diagonal contributions are washed out by parameter averaging), I(SC) ±→ 0and the ratio reduces to the classical one built from P(SC) diag (±). The explicit trajectory-pair expression (400) makes the geometric origin of the interference quotients clear: • their magnitude is controlled by the number of contributing orbit pairs, their stability factors and overlaps with the initial state and the entropy projector sectors; • their sign and fine structure are set by oscillatory phases exp [ i ( S(±) α−S(±) β ) /~ ], which depend on the areas enclosed by the corresponding trajectory loops in phase space; • in chaotic systems, families of correlated orbit pairs (such as Sieber–Richter pairs) can produce systematic, parameter-robust contributions to ∆ P(SC) int ( ± ), while uncorrelated pairs average out [73, 74]. These features explain, from a geometric point of view, why the interference factors I± in Sec. V can be both substantial and strongly dependent on the underlying classical dynamics (regular vs chaotic, mixed phase space, presence of transport barriers, etc.). Connection to the operator picture Finally, we emphasise the precise correspondence between the semiclassical quantities introduced here and the operator-level objects of Sec. IV. Identifying Pdiag(±)↔P(SC) diag (±) = X α∈P±C(±) α2,(407) and ∆Pint(±)↔∆P(SC) int (±) = X α,β∈P± α6=β C(±) αC(±)∗ βexpi ~S(±) α−S(±) β,(408) we see that the abstract diagonal/interference splitting of Sec. IV B acquires a concrete geometric meaning:
65 • the diagonal part corresponds to probabilities obtained by summing incoherently over classical trajectories in each entropy sector, • the interference part arises from coherent superpositions of contributions from distinct trajectories with nearly equal actions, whose phase-space geometry controls constructive or destructive interference. In this way, the semiclassical double-sum structure not only reproduces the operator-level modified fluctuation theorem, but also provides an intuitive picture of how classical dynamics and quantum coherence combine to shape the entropy-production statistics. D. Matching with operator derivation We now show how the semiclassical trajectory expansion of Secs. VI A–VI C follows from the operator expression for the entropy-sector probabilities via a stationary-phase approximation. This provides the precise bridge between the abstract operator framework of Sec. IV and the semiclassical picture of sums over classical trajectories, and clarifies the assumptions under which the identification of diagonal and interference contributions is valid. Operator expression in coordinate representation For a pure initial state ρ0 = |ψ0ihψ0| , the probability of being in the positive/negative entropy sector after time τis Pqm(±) = hψ0|U†(τ) Π±U(τ)|ψ0i,(409) cf. Sec. IV. In the position basis, inserting resolutions of the identity, we obtain Pqm(±) = Z···Zdqidq0 idqfdq0 fψ0(qi)ψ∗ 0(q0 i)K(qf, qi;τ)K∗(q0 f, q0 i;τ) Π±(q0 f, qf),(410) where K(qf, qi;τ) = hqf|U(τ)|qii,Π±(q0 f, qf) = hq0 f|Π±|qfi.(411) Equation (410) is an exact expression valid for any Hamiltonian and any choice of projectors Π±. To obtain the semiclassical form, we approximate the propagators by the Van Vleck–Gutzwiller kernel, Eq. (368), K(qf, qi;τ)≈X α:qi→qf C1/2 α(qf, qi;τ) expi ~Sα(qf, qi;τ)−iπ 2µα,(412) K∗(q0 f, q0 i;τ)≈X β:q0 i→q0 f C1/2 β(q0 f, q0 i;τ) exp−i ~Sβ(q0 f, q0 i;τ) + iπ 2µβ,(413) where α and β label classical trajectories with the indicated boundary conditions, Sα,β are the corresponding classical actions, Cα,β are Van Vleck stability determinants, and µα,β are Maslov indices [75, 76]. Stationary-phase conditions and classical trajectories Substituting Eqs. (412) and (413) into Eq. (410) yields Pqm(±)≈X α,β Z···Zdqidq0 idqfdq0 fψ0(qi)ψ∗ 0(q0 i) Π±(q0 f, qf)C1/2 α(qf, qi;τ)C1/2 β(q0 f, q0 i;τ) ×expi ~Sα(qf, qi;τ)−i ~Sβ(q0 f, q0 i;τ)−iπ 2(µα−µβ).(414)
66 The integrals over ( qi, q0 i, qf, q0 f )are oscillatory integrals with rapidly varying phases when ~ is small. In the semiclassical limit, the dominant contributions arise from stationary points of the total phase Φαβ(qi, q0 i, qf, q0 f) = Sα(qf, qi;τ)−Sβ(q0 f, q0 i;τ).(415) Stationarity with respect to variations of qi , q0 i , qf , and q0 f imposes classical equations of motion and matching conditions between the two trajectories: ∂Φαβ ∂qi =−pi,α = 0 (modulo phase from ψ0),∂Φαβ ∂qf =pf,α,(416) ∂Φαβ ∂q0 i = +pi0,β,∂Φαβ ∂q0 f =−pf0,β,(417) where ( qi,α, pi,α )and ( qf,α, pf,α )are the initial and final phase-space points of trajectory α , and similarly for β . When the projectors Π ± and the initial wavefunction ψ0 vary slowly on the scale of the oscillations of the exponential, the leading stationary-phase contributions satisfy, in particular, q0 i=qi, q0 f=qf,(418) and both αand βare classical trajectories connecting the same endpoints (qi, qf)in time τ. Under these conditions, the integrations over ( qi, q0 i, qf, q0 f )can be performed by expanding Φ αβ to second order around the stationary points and evaluating Gaussian integrals, exactly as in the derivation of the VVG propagator itself [ 75 ]. The result is a double sum over pairs of classical trajectories with the same endpoints, with prefactors given by products of Van Vleck determinants and overlaps with ψ0 and Π ± , and with phases determined by action differences Sα−Sβ . This is the origin of the semiclassical pair sum (388) and the decomposition (VI E). Selection of entropy sectors and classical entropy production The remaining ingredient is the implementation of the projectors Π ± . In the operator framework of Sec. IV, Π±are defined as spectral projectors of the entropy-production operator ˆ Σ, ˆ Σ = X s σs|sihs|,Π+=X σs>0|sihs|,Π−=X σs<0|sihs|.(419) In the semiclassical regime, the eigenstates |si can be represented by wave packets localised in phase space around classical trajectories α with a well-defined classical entropy production Σ cl [ α ], and the eigenvalues σs converge to Σ cl [ α ]in the classical limit. The Wigner function of Π ± is then supported on the regions of phase space where Σcl[α]is positive or negative. Assuming that Π ± are sufficiently coarse-grained that their matrix elements in the position basis are approximately diagonal on the semiclassical scale, Π±(q0 f, qf)≈χ±(qf)δ(q0 f−qf),(420) with χ± indicator (or smooth window) functions for the classical regions with Σ cl ≷ 0, the stationary-phase evaluation of Eq. (410) selects families of trajectories α , β whose classical entropy production has a definite sign. This is precisely the origin of the sets P± introduced in Eq. (380) and of the sector amplitudes (397) . The diagonal part of the semiclassical sum, corresponding to α = β , then reproduces the classical coarse-grained probabilities Pdiag ( ± ), while the off-diagonal part encodes interference between pairs of trajectories within the same entropy sector. Validity conditions and relation to Sec. 4 The stationary-phase derivation outlined above rests on several standard assumptions of semiclassical theory [75, 76]: • Isolated non-degenerate stationary points: for given boundary conditions ( qi, qf )and time τ , classical trajectories are isolated and the second derivatives of the action are non-singular. This ensures that the Hessian of Φ αβ is invertible and that Gaussian stationary-phase evaluation is justified. Near caustics (conjugate points), one must instead use uniform approximations (Airyor Pearcey-type integrals), which modify the prefactors but leave the basic double-sum structure intact.
67 • Semiclassical initial state: the initial state ψ0 has a well-defined classical limit, typically described by a smooth Wigner distribution localised on a classical manifold (e.g. an energy shell). This guarantees that ψ0 ( qi )varies slowly on the scale on which the action oscillates, so that it can be treated as a slowly varying prefactor in the stationary-phase analysis. • Coarse-grained projectors: the projectors Π ± are associated with phase-space regions defined by the sign of the classical entropy production and are smooth on the scale of a Planck cell. This justifies the diagonal approximation Π ± ( q0 f, qf ) ≈χ± ( qf ) δ ( q0 f−qf )and the identification of P± with sets of trajectories whose classical entropy production has definite sign. • Times shorter than the Ehrenfest time: the propagation time τ should not exceed the Ehrenfest time beyond which semiclassical approximations may break down due to exponential proliferation of trajectories in chaotic systems. Within this time window, the VVG approximation and the stationary-phase evaluation remain reliable. These conditions are fully compatible with the assumptions made in Sec. IV at the operator level, namely: • Microreversibility: the underlying Hamiltonian and system–bath couplings admit a classical limit with time-reversal symmetry (or more generally, a KMS structure). This ensures that the diagonal sector obeys the classical fluctuation theorem and that trajectory sets P± are related by appropriate time-reversal operations. • Well-defined entropy-production operator: the operator ˆ Σ has a semiclassical limit in which its eigenvalues converge to classical entropy-production functionals Σ cl [ α ]and its projectors Π ± become characteristic functions of the corresponding classical regions. • Coherence vs diagonal structure: the splitting of the initial state into diagonal and coherent parts in the eigenbasis of ˆ Σ translates, in the semiclassical limit, into a decomposition of contributions from individual trajectories (diagonal part) and from pairs of distinct trajectories (interference part). The semiclassical expressions (399) and (400) thus provide a concrete realisation of the abstract decomposition Pqm(±) = Pdiag(±)+∆Pint(±)of Sec. IV B. In this way, the semiclassical trajectory expansion not only matches the operator derivation of the modified fluctuation theorem but also gives it a geometric interpretation in terms of classical phase-space structures. The conditions under which the stationary-phase approximation is valid correspond precisely to those under which the diagonal sector reproduces the classical FT and the interference sector remains a controlled, well-defined correction. E. Semiclassical expressions for I±and their bounds In Sec. VI C we introduced the semiclassical interference quotients I(SC) ±≡∆P(SC) int (±) P(SC) diag (±), Pqm(±) = P(SC) diag (±)1 + I(SC) ±,(421) where P(SC) diag (±) = X α∈P±C(±) α2,(422) ∆P(SC) int (±) = X α,β∈P± α6=β C(±) αC(±)∗ βexpi ~S(±) α−S(±) β.(423) Here P± are the sets of classical trajectories with positive and negative classical entropy production, respectively, and C(±) α are the semiclassical coefficients given by Eq. (393) . In this subsection we derive explicit bounds on I(SC) ± and give them a geometric interpretation in terms of trajectory weights and action differences.
68 Exact algebraic relations from trajectory amplitudes It is convenient to write the sector amplitudes as A±≡X α∈P± C(±) αexpi ~S(±) α,(424) so that Pqm(±) = |A±|2.(425) Expanding the modulus, |A±|2=X α∈P±C(±) α2+X α,β∈P± α6=β C(±) αC(±)∗ βexpi ~S(±) α−S(±) β =P(SC) diag (±)+∆P(SC) int (±),(426) which reproduces Eq. (401). Thus we have the exact identity 1 + I(SC) ±=Pqm(±) P(SC) diag (±)=|A±|2 X α∈P±C(±) α2.(427) Since Pqm(±)≥0by definition, it follows immediately that 1 + I(SC) ±≥0 =⇒I(SC) ±≥ −1,(428) in agreement with the general operator-level bound of Sec. IV E. The equality I(SC) ± = − 1corresponds semiclassically to complete destructive interference within the sector, A±= 0, while P(SC) diag (±)>0. Bounds from Cauchy–Schwarz and effective number of trajectories To obtain upper bounds, we first factor out the overall scale of the coefficients. Define the normalised weights w(±) α=C(±) α2 X γ∈P±C(±) γ2,X α∈P± w(±) α= 1.(429) Write C(±) αin polar form, C(±) α=qP(SC) diag (±)qw(±) αeiφ(±) α,(430) with phases φ(±) α. Then the sector amplitude becomes A±=qP(SC) diag (±)X α∈P±qw(±) αeiφ(±) α+S(±) α/~,(431) and Eq. (427) simplifies to 1 + I(SC) ±=X α∈P±qw(±) αeiθ(±) α 2 , θ(±) α≡φ(±) α+S(±) α ~.(432) This writes the interference factor 1 + I(SC) ±as the squared modulus of a weighted phase sum.
69 Using the triangle inequality and the Cauchy–Schwarz inequality, we obtain X αqw(±) αeiθ(±) α≤X αqw(±) α ≤pN± X α w(±) α!1/2 =pN±,(433) where N±=|P±|is the number of trajectories in the sector. Consequently, 1 + I(SC) ±≤ X αqw(±) α!2 ≤N±, I(SC) ±≤N±−1.(434) This is a crude but general bound: if many trajectories contribute with comparable weight, the interference factor cannot grow arbitrarily, but is bounded by the effective number of paths. A more refined estimate uses the participation ratio N(eff) ±≡1 X αw(±) α2,(435) which quantifies the effective number of significantly contributing trajectories in sector ± . If all weights are equal, w(±) α = 1 /N± , then N(eff) ± = N± ; if one trajectory dominates, N(eff) ±≈ 1. While the exact functional dependence of I(SC) ± on N(eff) ± depends on the phase correlations between trajectories, one can show that [77, 78] |I(SC) ±| ≤ X αqw(±) α2 −1≤ N(eff) ±−1,(436) with the upper bound attained when all relevant phases θ(±) α are equal (maximally constructive interference). Thus the effective number of trajectories controls how large I(SC) ± can be, while the phase structure determines whether this bound is approached or interference is largely washed out. Random-phase estimates and typical magnitude The algebraic bounds above hold for any phase configuration. In chaotic systems, however, it is often reasonable to treat the phases θ(±) α as effectively random variables, at least for sufficiently long propagation times and away from special correlations [ 73 ]. In a simple random-phase model where the θ(±) αare independent and uniformly distributed on [0,2π), one finds I(SC) ±ph = 0,VarphI(SC) ±= 1 −X αw(±) α2= 1 −1 N(eff) ± ,(437) where h·iph denotes averaging over random phases. Thus the typical magnitude of the interference factor is |I(SC) ±|typ ∼s1−1 N(eff) ± .(438) For large effective numbers of trajectories, the variance saturates at a value of order unity, consistent with the simple algebraic bound Eq. (436) . In integrable or nearly integrable systems, phase correlations and families of degenerate orbits can lead to much larger and more structured interference effects, as is well known from periodic-orbit theory [77, 78]. In the presence of additional sources of decoherence or coarse-graining (for instance, weak coupling to an environment or averaging over an external parameter), each off-diagonal term in (423) is multiplied by a damping factor 0 ≤γαβ ≤ 1, which suppresses large phase differences. This further reduces |I(SC) ±| and, in the strong-decoherence limit, drives I(SC) ±→ 0in agreement with the operator-level analysis of Sec. IV E.
70 Semiclassical interpretation of the general bounds The general operator bounds on the interference quotients derived in Sec. IV E—in particular − 1 ≤ I±≤1/Pdiag(±)−1—can thus be given a clear semiclassical interpretation: • The lower bound I(SC) ±≥ − 1corresponds to complete destructive interference within the sector, where the sum over trajectory amplitudes A± vanishes even though individual classical contributions are non-zero. • The upper bounds (434) and (436) express the fact that the interference factor cannot exceed the effective number of coherently contributing classical trajectories minus one. In systems where only a few trajectories dominate, |I(SC) ±| can be large; in strongly chaotic systems with many weakly correlated trajectories, typical values of |I(SC) ±|are of order unity or smaller. • The random-phase estimate shows how, in the absence of special correlations, interference tends to fluctuate around zero, with fluctuations controlled by the participation ratio N(eff) ± . This links the magnitude of interference corrections in the fluctuation theorem directly to measures of complexity of the underlying classical dynamics. In summary, the semiclassical expressions for I± not only reproduce the operator-level structure of the modified fluctuation theorem but also reveal how the strength and sign of interference corrections are governed by the distribution of classical trajectory weights and by the statistics of action differences in phase space. VII. CHAOS-ASSISTED TUNNELLING AND A CONCRETE CHAOTIC MODEL A. Mixed phase space and chaos-assisted tunnelling (short review) In many low-dimensional Hamiltonian systems, classical phase space is neither fully integrable nor fully chaotic, but instead mixed: regular invariant tori coexist with chaotic seas and island chains. A paradigmatic example is a one-degree-of-freedom system with periodic driving, H(p, q, t) = p2 2m+V(q, λt), λt+T=λt,(439) for which the dynamics can be reduced to a symplectic Poincaré map M: (qn, pn)7→ (qn+1, pn+1),(440) obtained by stroboscopically sampling the flow every period T . For sufficiently weak driving, Kolmogorov– Arnold–Moser (KAM) theory implies that most non-resonant invariant tori of the integrable limit survive, while resonant tori break up into chains of stable and unstable periodic orbits, separatrices and thin stochastic layers. As the driving strength increases, these layers overlap, giving rise to extended chaotic seas, while islands of stability around elliptic periodic orbits persist [ 79 ]. The resulting phase space consists of: •regular islands, filled with invariant tori supporting quasi-periodic motion; • chaotic seas, where trajectories explore large regions of phase space in an apparently random (but deterministic) fashion; • partial barriers, such as cantori and broken tori, which restrict but do not completely prevent classical transport between different regions. Quantisation of such mixed systems leads to eigenstates and quasi-modes that are localised on these classical structures: some are concentrated on regular tori (“regular states”), others spread over the chaotic sea (“chaotic eigenstates”). Quantum transitions between regions that are dynamically disconnected at the classical level—for example, between two symmetry-related regular islands separated by a chaotic sea or by invariant tori—are referred to as dynamical tunnelling [80, 83].
71 Dynamical and chaos-assisted tunnelling In a near-integrable system with two symmetry-related regular islands and no intervening chaotic sea, the simplest scenario is direct dynamical tunnelling between the islands. Let |ψLi and |ψRi denote approximate quasi-modes supported on the left and right islands, respectively. In the quantum spectrum, these states typically form a quasi-degenerate doublet with energies E±=E0±∆/2, where ∆∼A(~eff) exp−Stun ~eff ,(441) is an exponentially small splitting determined by an imaginary classical action Stun associated with a complex tunnelling trajectory connecting the two islands, and ~eff is an effective Planck constant (e.g. proportional to 1 /N for a kicked rotor or kicked top). The tunnelling rate between the islands is then Γtun ∼∆/~eff. When the two regular islands are embedded in, or separated by, a chaotic sea, the situation changes qualitatively. The classical chaotic region supports a dense set of chaotic eigenstates |χµi with energies Eµ that mediate indirect couplings between |ψLi and |ψRi . In an appropriate basis, the Hamiltonian can be schematically written as [80, 81] H= EL0VL1VL2··· 0ERVR1VR2··· V∗ L1V∗ R1E10··· V∗ L2V∗ R20E2··· . . .. . .. . .. . .... ,(442) where EL≈ER≈E0 are the energies of the regular islands, {Eµ} are the energies of chaotic states, and VLµ,VRµ describe tunnel couplings between the regular and chaotic subspaces. To leading order in perturbation theory, the effective coupling Veff between the islands is obtained by eliminating the chaotic states, Veff ≈X µ VLµV∗ Rµ E0−Eµ ,(443) so that the chaos-assisted splitting is ∆cha ∼2|Veff |.(444) Because the chaotic energies {Eµ} and couplings {VLµ, VRµ} vary irregularly with system parameters (e.g. ~eff or a control parameter in the potential), Eq. (443) implies that ∆ cha displays strong, apparently erratic fluctuations as these parameters are varied. Statistical models based on random-matrix theory for the chaotic block of H reproduce these fluctuations and yield characteristic distributions of the tunnelling splittings [80, 81]. This mechanism is known as chaos-assisted tunnelling. A key point is that the presence of the chaotic sea can enhance tunnelling rates by many orders of magnitude compared to the direct integrable scaling (441) , because the regular islands are effectively coupled via a large manifold of intermediate chaotic states. At the same time, the splitting becomes extremely sensitive to small changes in ~eff or system parameters, leading to log-normal-like fluctuations and spiky structures in ∆cha as a function of control parameters. Resonance-assisted and chaos-assisted tunnelling In generic mixed systems, the regular islands themselves exhibit an intricate internal structure due to nonlinear resonances: around rational frequency ratios m : n , the motion on KAM tori resonates with the driving, leading to the formation of resonance chains and secondary islands within the main regular region. Quantum mechanically, these resonances open additional tunnelling pathways inside the regular island, which can strongly influence the overall tunnelling behaviour. This mechanism is known as resonance-assisted tunnelling [82]. A simplified picture is as follows. Consider a regular island supporting quantised tori labelled by an approximate action quantum number n , and suppose there is a prominent nonlinear r : s resonance inside
72 the island. The associated resonance chain couples states with actions differing by r quanta, |ni↔|n±ri , with a coupling strength that depends on the resonance Fourier component of the perturbation [82, 83]. States near the edge of the island, in turn, couple to the chaotic sea and ultimately to the other island, giving a multistep tunnelling pathway: (left core island) resonance-assisted −−−−−−−−−−−→ (outer island states) chaos-assisted −−−−−−−−−→ (right island).(445) The combined process—resonance-assisted tunnelling within the regular island followed by chaos-assisted tunnelling through the chaotic sea—leads to rich scaling laws for the average tunnelling splitting and its fluctuations. In particular, one finds plateaus and peaks in the average splitting as a function of ~eff , superimposed on an overall exponential decay related to the effective imaginary action of the dominant tunnelling path [82, 83]. Relevance for fluctuation theorems For our purposes, the importance of chaos-assisted tunnelling is twofold. First, it provides a concrete mechanism by which classical chaotic dynamics and quantum coherence combine to produce large, parameter-sensitive interference effects, in direct analogy with the interference quotients I± of Secs. IV and VI. Second, mixed phase-space systems with well-defined regular islands and chaotic seas offer natural examples where positive and negative entropy sectors correspond to distinct regions of phase space separated by transport barriers, but connected quantum mechanically via tunnelling. In the following subsections we will specify a concrete chaotic model with mixed phase space, quantise it, and compute the corresponding entropy-production statistics, thereby turning the qualitative discussion of chaos-assisted effects into a fully worked example compatible with the operator and semiclassical framework developed in the earlier sections. B. Choice of chaotic model To make the discussion of chaos-assisted tunnelling in Sec. VII A concrete and compatible with the semiclassical formalism of Sec. VI, we now choose a specific time-periodic Hamiltonian system with a well-understood mixed phase space. For definiteness, we work with the paradigmatic kicked rotor (also known as the standard map in its classical form) [84–86]. This model has the following advantages: • its classical dynamics is described by a simple two-dimensional symplectic map with a tunable parameter controlling the degree of chaos; • for intermediate parameter values, the phase space exhibits coexisting regular islands and chaotic seas, as required for chaos-assisted and resonance-assisted tunnelling; • its quantum Floquet operator has a compact and well-studied form, making numerical and semiclassical analysis tractable. Classical kicked rotor and standard map We consider the dimensionless kicked rotor Hamiltonian H(p, q, t) = p2 2+Kcos qX n∈Z δ(t−n),(446) where q∈ [0 , 2 π )is an angular coordinate, p∈R is its conjugate angular momentum, K is a dimensionless kick strength, and the kicks occur at integer times t = n . Between kicks ( t6 = n ), the system evolves freely under p2/2; at each kick it receives an instantaneous impulse proportional to sin q. The Hamiltonian (446) is 1-periodic in time and thus fits the general framework of Eq. (446) . The stroboscopic dynamics over one period from just after kick n to just after kick n + 1 is described by an area-preserving map (pn+1, qn+1) = F(pn, qn),(447)
73 which can be obtained by integrating Hamilton’s equations over one period. The update consists of a free rotation and a kick: kick: pn+1 =pn+Ksin qn,(448) free rotation: qn+1 =qn+pn+1 (mod 2π).(449) Equations (448) – (449) define the Chirikov standard map on the cylinder (or torus, if p is also taken modulo 2π). The Jacobian of this map has unit determinant, det∂(pn+1, qn+1) ∂(pn, qn)= 1,(450) so the map is symplectic and preserves phase-space area. The phase-space structure of the map depends sensitively on K: • For K 1, the dynamics is close to integrable. Most trajectories lie on slightly deformed invariant tori, with small chaotic layers near nonlinear resonances. • For K& 1, invariant tori begin to break up and large chaotic seas form. Stable periodic orbits give rise to regular islands embedded in the chaotic background. • For K& 6, the chaotic sea occupies most of phase space, with only small regular islands surviving around a few stable periodic orbits. In the parameter regime K∼ 2–4, the map exhibits pronounced mixed phase space: sizeable regular islands coexist with an extended chaotic sea, and resonance chains are clearly visible around the islands [ 84 ]. This is the regime of interest for chaos-assisted tunnelling and for the semiclassical analysis of interference corrections in fluctuation relations. Quantum kicked rotor and Floquet operator Quantisation proceeds by promoting ( q, p )to operators obeying [ ˆq, ˆp ] = i~eff , where ~eff is an effective (dimensionless) Planck constant. For concreteness, we impose periodic boundary conditions in q on the interval [0 , 2 π ), which implies a discrete momentum spectrum pm = m~eff with m∈Z . If ~eff is chosen such that 2 π/~eff = N is an integer, the Hilbert space can be restricted to an N -dimensional space spanned by the angular momentum eigenstates {|mi}N−1 m=0 , effectively corresponding to motion on a torus [86]. The time evolution over one period (from just after kick n to just after kick n + 1) is governed by the unitary Floquet operator UF= exp−i ~eff ˆp2 2exp−iK ~eff cos ˆq,(451) where we have ordered the free rotation after the kick (other conventions differ by a similarity transformation and do not affect the quasienergy spectrum). The eigenstates and eigenvalues of UF, UF|φki= e−iεk|φki,(452) define the quasienergy spectrum {εk} in [ −π, π )and a basis of Floquet modes {|φki} that reflect the mixed classical phase-space structure: some modes are localised on regular islands, others are delocalised over the chaotic sea, and some hybridise these behaviours [85, 86]. In the semiclassical limit ~eff → 0with K fixed, the matrix elements of UF in the angle basis {|qji} can be approximated by a Van Vleck–Gutzwiller-type sum over classical trajectories of the standard map, as discussed in Sec. VI. Tunnelling between symmetry-related islands in the standard-map phase space, mediated by the chaotic sea, realises the chaos-assisted tunnelling mechanism reviewed in Sec. VII A. In subsequent subsections we will specify how entropy-production sectors and projectors Π ± are defined for this model and how the interference factors I± can be computed semiclassically from trajectory sums guided by the classical standard map.
80 so that Pqm(±) = P(SC) diag (±)1 + I(SC) ±. Let us introduce the effective number of paths (“participation ratio”) Neff ±≡1 X α∈P±w(±) α2,(477) which counts, in a coarse sense, how many trajectory families contribute significantly to the sector amplitude. If every path carries the same weight w(±) α = 1 /N± for N± = |P±| trajectories, then Neff ±=N±. From Eq. (475) we immediately obtain two key properties: 1. Positivity and lower bound: since Pqm(±)≥0, we have 1 + I(SC) ±≥0, i.e. I(SC) ±≥ −1.(478) The limiting value I(SC) ± = − 1corresponds to complete destructive interference within the sector (the amplitude sum vanishes, even though individual classical contributions are non-zero). 2. Maximal coherent enhancement: by the triangle inequality, 1 + I(SC) ±1/2=X αqw(±) αeiθ(±) α≤X αqw(±) α.(479) Using Cauchy–Schwarz, Pαqw(±) α≤qNeff ±, we obtain the upper bound 1 + I(SC) ±≤ Neff ±, I(SC) ±≤ Neff ±−1.(480) This bound is saturated when all phases θ(±) α are equal, so that all trajectories interfere fully constructively. Equation (480) makes precise the statement that the maximal possible interference enhancement grows with the effective number of coherent pathways. In an integrable or nearly integrable system, Neff ± is often of order unity, so that |I(SC) ±| is modest. In a chaotic system with mixed phase space, the number of distinct trajectory families connecting the left and right islands through the chaotic sea can grow rapidly with time, and Neff ± can become large, opening the possibility of |I(SC) ±| 1in finely tuned situations (e.g. at chaos-assisted tunnelling resonances). On the other hand, in the generic situation where the phases θ(±) α behave effectively randomly (for given control parameters and sufficiently long times), one finds that hI(SC) ±iph = 0 and Var ( I(SC) ± ) = 1 − 1 /Neff ± for equal weights. Thus the typical size of |I(SC) ±| remains of order unity even for large Neff ± . Large interference enhancements |I(SC) ±| ∼ Neff ± require special phase correlations, which in our context are provided by the small number of chaos-assisted and resonance-assisted paths that take the system through the chaotic sea in a correlated way. Timescales: Ehrenfest time and dephasing time The effective number of coherent pathways is not just a property of the classical dynamics; it also depends on how long we allow the system to evolve and on how long quantum coherence is preserved. Two timescales are particularly important [98, 99]: • the Ehrenfest time tE , which marks the crossover from quasi-classical wavepacket dynamics to fully delocalised quantum evolution in a chaotic system; • the dephasing time τϕ , which characterises the decay of phase coherence due to coupling to an environment or external noise.
81 Ehrenfest time. Consider two nearby classical trajectories with initial phase-space separation δz (0) in a region with Lyapunov exponent λL>0. Their separation grows as |δz(t)| ∼ |δz(0)|eλLt.(481) In quantum mechanics, the minimal resolvable phase-space volume is set by ~eff . If the relevant classical phase-space scale for coarse-graining is Acl (e.g. the area of an island or of a chaotic band), an initially localised minimal wavepacket of size √~eff requires a time tE∼1 λL ln Acl ~eff (482) to spread over the classical structure in question. For times ttE , the quantum evolution is effectively quasi-classical: a single “bundle” of trajectories dominates and Neff ± remains small, so that I(SC) ± is close to its classical value (typically zero). For times t&tE , the wavepacket has explored a large portion of the chaotic sea, and the number of distinguishable trajectory families grows rapidly, roughly as Neff ±(t)∼ehKS(t−tE),(483) where hKS is the Kolmogorov–Sinai entropy rate of the chaotic region. This is the regime where chaos most strongly multiplies the number of coherent pathways, and where large interference corrections to the FT are in principle possible. Dephasing time. Coupling to an environment (e.g. a thermal bath or technical noise) introduces dephasing, which damps the off-diagonal elements of the density matrix in a suitable basis [ 100 ]. A simple model is a pure dephasing Lindblad equation ˙ρ=−i[H, ρ]−1 2τϕ [ˆ A, [ˆ A, ρ]],(484) where ˆ A is an observable whose eigenbasis defines the preferred classical states. In that basis, off-diagonal elements decay as ρmn(t) = ρmn(0) e−t/τϕ, m 6=n, (485) so that the interference terms in Pqm ( ± )acquire an overall factor e −t/τϕ , while the diagonal terms are approximately unaffected. In the semiclassical pair-sum representation, ∆P(SC) int (±;t)∼X α,β∈P±(t) α6=β C(±) αC(±)∗ βei(Sα−Sβ)/~eff e−t/τϕ,(486) each pair contribution is suppressed by e −t/τϕ , but the number of contributing pairs grows roughly as Neff ± ( t ) 2 in the absence of correlations. A simple random-phase estimate then gives a typical interference magnitude scaling as |I(SC) ±(t)|typ ∼e−t/τϕqNeff ±(t),(487) illustrating the competition between chaos-induced pathway multiplication (which increases Neff ± ( t )) and dephasing (which suppresses coherence). Combining these estimates, we see that sizeable interference corrections |I(SC) ±|∼O (1) are expected primarily in the window tE.t.τϕ,(488) where the wavepacket has had enough time to explore multiple chaotic pathways, but not long enough for decoherence to fully wash out phase correlations. For ttE , there are too few effective pathways; for tτϕ, coherence is lost and the quantum FT reduces to its classical form.
82 Chaos as a pathway multiplier for FT corrections The picture that emerges is the following: • Classical chaos endows the system with an exponentially large number of possible trajectories connecting a given initial coarse-grained region (e.g. the left island) to a given final one (e.g. the right island). This is the “pathway multiplier” effect, quantified by the growth of Neff ± ( t )with time. • Quantum coherence allows these paths to interfere, producing interference quotients I± that directly modify the fluctuation-theorem ratio via ln Pqm(+) Pqm(−)=Σcl kB + ln 1 + I+ 1 + I− .(489) The maximal magnitude of I± scales with Neff ± [Eq. (480) ], and large constructive interference corresponds to rare parameter values where chaos-assisted and resonance-assisted trajectories become phase-aligned. • Ehrenfest and dephasing times delimit the interval where chaos can act as an effective pathway multiplier for FT corrections: tE marks the onset of multiple coherent pathways, while τϕ marks the loss of phase coherence. In between, especially near chaos-assisted tunnelling resonances, the quantum FT can exhibit pronounced, parameter-sensitive deviations from the classical steady-state relation. This completes the conceptual link between the operator-level modified fluctuation theorem of Sec. IV, the semiclassical trajectory-sum picture of Sec. VI, and the chaos-assisted tunnelling phenomenology of Sec. VII: chaos does not change the form of the fluctuation theorem, but it substantially enriches the space of coherent pathways that contribute to the entropy-production statistics, thereby enhancing and structuring the interference corrections encoded in the factors I±. VIII. PURIFICATION AND PHASE-ENGINEERING AS A CONTROL RESOURCE A. Purification basics and phase freedom In this section we view quantum coherence and interference as resources that can be prepared and tuned at the level of initial states by using an ancilla. The natural language for this is the purification of mixed states. We recall the basic structure of purifications, emphasise their isometric freedom on the ancilla, and show how this freedom can be used to introduce controllable phases that subsequently appear in the interference factors I±. Purification of a mixed state Let ρS be a density operator on a finite-dimensional system Hilbert space HS . A purification of ρS is a pure state |ΨSAion an enlarged Hilbert space HSA =HS⊗HAsuch that ρS= TrA[|ΨSAihΨSA|].(490) Here HAis the Hilbert space of an ancilla Awhose dimension is at least the rank of ρS. Let the spectral decomposition of ρSbe ρS=X i λi|siihsi|, λi≥0,X i λi= 1,(491) where {|sii} form an orthonormal eigenbasis of ρS . A canonical purification is obtained by introducing an orthonormal basis {|aii} of an ancilla Hilbert space HAwith dim HA≥rank(ρS)and defining |Ψ(0) SAi=X ipλi|sii⊗|aii.(492)
83 Then TrAh|Ψ(0) SAihΨ(0) SA|i=X i,j pλiλj|siihsj| haj|aii | {z } =δij =X i λi|siihsi|=ρS,(493) so that Eq. (490) is satisfied. The Schmidt coefficients of | Ψ (0) SAi are {√λi} , which are uniquely determined (up to degeneracies) by ρS[101]. It is often convenient to reparametrise the eigenvalues as λi=p2 iwith pi≥0and Pip2 i= 1, so that ρS=X i p2 i|siihsi|,|Ψ(0) SAi=X i pi|sii⊗|aii.(494) This notation will be used below when we introduce phase parameters. Isometric freedom on the ancilla Purifications of a given mixed state are not unique: if | Ψ SAi and | Φ SAi are two purifications of ρS , they must be related by an isometry acting on the ancilla alone. More precisely, if HA has dimension exactly equal to the rank of ρS, there exists a unitary VAon HAsuch that [101] |ΦSAi= (IS⊗VA)|ΨSAi.(495) If dim HAis larger, VAis an isometry whose range lies in the support of the original purification. A simple way to see this is to consider the Schmidt decompositions of the two purifications, |ΨSAi=X ipλi|sii⊗|aii,|ΦSAi=X ipλi|sii⊗|˜aii,(496) where {|aii} and {|˜aii} are orthonormal sets in HA and the Schmidt coefficients √λi are the same because both purifications have ρS as their reduced state. Then there exists a unitary VA such that |˜aii = VA|aii for all i , and Eq. (495) follows. In other words, all purifications of ρS are related by unitary transformations on the ancilla alone. This isometric freedom is the key structural fact that allows us to regard phases introduced on the ancilla as genuine control parameters: changing VA does not affect ρS , but it changes the global pure state |ΨSAiand hence the phases with which different eigencomponents |siiare coherently superposed. Phase-parameterised family of purifications A particularly simple and physically transparent subset of the full isometric freedom is obtained by restricting to diagonal unitaries on the ancilla. Starting from the canonical purification (494) , we define VA(φ) = X i eiφi|aiihai|,φ= (φ1, φ2, . . . ),(497) and act with it on the ancilla only: |ΨSA(φ)i= (IS⊗VA(φ))|Ψ(0) SAi.(498) Using Eq. (494), we obtain the phase-parameterised family of purifications |ΨSA(φ)i=X i pieiφi|sii⊗|aii,(499) which is the form anticipated in the outline, |ΨSA(φ)i=X i pieiφi|sii|aii.(500)
84 By construction, TrA[|ΨSA(φ)ihΨSA(φ)|] = X i p2 i|siihsi|=ρS,(501) for all choices of phases φ : the reduced state on S is insensitive to these phases, but the global state on SA is not. The set of purifications (499) thus parametrises a family of phase-engineered initial conditions, all of which realise the same mixed state ρS on the system but differ by coherent relative phases between the components |sii⊗|aii . In the context of the ancilla-based measurement schemes of Sec. III D, these phases are experimentally tunable via unitary control on the ancilla alone. Physical meaning of phase freedom The phases {φi} in Eq. (499) do not affect static properties that depend only on ρS , such as von Neumann entropy, eigenvalues, or diagonal expectation values of observables commuting with ρS . However, they do affect dynamical quantities that depend on the full pure state | Ψ SAi and its subsequent joint evolution under a unitary on S+Aor S+E+A. In particular: • When the composite state | Ψ SA ( φ ) i is evolved under a unitary USA or USEA , the phases φi enter the interference pattern between different eigencomponents |sii ; they can therefore modify trajectory-level amplitudes and the resulting entropy-production statistics. • From the point of view of the system alone, different choices of φ correspond to different coherent embeddings of the same mixed state into a larger Hilbert space. These embeddings may lead to different off-diagonal contributions in the operators that enter the interference factors I± in Secs. IV and V. • Operationally, the phases φi are control knobs that can be tuned by applying simple diagonal unitaries on the ancilla prior to the main thermodynamic protocol. Since such unitaries are typically easy to implement in platforms like trapped ions or superconducting qubits, the phase freedom of purifications provides a realistic resource for manipulating interference corrections to fluctuation relations. In summary, purification expresses a mixed state ρS as the marginal of a pure entangled state | Ψ SAi ; the isometric freedom on the ancilla implies that there is a whole manifold of such purifications related by unitaries on A . The phase-parameterised family (499) is a particularly simple and useful subset of this manifold, in which the system eigenstates |sii are entangled with ancilla states |aii with tunable relative phases. These phases will appear explicitly in the interference quotients I± and can therefore be used as a resource for phase engineering of entropy-production statistics, as developed in the following subsections. B. Transition probability into the negative-entropy sector We now connect the phase freedom of the purification | Ψ SA ( φ ) i introduced in Sec. VIII A to the probability of landing in the negative-entropy sector. The key point is that, for any fixed protocol and measurement scheme, the transition amplitude into the negative-entropy sector is a linear functional of the initial pure state on SA ; when this state is parametrised by ancilla phases, the amplitude becomes a coherent sum of phase-weighted complex coefficients. We can then derive rigorously the phase-alignment condition that maximises the corresponding probability. Amplitude into the negative-entropy sector Let Π − denote the projector onto the negative-entropy sector in the extended Hilbert space (system plus environment, and possibly ancilla) as defined in Sec. III C. For definiteness, we consider a fixed global unitary evolution U acting on S (and on E and A if present), and an initial purified state | Ψ SA ( φ ) i of the form |ΨSA(φ)i=X i √pieiφi|sii|aii,(502)
85 as in Eq. (499), with pi>0,Pipi= 1 and {|sii} the eigenbasis of ρS. Let us denote by | Ω −i a normalised vector in the support of Π − , representing the “negative-entropy outcome” in the extended space. The transition amplitude into this sector, starting from | Ψ SA ( φ ) i , is of the general form A−(φ)≡ hΩ−|U|ΨSA(φ)i.(503) Linearity of U and of the inner product implies that A− ( φ )is a linear combination of the phase factors e iφi with complex coefficients determined by the dynamics. Inserting Eq. (502) into Eq. (503) , we obtain A−(φ) = X i √pieiφihΩ−|U|si, aii ≡X i pieiφici,(504) where, in the second line, we have simply redefined the coefficients by absorbing √piinto them, so that ci≡1 pi √pihΩ−|U|si, aii=1 √pihΩ−|U|si, aii,(505) for those i with pi> 0(if some pi = 0, the corresponding term does not contribute and ci can be chosen arbitrarily). Thus, without loss of generality, we arrive at the compact expression requested in the outline: A−(φ) = X i pieiφici,(506) with real non-negative weights pi (not necessarily equal to the eigenvalues of ρS anymore, after redefinition) and fixed complex coefficients ci , which incorporate all details of the dynamics, measurement, and embedding into the negative-entropy sector, but are independent of the phases φ. The corresponding transition probability into the negative-entropy sector is P−(φ) = |A−(φ)|2=X i pieiφici 2 .(507) Our goal is to determine the choice of phases φthat maximises P−(φ). Phase alignment condition for maximising P−(φ) We now derive rigorously the phase-alignment condition that yields the maximum of P− ( φ )over all possible phase choices φ= (φ1, φ2, . . .). Write each complex coefficient in polar form, ci=|ci|eiγi,|ci| ≥ 0, γi∈R.(508) Then A−(φ) = X i pi|ci|ei(φi+γi)=X i ai(φ),(509) where we defined individual contributions ai(φ)≡pi|ci|ei(φi+γi).(510) The magnitude of the total amplitude is bounded by the triangle inequality: |A−(φ)|=X i ai(φ)≤X i|ai(φ)|=X i pi|ci|.(511) The right-hand side is independent of φ . Therefore we have an upper bound on the transition probability, P−(φ) = |A−(φ)|2≤ X i pi|ci|!2 ≡Pmax −,(512)
86 for all choices of the phases. Moreover, the triangle inequality is saturated if and only if all the complex numbers ai(φ)have the same phase, i.e. there exists a global angle θ∈Rsuch that ai(φ) = pi|ci|ei(φi+γi)= eiθ pi|ci|for all iwith pi|ci| 6= 0.(513) Equivalently, φi+γi=θ(mod 2π),∀iwith pi|ci| 6= 0.(514) Since a global phase θ does not affect the probability P− , we may choose it arbitrarily (e.g. θ = 0). The phase alignment condition that maximises P−(φ)is therefore φi=−γi+θ(mod 2π),∀iwith pi|ci| 6= 0,(515) or, in words: Choose the phases φi so that all contributions pieiφici to the amplitude have the same complex phase. Any set of phases satisfying Eq. (515) yields the same maximal transition probability Pmax −= X i pi|ci|!2 .(516) Conversely, any deviation from perfect phase alignment necessarily decreases |A− ( φ ) | and hence P− ( φ ). Geometric and physical interpretation The structure of Eq. (507) is mathematically identical to multi-path interference in an optical interferometer: the pi play the role of path weights, the |ci| encode path-dependent coupling strengths into the negative-entropy “detector”, and the phases φi + γi represent total phase shifts accumulated along each path. The triangle inequality (511) states that the intensity (probability) at the detector is bounded by the square of the sum of moduli of the path amplitudes; the bound is attained when all paths interfere fully constructively. The phase-alignment condition (515) is the precise analogue of adjusting phase shifters in an interferometer to maximise the output intensity in a given port. In the present context, the phases φi are not fixed by the dynamics but are external control parameters acting on the ancilla only. By changing φi , we effectively rotate the purification within the manifold of states that share the same reduced ρS . Equation (515) then shows that we can steer the negative-entropy transition probability P− ( φ )between its minimum and maximum possible values (subject to the fixed weights pi and the dynamical coefficients ci ) by aligning or misaligning the phases of the different components. In subsequent subsections we will exploit this freedom to design phase-engineering protocols that selectively enhance or suppress negative-entropy events and thereby tune the interference quotients I±entering the modified fluctuation relations. C. Relation to I−and FT ratio In Sec. VIII B we expressed the transition amplitude into the negative-entropy sector as A−(φ) = X i pieiφici, P−(φ) = |A−(φ)|2,(517) where the weights pi≥ 0are fixed by the initial mixed state ρS , the complex coefficients ci are fixed by the dynamics and measurement scheme (negative-entropy projector and global unitary), and the phases φ = ( φ1, φ2, . . . )are controllable via local unitaries on the ancilla. In this subsection we make explicit how the ci enter the more microscopic semiclassical coefficients C(−) α , how they feed into the interference factor I−, and how phase control on the ancilla reshapes the fluctuation-theorem ratio.
87 From cito trajectory-level coefficients C(−) α In the semiclassical / trajectory picture of Sec. VI, the negative-entropy sector is described by a sector amplitude A(SC) −(φ) = X α∈P− C(−) α(φ) expi ~eff S(−) α,(518) where P− is the set of classical trajectories in the backward/negative-entropy sector, S(−) α are the associated classical actions, and C(−) α ( φ )are complex prefactors encoding stability determinants, Maslov indices and overlaps with the initial state and the negative-entropy projector. To connect Eq. (518) with the coarse-grained amplitude (517) , we insert a decomposition of the matrix elements, ci=hχ−|U|si, aii=X α∈P− d(−) iα expi ~eff S(−) α,(519) where d(−) iα are trajectory-dependent coefficients capturing the contribution of the classical path α when the initial component is |sii|aii. Substituting (519) into (517) gives A−(φ) = X i pieiφiX α∈P− d(−) iα expi ~eff S(−) α =X α∈P−"X i pieiφid(−) iα #expi ~eff S(−) α.(520) Comparing with Eq. (518), we identify C(−) α(φ) = X i pieiφid(−) iα .(521) Thus, the phase-controlled coefficients pi e iφi do not enter the semiclassical expansion in a trivial way: they provide coherent weights for how the different eigencomponents |si, aii feed into each classical trajectory α∈ P− . The same set of phases φ appears simultaneously in all C(−) α , so phase control on the ancilla re-weights both the diagonal and interference trajectory contributions in a correlated fashion. The negative-sector probability then takes the pair-sum form P−(φ) = |A−(φ)|2≈X α,β∈P− C(−) α(φ)C(−) β(φ)∗expi ~eff (S(−) α−S(−) β),(522) which is the semiclassical analogue of Eq. (97) . The diagonal part ( α = β ) defines P(SC) diag ( − ); the off-diagonal terms (α6=β) define ∆P(SC) int (−)and thus the interference quotient I(SC) −(φ) = ∆P(SC) int (−;φ) P(SC) diag (−).(523) Equation (521) shows explicitly how the coarse-grained coefficients ci entering A− ( φ )are resolved into trajectory-level coefficients C(−) α , and how phase control on the ancilla modulates the interference structure of the negative-entropy sector. Functional J(φ)and its gradient At the operator level, the modified steady-state fluctuation theorem takes the form Pqm(+; φ) Pqm(−;φ)= expΣ kB1 + I+ 1 + I−(φ),(524)
88 where Σis the classical entropy production, I+ is the interference quotient in the positive sector (independent of φ for the present class of ancilla operations), and I− ( φ )is the negative-sector interference factor modified by the ancilla phases. We define the functional J(φ)≡ln Pqm(+; φ) Pqm(−;φ)−Σ kB = ln1 + I+−ln1 + I−(φ),(525) which measures the deviation of the quantum FT ratio from its classical value. Minimising J ( φ ) corresponds to choosing phases such that the FT ratio is reduced as much as possible (for fixed Σand I+), without changing the underlying classical microreversibility encoded in Σand the diagonal sectors. Using the definition P−(φ) = Pdiag(−)1 + I−(φ),(526) with Pdiag(−)independent of φ, we have 1 + I−(φ) = P−(φ) Pdiag(−),ln1 + I−(φ)= ln P−(φ)−ln Pdiag(−).(527) Substituting into Eq. (525) gives J(φ) = const −ln P−(φ),(528) where const = ln (1 + I+ ) + ln Pdiag ( − )does not depend on φ . Thus minimising J ( φ )is equivalent to maximising the negative-sector probability P−(φ)for fixed piand ci. To obtain the gradient of J(φ)with respect to the phases, we start from P−(φ) = |A−(φ)|2, A−(φ) = X i zieiφi, zi≡pici.(529) Then ∂A− ∂φk =izkeiφk,∂A∗ − ∂φk =−iz∗ ke−iφk,(530) and hence ∂P− ∂φk =∂ ∂φkA−A∗ −=∂A− ∂φkA∗ −+A−∂A∗ − ∂φk =izkeiφkA∗ −−iz∗ ke−iφkA− = 2 ReizkeiφkA∗ −=−2 ImzkeiφkA∗ −.(531) Using Eq. (528), ∂J ∂φk =−1 P−(φ) ∂P− ∂φk =2 P−(φ)ImzkeiφkA∗ −(φ).(532) Thus the gradient components are proportional to the imaginary parts of the pairwise products between each path amplitude zkeiφkand the total amplitude A−(φ). Stationary points and phase alignment. Stationary points of J(φ)satisfy ∂J ∂φk = 0 ∀k⇐⇒ ImzkeiφkA∗ −= 0 ∀k. (533) This condition implies that, for each k with zk6 = 0, the complex number zk e iφkA∗ − is real, i.e. the phase of zk e iφk is either aligned or anti-aligned with the phase of A− ( φ ). Among all such stationary points, the global maximum of P− ( φ )(and hence the global minimum of J ( φ )) occurs when all path amplitudes are aligned with A−(φ), which is exactly the phase-alignment condition derived in Sec. VIII B, φopt i=−arg ci+α0(mod 2π),∀iwith pi|ci| 6= 0.(534) At this point, P−(φ)attains its maximum Pmax −=Pipi|ci|2, and J(φ)attains its minimum Jmin =const −ln Pmax −.(535) In other words, the same phase choices that produce fully constructive interference in the negative-entropy sector also minimise the FT deviation functional J ( φ ), thereby reducing the FT ratio Pqm (+) /Pqm ( − )as much as is consistent with the fixed classical entropy production Σand with positivity (P−≥0).
89 Microreversibility and control without violating the FT structure It is important to stress that phase engineering on the ancilla does not change the diagonal probabilities Pdiag ( ± )or the classical entropy production Σ. These quantities are determined solely by the spectrum of ρS and by the microreversible dynamics and are independent of φ . Consequently, the classical FT relation for the diagonal sectors, Pdiag(+) Pdiag(−)= expΣ kB,(536) remains intact. Phase control only redistributes probability between diagonal and interference contributions in each sector, modifying I±while preserving the overall FT structure (524) [105]. The functional J ( φ )thus defines a natural control landscape on the space of ancilla phases: gradientbased methods (analogous to those used in quantum optimal control [ 106 , 107 ]) can be employed to steer the system towards phase configurations that minimise J ( φ ), i.e. that optimally suppress the FT ratio for a given protocol without altering Σor violating microreversibility. The structure of the gradient (532) shows that the landscape is smooth and that the global optimum corresponds to full phase alignment in the negative-entropy sector, realising in a precise quantitative sense the idea that coherent phase control can be used to shape quantum corrections to classical fluctuation relations. D. Coherence as a resource and robustness The previous subsections showed that the probability of entering the negative-entropy sector and the corresponding interference quotient I− ( φ )depend sensitively on the controllable phases φ in a purification of the initial mixed state. In this subsection we connect these phase-engineering effects to the resource theory of quantum coherence, and we analyse the robustness of the resulting interference corrections under phase noise and dephasing. This provides a quantitative notion of “coherence as a resource” for modifying fluctuation relations. Resource-theoretic measures of coherence Let {|sii} be a fixed reference basis on HS , which for concreteness we take to be the eigenbasis of the entropy-production operator ˆ Σ or of the relevant initial observable, as in Sec. III C. The resource theory of coherence [ 108 , 109 ] characterises the coherence of a state ρ with respect to this basis via quantities such as: •the `1-norm of coherence, C`1(ρ)≡X i6=j|ρij|,(537) where ρij =hsi|ρ|sji; •the relative entropy of coherence, Crel(ρ)≡Sρdiag−S(ρ),(538) with S(ρ) = −Tr(ρln ρ)and ρdiag =Piρii|siihsi|. Both C`1 and Crel vanish if and only if ρ is diagonal in the reference basis, and they are nonincreasing under incoherent operations. In our setting, the reduced system state ρS is diagonal in its eigenbasis by construction, ρS = Pipi|siihsi|, so C`1(ρS)=0in that basis. However, coherence re-enters at two levels: 1. in the joint purified state | Ψ SA ( φ ) i on HS⊗HA , which contains off-diagonal coherences in the product basis {|sii|aji}; 2. in the representation of ρS and ρSA in the eigenbasis of ˆ Σ (or of the measurement operators), where the eigenvectors of ρSneed not coincide with those of ˆ Σ. Coherence in these bases is precisely what fuels the interference terms ∆ Pint ( ± )and hence the interference quotients I± , cf. Eq. (A35) . We now show how the magnitude of I− can be bounded in terms of C`1.
96 B. Specific toy model: 6-state ATP synthase substep To illustrate the interplay of chemical driving, mechanical rotation and quantum coherence in a minimal setting, we now introduce a six-state toy model inspired by a single 120 ◦ substep of the rotary F1 motor in ATP synthase [ 122 – 124 ]. The model is not meant to reproduce the detailed biochemistry of ATP synthase, but rather to capture the essential structure: three coarse-grained chemical states (binding, synthesis,release) combined with two mechanical orientations of the rotor, coupled coherently and driven by chemostats and an external torque [125]. Basis states and Hamiltonian The Hilbert space of the motor substep is spanned by six orthonormal basis states |B, θ+i,|S, θ+i,|R, θ+i,|B, θ−i,|S, θ−i,|R, θ−i,(569) where: •B, S, R label coarse-grained chemical conformations of one catalytic site: a binding-like state B , a synthesis (or high-energy intermediate) state S , and a release state R in which ATP is ready to be released. •θ± denote two neighbouring angular positions of the rotor differing by a substep ∆ θ > 0: for concreteness we take θ±=θ0±∆θ/2, so that θ+−θ−= ∆θ. The system Hamiltonian consists of chemical energy levels, mechanical coupling to an external torque τand coherent tunnelling between the two angular positions: ˆ HS=X X∈{B,S,R}X σ=± [EX−τ θσ]|X, θσihX, θσ| +~ΩX X∈{B,S,R}|X, θ+ihX, θ−|+|X, θ−ihX, θ+|,(570) where: •EXis the internal chemical energy of conformation X; •the term −τ θσrepresents the mechanical potential energy due to the load torque τ; •Ωis a coherent tunnelling frequency that mixes the θ±orientations for a fixed chemical state X. The coherent term in Eq. (570) is the origin of quantum superpositions between positive and negative entropy sectors (associated with forward and backward substeps), and will be responsible for nontrivial interference corrections to the fluctuation relation. Lindblad jumps for binding, synthesis and release The motor interacts with chemostats of ATP, ADP and inorganic phosphate Pi at fixed chemical potentials µATP , µADP and µPi . We model the chemical part of the dynamics by Lindblad jump operators associated with binding, synthesis and release reactions, each acting locally in the angle index θ±. For each mechanical orientation θσ(σ=±) we introduce the following forward jumps: ˆ L(bind,σ) B→S=pΓbind |S, θσihB, θσ|,(571) ˆ L(syn,σ) S→R=pΓsyn |R, θσihS, θσ|,(572) ˆ L(rel,σ) R→B=pΓrel |B, θσihR, θσ|,(573)
97 and corresponding backward jumps ˆ L(bind,σ) S→B=q˜ Γbind |B, θσihS, θσ|,(574) ˆ L(syn,σ) R→S=q˜ Γsyn |S, θσihR, θσ|,(575) ˆ L(rel,σ) B→R=q˜ Γrel |R, θσihB, θσ|.(576) Here: • the (forward) binding jump |B, θσi → |S, θσi is associated with binding of ADP and Pi to the catalytic site; • the (forward) synthesis jump |S, θσi→|R, θσi corresponds to the chemical step in which ATP is formed from bound ADP and Pi; • the (forward) release jump |R, θσi→|B, θσi represents release of ATP back into the solution, returning the site to a binding-ready state. The Lindblad master equation for the reduced density operator ρof the motor is therefore ˙ρ=−i ~[ˆ HS, ρ] + X σ=±X α∈{bind,syn,rel}D[ˆ L(α,σ) fwd ]ρ+X σ=±X α∈{bind,syn,rel}D[ˆ L(α,σ) bwd ]ρ, (577) with dissipator D[ˆ L]ρ=ˆ Lρˆ L†−1 2{ˆ L†ˆ L, ρ}.(578) Local detailed balance and chemical affinity for the substep The six rates Γ bind , ˜ Γbind ,Γ syn , ˜ Γsyn ,Γ rel , ˜ Γrel are constrained by local detailed balance in contact with the ATP/ADP/Pichemostats at temperature T[118, 125]. For each reaction channel we impose ln Γbind ˜ Γbind =βµADP +µPi−∆ES,B,(579) ln Γsyn ˜ Γsyn =βµATP −µADP −µPi−∆ER,S,(580) ln Γrel ˜ Γrel =β−µATP −∆EB,R,(581) where ∆ EY,X = EY−EX and β = ( kBT ) −1 . Summing the log-ratios around a full forward chemical cycle B→S→R→Byields ln ΓbindΓsynΓrel ˜ Γbind ˜ Γsyn ˜ Γrel =βµATP −µADP −µPi≡β∆µATP,(582) independent of the internal energies EX . Equation (582) identifies ∆ µATP as the chemical affinity per completed substep. The mechanical torque τ enters through the Hamiltonian (570) and contributes a mechanical work Wsub mech = τ ∆ θ per forward substep. The net medium entropy production associated with one forward chemo–mechanical substep is therefore σsub =β∆µATP −τ∆θ,(583) in agreement with the classical expression (568) for a unicyclic motor, but now at the level of a single 120◦substep.
98 Entropy-production operator for the substep To connect this model to the general operator framework of Sec. III C, we define a Hermitian entropyproduction operator for the substep whose eigenvalues coincide with the stochastic entropy production associated with forward/backward rotation of the rotor during the substep. We group the six basis states into positive and negative entropy sectors according to their mechanical orientation: Πsub +=X X∈{B,S,R}|X, θ+ihX, θ+|,(584) Πsub −=X X∈{B,S,R}|X, θ−ihX, θ−|.(585) By construction, Π sub ± are orthogonal projectors satisfying Π sub + + Π sub − = I6 , and they define a coarsegrained partition of the Hilbert space into positiveand negative-entropy sectors. We then define the dimensionless entropy-production operator for the substep as ˆ Σsub =σsubΠsub +−Πsub −,(586) with σsub given by Eq. (583). Its spectral decomposition is ˆ Σsub =σsub X X|X, θ+ihX, θ+|−σsub X X|X, θ−ihX, θ−|,(587) so that the eigenvalues are ±σsub with threefold degeneracy each. In the classical limit where coherences between |X, θ+i and |X, θ−i are suppressed (e.g. by strong dephasing in the θ -basis), ˆ Σsub reduces to a diagonal operator whose eigenvalues coincide with the classical entropy-production values assigned to forward and backward substeps in Sec. IX A. In the fully quantum regime, the coherent couplings in ˆ HS generate superpositions between positive and negative sectors, and the projectors Πsub ±enter directly in the definition of the sector probabilities and interference factors I± for this specific chemo–mechanical model. C. Computation of entropy-production statistics We now show how the six-state chemo–mechanical model of Sec. IX B can be used to compute the statistics of entropy production for a single substep (or for repeated substeps composing a full cycle). We proceed in three stages: 1. solve the Lindblad master equation for the reduced state of the motor, both in the classical (diagonal) and fully quantum cases; 2. implement the ancilla-based measurement scheme of Sec. III D to obtain the characteristic function and distribution of the entropy-production operator ˆ Σsub defined in Eq. (586); 3. extract the diagonal and full sector probabilities Pdiag ( ± )and Pqm ( ± )and hence the interference quotients I±and the modified FT ratio. Classical (diagonal) dynamics and Pdiag(±) If we suppress quantum coherences between the angular states |θ+i and |θ−i —for instance by setting the coherent couplings JX in Eq. (570) to zero or by introducing strong dephasing in the {θ±} basis—the density matrix remains diagonal at all times and the dynamics reduces to a classical Markov jump process on the six states {|B, θ±i,|S, θ±i,|R, θ±i} with transition rates determined by the Lindblad operators (571)–(576). Denoting by p(t) = pB,+(t), pS,+(t), pR,+(t), pB,−(t), pS,−(t), pR,−(t)T(588) the vector of populations, the classical master equation reads ˙ p(t) = Wp(t),(589)
99 where Wis a 6×6rate matrix whose non-zero off-diagonal elements are, for each σ=±, WB,σ←R,σ =k+ b, WR,σ←B,σ =k− b, WS,σ←B,σ =k+ s, WB,σ←S,σ =k− s, WR,σ←S,σ =k+ r, WS,σ←R,σ =k− r,(590) and Wii =−Pj6=iWj←i. For a given initial distribution p(0) (e.g. localised in |B, θ−i), the solution is p(t)=eW t p(0),(591) which can be obtained analytically by diagonalising W or numerically by standard matrix-exponential methods. The coarse-grained positive and negative sectors are defined by the projectors Π sub ± introduced in Sec. IX B. In the classical limit, the probabilities of being in these sectors at time τsub are Pdiag(+) = X X∈{B,S,R} pX,+(τsub), Pdiag(−) = X X∈{B,S,R} pX,−(τsub).(592) These are precisely the classical sector probabilities entering the steady-state FT for the substep, with entropy production σsub given by Eq. (583). Full quantum dynamics and Pqm(±) In the fully coherent case, with finite tunnelling amplitudes JX , the density matrix ρ ( t )evolves according to the Lindblad equation (577). For a time-independent generator L, the formal solution is ρ(t)=eLtρ(0),(593) where L acts linearly on the space of 6 × 6matrices. In practice, one vectorises ρ to a 36-component column vector |ρii and rewrites Eq. (577) as d dt|ρ(t)ii =L|ρ(t)ii,(594) with a 36 × 36 Liouvillian matrix L . The propagator e Lt is then computed numerically (or analytically in special parameter regimes). The initial condition ρ (0) incorporates the phase-engineered purification of Sec. VIII restricted to the six-dimensional system subspace. The quantum sector probabilities at time τsub are Pqm(±) = TrΠsub ±ρ(τsub)= TrΠsub ±eLτsub ρ(0).(595) The interference contributions are then ∆Pint(±) = Pqm(±)−Pdiag(±), I±=∆Pint(±) Pdiag(±),(596) as in Sec. IV B. By varying the initial ancilla phases φ that define the purification | Ψ SA ( φ ) i , one explores the control landscape I±(φ)and the functional J(φ)of Sec. VIII C for this concrete model. Characteristic function and distribution of entropy production To obtain the full statistics of the substep entropy-production operator ˆ Σsub , we use the ancilla-based scheme of Sec. III D. An ancilla qubit A is coupled to the motor such that, in the interaction picture, the joint evolution over the substep interval [0, τsub]implements the unitary [127, 141] U(u) SA = exp−iu 2ˆ Σsub ⊗σA zUSE exp−iu 2ˆ Σsub ⊗σA z,(597)
100 where USE is the system–environment propagator generated by ˆ HS and the Lindblad couplings, and σA z is the Pauli operator acting on the ancilla. The ancilla is prepared in a superposition state ( | 0 iA + | 1 iA ) /√2 , and its coherence after the protocol encodes the characteristic function χ(u) = TrS,E,A hU(u) SA ρ0⊗|+iAh+|⊗ρEU(u)† SA i,|+iA=|0iA+|1iA √2.(598) Tracing out Sand Eand reading out the ancilla in the x-yplane yields χ(u)experimentally. From a computational point of view, it is convenient to work with an equivalent tilted Liouvillian Lu acting on the system density matrix [ 141 ]. For each Lindblad jump operator Lr (with label r running over the six reactions in Eqs. (571) – (576) at both angles), we assign a dimensionless entropy increment σrgiven by the log-ratio of rates, σr= ln Γfwd r Γbwd r =βQr/T,(599) consistently with the local detailed balance relations (E9). The tilted generator is defined by Lu[ρ] = −i[ˆ HS, ρ] + X reiuσr/2Lrρ L† reiuσr/2−1 2{L† rLr, ρ},(600) so that the u-dependent state ρu(t)obeys ˙ρu(t) = Lu[ρu(t)], ρu(0) = ρ(0),(601) and the characteristic function is χ(u) = Tr ρu(τsub).(602) In the eigenbasis of ˆ Σsub , the Fourier transform of χ ( u )yields the full probability distribution of entropy production, P(Σ) = 1 2πZπ −π due−iuΣχ(u).(603) Extraction of FT ratio and interference quotients The distribution P(Σ) and the projectors Πsub ±allow us to compute the sector probabilities Pqm(+) = ZΣ>0 dΣ P(Σ), Pqm(−) = ZΣ<0 dΣ P(Σ),(604) which coincide with the quantum sector probabilities (595) evaluated with the projectors onto positive and negative eigenvalues of ˆ Σsub . The diagonal sector probabilities Pdiag ( ± )are obtained by repeating the same analysis with the diagonalised dynamics of Eq. (589) (i.e. by setting JX = 0 or adding strong dephasing). Once Pdiag ( ± )and Pqm ( ± )are known, the interference quotients I± follow from Eq. (596) . The fluctuation-theorem ratio for the substep is then Pqm(+) Pqm(−)= expσsub kB1 + I+ 1 + I− ,(605) with σsub given by Eq. (583) . Numerical evaluation of this expression as a function of the coherent coupling strengths JX , the ancilla phases φ and the chemical and mechanical affinities provides a fully worked, finite-dimensional example of the general modified quantum fluctuation theorem developed in Secs. IV and VI.
101 D. Plots and physical interpretation We now summarise how the entropy-production statistics of the six-state chemo–mechanical model manifest in explicit plots, and how these plots encode the role of coherence and phase engineering. We focus on three closely related objects: • the fluctuation-theorem (FT) log-ratio as a function of the substep entropy production Σ sub , for different ancilla phase settings; •the corresponding phase-dependent correction δR(φ) = Rqm(φ)−Rcl; • the mean entropy production h Σ i as a function of phase, and the associated efficiency η = 1 − hΣi/(β∆µATP). Throughout this subsection we use the notation and results of Secs. IX B and IX C, in particular the substep entropy-production operator ˆ Σsub with eigenvalues ± Σ sub and projectors Π ± , and the classical vs quantum sector probabilities Pdiag ( ± ; τ )and Pqm ( ± ; τ )obtained from the Lindblad dynamics for the six-state model. FT log-ratio vs entropy production and phase For the six-state model, a single chemo–mechanical substep has only two possible values of medium entropy production, Σ = ±Σsub, where (see Eq. (583)) Σsub =β∆µATP −τ∆θ,(606) with ∆ µATP the chemical potential drop per ATP and τ ∆ θ the mechanical work against the load during the substep. The classical steady-state FT for this coarse-grained substep reads Pdiag(+; τ) Pdiag(−;τ)= expΣsub kB,(607) cf. Eq. (639) . In the presence of quantum coherence and phase-engineered initial conditions, the corresponding quantum FT ratio takes the form (see Eq. (524)) Pqm(+; τ) Pqm(−;τ)= expΣsub kB1 + I+(τ) 1 + I−(τ;φ),(608) where the interference quotients I± ( τ )are defined by Eq. (651) . For the class of phase operations considered in Sec. VIII, I+ ( τ )is phase independent, whereas I− ( τ ; φ )depends on a single control phase φin the purification. It is convenient to introduce the FT log-ratio R(Σsub;φ)≡ln Pqm(+; τ) Pqm(−;τ)=Σsub kB + ln1 + I+(τ)−ln1 + I−(τ;φ).(609) For fixed microscopic parameters and fixed protocol duration τ , one may regard Σ sub as a control parameter tuned by changing the chemical affinity ∆ µATP or the torque τ . The classical FT prediction corresponds to the straight line Rcl(Σsub)=Σsub/kB. Figure 3 shows the numerically obtained log-ratio R (Σ sub ; φ )for the six-state model, using the thermodynamically consistent rates of Sec. IX B, as a function of Σ sub for three representative choices of the ancilla phase φ . The solid black curve displays the classical prediction Rcl (Σ sub )obtained from the diagonal dynamics, while the coloured curves show the full quantum log-ratio for φ = 0, φ≃π/ 2 and φ≃π . The quantum curves remain close to the classical line, reflecting the fact that the substep operates in a strongly driven, strongly dissipative regime where classical drift dominates, but they exhibit systematic, phase-dependent shifts consistent with Eq. (609).
102 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 sub 2 4 6 8 R ( ) = ln[ P qm(+)/ P qm( )] classical R cl =sub quantum, = 0.00 quantum, = 1.57 quantum, = 3.14 Figure 3. Fluctuation-theorem log-ratio R (Σ sub ; φ ) = ln [ Pqm (+; τ ) /Pqm ( − ; τ )] for the six-state chemo–mechanical molecular machine as a function of the substep entropy production Σ sub . The solid black curve shows the classical FT prediction Rcl (Σ sub )=Σ sub/kB obtained from the diagonal dynamics. The coloured curves show the full quantum log-ratio for three ancilla phases φ = 0 (blue), φ≃π/ 2(orange) and φ≃π (green). The phase dependence produces modest but systematic deviations from the classical line, in agreement with the interference-corrected FT relation (608). Interference correction to the FT ratio The structure of Eq. (609) suggests isolating the interference correction δR(φ)≡R(Σsub;φ)−Rcl(Σsub) = ln1 + I+(τ)−ln1 + I−(τ;φ),(610) which, within the present model, depends on φ but not on Σ sub for fixed microscopic parameters. In the finite-time Lindblad dynamics of the six-state machine, small residual dependencies on Σ sub arise from the interplay of chemical and mechanical rates, but the overall magnitude of δR ( φ )remains bounded by the general inequalities of Sec. IV E. Figure 4 shows δR ( φ )as a function of Σ sub for the same three ancilla phases as in Fig. 3. The correction is small compared to the classical contribution, as expected in a parameter regime where the motor is strongly driven and entropy production per substep is large, but it is clearly phase dependent and changes sign as φ is varied. Phase choices that enhance interference in the negative-entropy sector (as discussed in Sec. VIII B) yield negative δR ( φ ), while phase choices that favour the positive sector yield positive corrections. The interference contribution thus acts as a thermodynamically consistent, phase-tunable “offset” to the FT ratio. Mean entropy production, phase control, and efficiency For the single-substep distribution supported on Σ = ± Σ sub , the mean entropy production at time τ is hΣi(φ)=Σsub Pqm(+; φ)−Pqm(−;φ)= Σsub 2Pqm(+; φ)−1,(611) with Pqm(+; φ) + Pqm(−;φ)=1. In terms of the diagonal probabilities and interference quotients, Pqm(±;φ) = Pdiag(±)1 + I±(φ),(612) we obtain the decomposition hΣi(φ) = hΣicl + ∆hΣi(φ),(613)
103 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 sub 8 6 4 2 0 R ( ) = R qm( ) R cl R ( = 0.00) R ( = 1.57) R ( = 3.14) Figure 4. Interference correction δR ( φ ) = Rqm (Σ sub ; φ ) −Rcl (Σ sub )for the six-state chemo–mechanical model, plotted as a function of Σ sub for three ancilla phases φ = 0 (blue), φ≃π/ 2(orange) and φ≃π (green). The correction is small in magnitude compared to the classical FT contribution, consistent with the strong-dissipation regime of the motor, but depends systematically on φ and can be either positive or negative. This behaviour is fully consistent with the bounds on the interference quotients I±derived in Sec. IV E. with hΣicl = Σsub Pdiag(+) −Pdiag(−),(614) ∆hΣi(φ)=Σsub Pdiag(+)I+(φ)−Pdiag(−)I−(φ).(615) The first term (614) coincides with the classical mean entropy production for the substep, determined solely by the chemical and mechanical affinities. The second term (615) encodes the coherent correction due to interference. Figure 5 shows the numerically obtained mean entropy production as a function of the ancilla phase φ for fixed (∆ µATP, τ ∆ θ )and coupling parameters. The dotted curve corresponds to h Σ icl , which is essentially independent of φ , while the solid curve shows the full quantum mean entropy production h Σ iqm ( φ ). The dashed curve represents the excess ∆ h Σ i ( φ ) = h Σ iqm ( φ ) −h Σ icl . As anticipated from Eq. (615) , the excess entropy production changes sign as φ is varied: for some phases coherence reduces dissipation relative to the classical baseline, for others it increases it. Importantly, h Σ iqm ( φ )remains strictly non-negative for all phases, so the second law is always satisfied. For a chemo–mechanical motor driven by an ATP hydrolysis free energy ∆ µATP and working against a torque τ , the mechanical work performed per forward substep is Wsub mech = τ ∆ θ . The medium entropy production per substep is related to the chemical and mechanical energetics via Eq. (606) , so that, at the level of averages, hWsub mechi= ∆µATP −hΣi β.(616) The energetic efficiency of the motor per substep, defined as useful mechanical work divided by chemical free-energy input [128–130], is then η(φ)=1−hΣi(φ) β∆µATP .(617) Equation (617) makes transparent why h Σ i ( φ )is the natural diagnostic quantity: for fixed chemical driving ∆ µATP , any reduction of mean entropy production directly translates into an increase of efficiency. In the six-state model, Fig. 5 shows that phase engineering can shift h Σ i ( φ )by a modest but measurable
104 0123456 ancilla phase 0 2 4 6 8 entropy production diag( ) qm( ) ( ) ( ) 5.5 6.0 6.5 7.0 7.5 ( ) 1e 13+1.177117118e 1 Figure 5. Mean entropy production per substep h Σ i ( φ )for the six-state chemo–mechanical model as a function of the ancilla phase φ . The dotted curve shows the classical (diagonal) mean entropy production h Σ icl , which is essentially independent of φ . The solid curve shows the full quantum mean entropy production h Σ iqm ( φ )obtained from phase-engineered initial states. The dashed curve plots the excess entropy production ∆ h Σ i ( φ ) = h Σ iqm ( φ ) −h Σ icl , which changes sign with φ . Coherence and phase control therefore act as a thermodynamic resource that can either reduce or increase dissipation relative to the classical baseline, while the total mean entropy production remains non-negative. amount around its classical value, and thus modulate the efficiency within the universal constraint η ( φ ) ≤ 1. In this sense the six-state chemo–mechanical motor provides a realistic example in which the interference-corrected FT structure and the resource character of coherence, developed abstractly in Secs. IV and VIII, remain visible even in a strongly dissipative, biophysically motivated setting. E. Second example: excitonic dimer As a second, conceptually distinct application of the general framework developed in Secs. III C–IX, we consider a minimal model of coherent excitonic transport through a molecular dimer. The goal is to illustrate how the same operator-level fluctuation theorem and interference quotients introduced earlier can be applied to a different physical setting, in which the relevant “machine” does not convert chemical free energy into mechanical work, but instead channels electronic excitations between reservoirs in the presence of strong vibronic coupling. In particular, we focus on how coherent excitonic superpositions and vibronic interference pathways modify the tails of the entropy-production distribution, and how these modifications show up as controlled deviations from the classical fluctuation-theorem line. Model definition and Hamiltonian We consider a three-level system consisting of a ground state |gi and two single-exciton states |1i and |2ithat form an excitonic dimer. In the site basis, the system Hamiltonian reads ˆ HS=Eg|gihg|+E1|1ih1|+E2|2ih2|+Jex (|1ih2|+|2ih1|),(618) with Eg the ground-state energy (taken as zero reference without loss of generality), E1,2 the on-site exciton energies, and Jex the coherent electronic coupling between the two sites. The eigenstates of ˆ HS in the single-excitation subspace, |+i= cos θ|1i+ sin θ|2i,|−i =−sin θ|1i+ cos θ|2i,(619)
105 with mixing angle tan(2θ) = 2Jex E1−E2 ,(620) provide a natural basis in which the role of coherence is particularly transparent: for Jex 6 = 0 the energy eigenstates are delocalized over both sites and transport can proceed along multiple interfering pathways. The dimer is coupled to two electronic or excitonic reservoirs (e.g. a “source” antenna and a “drain” reaction centre) and to a structured phonon environment that induces dephasing and relaxation. For concreteness, we model the source and drain as Markovian bosonic or fermionic reservoirs at inverse temperatures βL and βR and chemical potentials µL and µR . The system–bath coupling is chosen such that the left reservoir predominantly injects excitations into site |1i , while the right reservoir predominantly removes excitations from site |2i. Quantum master equation and local detailed balance Under the usual Born–Markov and secular approximations, the reduced dynamics of the dimer is described by a Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) master equation d dtˆρ(t) = −i ~hˆ HS+ˆ HLS,ˆρ(t)i+X rDrˆρ(t),(621) where ˆ HLS is a Lamb-shift term and each dissipator Dris associated with a specific jump channel r, Dr[ˆρ] = ˆ Lrˆρˆ L† r−1 2nˆ L† rˆ Lr,ˆρo.(622) For the source and drain reservoirs, we introduce the jump operators ˆ LL,+=qγ+ L|1ihg|,ˆ LL,−=qγ− L|gih1|,(623) ˆ LR,+=qγ+ R|2ihg|,ˆ LR,−=qγ− R|gih2|,(624) representing, respectively, excitation and de-excitation events on sites |1i and |2i . The rates obey local detailed balance, γ+ L γ− L = exp[−βL(E1−Eg−µL)] ,γ+ R γ− R = exp[−βR(E2−Eg−µR)] ,(625) which ensures that in the absence of coherent coupling and other baths each site would relax to the grand-canonical equilibrium state of its local reservoir. In addition, we include pure-dephasing and intraband relaxation channels due to the vibrational environment. A minimal choice is ˆ Lφ,1=√γφ,1|1ih1|,ˆ Lφ,2=√γφ,2|2ih2|,(626) ˆ Lrel,+=qγ+ rel |+ih−|,ˆ Lrel,−=qγ− rel |−ih+|,(627) with relaxation rates γ± rel satisfying γ+ rel γ− rel = exp[−βvib (E+−E−)] ,(628) where βvib is an effective inverse temperature for the phonon bath and E± are the excitonic eigenenergies of ˆ HS . The pure-dephasing rates γφ,1 and γφ,2 do not contribute directly to entropy production, but they control the degree to which excitonic coherences survive long enough to influence transport.
112 provides a useful bridge between rigorous fluctuation relations and concrete models of molecular and excitonic machines. B. Relation to existing quantum thermodynamics frameworks The framework developed in this work sits at the intersection of three major strands of quantum thermodynamics: (i) fluctuation relations based on two-point measurements (TPM) of energy or entropylike observables; (ii) quantum-jump and full-counting-statistics approaches to entropy production in open quantum systems; and (iii) resource-theoretic formulations which treat athermality and coherence as explicit resources under thermal operations. In this subsection we clarify how our results relate to, and extend, each of these perspectives. Standard TPM-based fluctuation relations. Most quantum fluctuation relations for work and entropy production start from a TPM protocol on the system. Here we focus on the conceptual structure rather than on a particular implementation. See, e.g., Ref. [ 141 ] for a comprehensive review. For a closed system with Hamiltonian ˆ H ( t )driven by an external protocol from t = 0 to t = τ , one measures the initial energy ˆ H (0) projectively, obtaining an eigenvalue E0 n and collapsing the state to n0n0 , then evolves unitarily and measures the final energy ˆ H ( τ )with outcome Eτ m . The stochastic work is defined as W = Eτ m−E0 n , and the corresponding forward and backward work distributions pF ( W )and pR ( W )obey Crooksand Jarzynski-type relations of the form pF(W) pR(−W)= eβ(W−∆F),he−βW iF= e−β∆F,(658) where ∆ F is the equilibrium free-energy difference between the initial and final Hamiltonians. In open systems, analogous TPM-based constructions lead to fluctuation relations for entropy production, again expressed as log-ratios of forward and backward TPM path probabilities.[141] Our operator-level construction can be viewed as a generalisation of this setting. If we choose the entropy-production operator ˆ Σ so that its eigenbasis coincides with the energy eigenbasis at t = 0 and define it via the TPM log-likelihood, then the projectors ˆ Π± onto positive and negative σs correspond precisely to coarse-grained TPM sectors, and the diagonal contribution Pdiag ( ± )reproduces the TPMbased positive/negative probabilities. In this case, the interference quotients I± vanish because initial coherences are destroyed by the first projective measurement, and the modified FT reduces to the standard TPM FT. By contrast, in the coherence-preserving ancilla scheme of Sec. 3.4, the characteristic function of ˆ Σ is accessed interferometrically without an initial projective measurement on S , the coherences in ˆρ0 are preserved, and the same operator ˆ Σ generates non-trivial I± . Thus the TPM framework appears in our approach as the special case where Pqm ( ± ) = Pdiag ( ± )and the fluctuation relation is purely classical, while our general result explicitly tracks the additional interference corrections that arise when initial coherences are not discarded by measurement. Quantum-jump trajectories and counting statistics. A second major line of work studies fluctuation theorems for open quantum systems described by Markovian master equations and their quantum-jump unravellings.[141, 142] In this setting one considers a GKLS generator d dtˆρ(t) = −i ~[ˆ H, ˆρ(t)] + X rDr[ˆρ(t)],(659) with jump operators ˆ Lr obeying local detailed balance with respect to one or several reservoirs. Along each quantum trajectory Γgenerated by the stochastic sequence of jumps, the entropy production is defined as Σ[Γ] = X r σrNr[Γ],(660) where Nr [Γ] counts the jumps of type r and σr is the entropy increment associated with that jump, given by the log-ratio of its forward and backward rates. The corresponding generating function and full counting statistics of Σcan be obtained by tilting the Liouvillian with counting fields; the resulting CGF satisfies symmetries that encode detailed and integral fluctuation theorems.[141] Our Lindblad-based models in Secs. 5, 7 and 9 are fully compatible with this quantum-jump picture. We explicitly assigned to each jump operator ˆ Lr an entropy increment σr from local detailed balance and
113 computed the mean entropy production via hΣiτ=X r σrZτ 0 dtTrhˆ L† rˆ Lrˆρ(t)i,(661) exactly as in the quantum-jump literature. The novelty here is not the definition of Σ[Γ] or its counting statistics, but rather the way in which we coarse-grain the trajectory ensemble into positive and negative entropy sectors via projectors onto the spectrum of ˆ Σ , and then isolate the interference corrections I± to the FT ratio. In other words, our operator-level FT, Pqm(+) Pqm(−)= eΣst/kB1 + I+ 1 + I− ,(662) reorganises information that is already present in the full counting statistics of the quantum-jump process into a form that makes the role of coherence and interference explicit. The ancilla-based measurement scheme can be understood as a concrete interferometric implementation of the tilted-generator formalism for ˆ Σ, akin to interferometric measurements of work or charge-transfer statistics. Resource-theoretic coherence and thermal operations. A third major framework treats quantum thermodynamics as a resource theory.[ 143 , 144 ] In that approach, one considers a fixed background temperature T and defines thermal operations as those quantum channels that can be implemented by coupling the system to a heat bath in the Gibbs state ˆγ∝ e −βˆ H via energy-conserving unitaries. States that are not Gibbsian at temperature T are athermal and can serve as resources for tasks such as work extraction or cooling. A hierarchy of second laws arises in the single-shot regime: a family of generalized free energies Fα must decrease under thermal operations, and quantum coherence in the energy basis leads to additional constraints and to a distinction between “classical” athermality (populations) and coherence-related resources.[143, 144] Our results are dynamical and model-specific rather than axiomatic, but they are conceptually aligned with this resource-theoretic viewpoint. On the one hand, we restrict attention to dynamics generated by microreversible unitaries or GKLS generators with local detailed balance, which are natural continuoustime analogues of thermal operations.[ 142 ] On the other hand, we explicitly identify coherence (and its controllable phases) as a resource that can be used to modulate the probabilities of positive and negative entropy sectors and hence the FT ratio. The interference quotients I± behave, in many respects, like coherence monotones: they vanish for diagonal states, they are suppressed by dephasing, and their magnitude is constrained by operator inequalities that can be expressed in terms of coherence measures on the initial state. There is also a direct link to resource-theoretic work extraction. Horodecki and Oppenheim showed that, in the nanoscale regime, work extraction from a single non-equilibrium state is limited by finitesize and coherence effects, leading to distinct “one-shot” free energies and generically irreversible state transitions.[ 143 ] In our language, the same physical limitations appear as bounds on how far the FT ratio can be shifted by manipulating I± through purification and phase engineering: negative-entropy events can be enhanced or suppressed, but the interference factors remain bounded and cannot overturn the overall directionality encoded in Σ st > 0. Similarly, the resource theory of athermal states developed in Ref. [ 144 ] formalises which state transformations are possible under thermal operations; our trajectory-based results provide explicit dynamical examples where these abstract constraints manifest as interference-controlled modifications of entropy-production statistics, without contradicting the underlying second laws. In summary, our work does not replace TPM-based fluctuation relations, quantum-jump approaches or resource-theoretic thermodynamics. Rather, it provides a unifying operator framework and a set of concrete models that (i) reduce to standard TPM and quantum-jump FTs in appropriate limits, and (ii) make the contributions of coherence and classical chaos to fluctuation relations quantitatively transparent, in a manner that resonates with the resource-theoretic separation between classical and coherent thermodynamic resources. C. Limitations The framework developed in this work relies on a number of structural assumptions and idealizations. In this subsection we make these limitations explicit. This is important both to avoid over-interpreting our results and to indicate where future work is needed to generalise the formalism beyond the idealised regimes considered here.
114 Measurement protocols and entropy-production operator. A first limitation concerns the measurement schemes used to define and access the entropy-production operator ˆ Σ and the associated projectors ˆ Π± . Throughout the paper we have worked with two concrete classes of protocols: • A TPM-type scheme, where ˆ Σ is defined in terms of projective measurements at the beginning and the end of the protocol, and where the initial measurement diagonalises ρ0 in a chosen basis. In this case the interference corrections I±vanish by construction. • A coherence-preserving ancilla-based Ramsey scheme, where an ancilla is entangled with the system through controlled unitaries implementing e iuˆ Σ , and the characteristic function of ˆ Σ is read out from interference fringes in the ancilla. Both schemes implicitly assume ideal projective measurements (for TPM) or idealised control unitaries on a finite-dimensional ancilla (for the Ramsey scheme). Realistic detectors are described by generalised measurements (POVMs) and quantum instruments, with finite efficiency, thermal noise and back-action that may itself contribute to entropy production. Our derivations do not explicitly incorporate these imperfections, nor do they treat the detector and ancilla as dynamical thermodynamic systems with their own entropy budgets. In particular, we treat the ancilla as a perfectly coherent and cost-free resource. Mathematically, the operator ˆ Σ is assumed to be Hermitian with a discrete spectrum and to admit a spectral decomposition ˆ Σ = X s σs|sihs|,(663) with projectors |sihs| that can in principle be measured or coupled to via controlled unitaries. For generic POVMs or continuous-spectrum operators, this discrete spectral picture must be generalised, and it is not guaranteed that the interference quotients I± retain such a simple algebraic structure. Extending the present analysis to arbitrary instruments and continuous spectra is therefore a non-trivial open problem. Semiclassical approximation and chaotic dynamics. The semiclassical Van Vleck–Gutzwiller representation used in Sec. 6 and in the chaos-assisted tunnelling analysis of Sec. 7 relies on the standard stationary-phase approximation applied to Feynman path integrals and on hyperbolic classical dynamics with well-separated trajectories.[145] Concretely, we assume that: 1. The relevant actions Sα/~ are large, so that phases e iSα/~ oscillate rapidly and stationary-phase methods are justified. 2. Stationary points of the action are non-degenerate, i.e. det ∂2Sα ∂qi∂qj6= 0,(664) which ensures that the stability amplitudes Cα are finite and that the semiclassical propagator is well-defined away from caustics.[145] 3. The observation time τ is not too large compared to the Ehrenfest time tE∼λ−1ln ( Scl/~ ), where λ is the largest Lyapunov exponent and Scl is a characteristic classical action. For τtE , quantum interference becomes highly intricate and semiclassical trajectory sums may require uniform approximations or resummation techniques not considered here. Our semiclassical expressions for Pqm ( ± )and the interference quotients I± are therefore reliable only in the regime where these conditions hold. In regimes with strong diffraction, mixed regular–chaotic phase space near island boundaries, or strong coupling to slow environmental modes, the simple trajectory picture may break down. Moreover, the chaos-assisted tunnelling analysis assumes an underlying classical system with mixed phase space in the sense of Gutzwiller.[ 145 ] Quantum systems without a well-defined classical limit, or with many-body localisation, will require different tools. Finite time windows, decoherence and Markovianity. Another key assumption is that entropy production is monitored over a finite time window [0 , τ ]that is short compared to certain decoherence and relaxation times, but long enough for the relevant steady-state or large-deviation regime to emerge. In practice, there are several competing timescales: coherent evolution times tcoh ∼~/J , decoherence times tdec set by environment-induced dephasing, relaxation times trel towards local or global stationary
115 states, and driving timescales associated with external protocols.[ 146 , 147 ] Our idealised models implicitly assume a separation of these scales such that: tcoh .τtdec, trel,(665) for the ancillary phase-engineering protocols, so that coherence has time to influence the dynamics but is not completely washed out by decoherence. In addition, the Lindblad generators used in Secs. 5, 7 and 9 are Markovian and of GKLS form, corresponding to weak system–bath coupling, short bath correlation times and factorized initial conditions.[ 147 ] Strong coupling, structured reservoirs and non-Markovian memory effects can qualitatively alter entropyproduction statistics and even invalidate simple detailed-balance relations for jump rates. Our operatorlevel fluctuation theorem assumes that a microreversible dilation exists and that reduced dynamics can be consistently described by such GKLS generators; extending the analysis to genuinely non-Markovian environments remains an open and challenging problem. Control overhead and thermodynamic cost of ancilla operations. A final, and conceptually important, limitation concerns the treatment of control operations. In the purification and phase-engineering protocols, as well as in the ancilla-based measurement scheme, we effectively assume that: •Ancilla systems can be prepared in pure reference states at negligible thermodynamic cost. • Controlled unitaries implementing e iuˆ Σ and phase rotations e iφi can be performed without dissipation. • Measurements on the ancilla can be read out and post-processed without accounting for the associated entropy and work cost. This is a common idealisation in fluctuation-theorem and quantum-information protocols, but it is not realistic in the strict thermodynamic sense. The thermodynamic cost of measurement and feedback control has been analysed in detail by Sagawa and Ueda and others,[ 148 , 149 ] leading to generalised second laws that include information-theoretic terms and explicit work–information trade-offs. By neglecting these costs, our analysis effectively treats the control layer as an external resource with unbounded free energy and coherence. From a pragmatic perspective, this is acceptable when the main goal is to characterise what is in principle possible within quantum mechanics, and when the energetic cost of control is small compared to the thermodynamic scales of interest (e.g. ATP hydrolysis in molecular machines). However, for a fully self-contained thermodynamic treatment in which the controller, ancilla and measuring apparatus are explicitly included in the entropy budget, the present framework must be extended to include the work and heat associated with control operations and the back-action of measurements, along the lines of Refs. [148, 149]. Summary of limitations. To summarise, the main limitations of the present work are: (i) the restriction to specific, idealised measurement protocols (TPM and ancilla-based Ramsey schemes) and Hermitian entropy-production operators with discrete spectra; (ii) the dependence of the semiclassical analysis on standard stationary-phase conditions and on a well-defined chaotic classical limit; (iii) the reliance on Markovian GKLS master equations with local detailed balance and on a controlled separation of timescales between coherence, decoherence and relaxation; and (iv) the neglect of the explicit thermodynamic cost of control, measurement and ancilla manipulation. Within these assumptions our results are internally consistent and capture genuine quantum-coherent and chaos-assisted corrections to fluctuation relations, but they should be interpreted as a first step towards a more complete and experimentally realistic theory rather than a final description of thermodynamics in complex quantum environments. D. Experimental prospects The formalism developed in this work has been deliberately anchored in models that are close to platforms where fluctuation theorems and quantum coherence are already being investigated experimentally. Here we outline several concrete directions in which the main ingredients of our framework—operator-level entropy production, interference corrections to fluctuation relations, chaos-assisted amplification, and purification-based phase control—could be probed in the laboratory.
116 Chaotic cold-atom and microwave-billiard platforms. The semiclassical and chaos-assisted aspects of our theory (Secs. VI and VII) are naturally connected to systems where classically chaotic dynamics has already been realised and characterised at the quantum level. Prominent examples include cold-atom realizations of the kicked rotor, where dynamical localization and quantum signatures of classical chaos have been observed,[ 150 ] and microwave or superconducting billiards, where the stationary spectra and wavefunctions of classically chaotic cavities can be mapped with high precision.[151] In such setups, one can envisage implementing an effective “machine” degree of freedom (for instance, a rotor angle or a discrete set of angular sectors) coupled to engineered reservoirs realised either by auxiliary atomic modes or by external fields with controlled noise. The Hamiltonians underlying our semiclassical analysis—with mixed phase space and chaos-assisted tunnelling between stable islands—can be closely mimicked by suitably driven optical-lattice potentials or microwave resonators. To probe fluctuation relations, one would use time-dependent driving protocols and repeated preparations of the same initial state, reconstructing coarse-grained probabilities Pqm ( ± )for entropy-production sectors from measured energy or particle-transfer statistics. Although a full implementation of the entropy-production operator ˆ Σ and its projectors ˆ Π± is experimentally demanding, partial tests—for example, measuring the dependence of a coarse-grained FT ratio on control parameters that modify the number of interfering trajectories—are within reach of current cold-atom and microwave-physics technology. Trapped ions, NMR and superconducting qubits for ancilla-based FTs. The ancilla-based Ramsey scheme we propose to access the characteristic function of ˆ Σ is closely related to experimental techniques already used to reconstruct work distributions and verify quantum fluctuation relations. In particular, Batalhão et al. experimentally reconstructed the nonequilibrium work distribution and tested Crooks and Jarzynski relations in a liquid-state NMR setup, using an interferometric protocol in which an ancilla spin encodes the characteristic function of work.[ 152 ] Similarly, An et al. performed a direct test of the quantum Jarzynski equality using a single trapped-ion system, implementing projective energy measurements and coherent control over the motional state of the ion.[153] These platforms are ideal candidates to implement our entropy-production protocol. Instead of coupling the ancilla to the system Hamiltonian to measure work, one would couple it to the entropy-production operator ˆ Σ defined by the relevant reservoirs and jump channels in a driven open-system dynamics (as in our Lindblad models). The same Ramsey-type pulse sequences used to reconstruct characteristic functions of work can, in principle, be adapted to reconstruct the characteristic function of ˆ Σ , and therefore the full distribution P (Σ) in both TPM and coherence-preserving schemes. Superconducting qubit architectures in circuit QED, which combine strong coupling, fast control and high-fidelity readout, offer another promising platform for such ancilla-based measurements and for monitoring entropy production at the single-quanta level. Synthetic nanomachines and biomimetic rotors. The chemo–mechanical molecular machine discussed in Sec. IX was inspired by ATP synthase, but its parameters and couplings were chosen for conceptual clarity rather than quantitative realism. On the experimental side, single-molecule studies of the F 1 sector of ATP synthase have long demonstrated that individual enzymes act as rotary motors, with discrete 120 ◦ or 80 ◦ + 40 ◦ steps per ATP hydrolysis, measurable via attached beads or fluorescent markers.[ 154 ] These experiments routinely resolve the torque–speed relationship, stall behaviour and stochastic stepping statistics, and more recent work has extended such measurements to reconstituted F O F 1 complexes in lipid membranes, bringing the full chemo–mechanical cycle into view. At present, most single-molecule studies of ATP synthase can be described to good approximation by classical Markov jump models with thermodynamically consistent rates, and there is little direct evidence for robust quantum coherence in the rotor dynamics at physiological conditions. However, synthetic and biomimetic nanomachines—including light-driven molecular rotors and chemically powered artificial molecular motors—offer more flexibility in designing and probing coherent transitions.[ 155 ] In such systems, electronic or vibronic coherences can be created and manipulated by ultrafast pulses, while mechanical motion is read out through fluorescence or mechanical deflection. By designing a rotor with a small number of well-separated angular states and tunable coherent couplings between them, one could realise a close analogue of our finite-dimensional chemo–mechanical model, and test how phase control and coherence modify the trajectories of mechanical steps and the associated entropy-production statistics. Biological and synthetic excitonic complexes. The excitonic dimer example of Sec. IX E is structurally close to minimal models used to interpret ultrafast spectroscopy experiments on photosynthetic lightharvesting complexes. Two-dimensional electronic spectroscopy has provided strong evidence for coherent excitonic dynamics in pigment-protein complexes such as the Fenna–Matthews–Olson (FMO) complex at cryogenic temperatures,[ 156 ] and for long-lived quantum coherence in marine cryptophyte algae even at ambient conditions.[157]
117 In these experiments, coherent superpositions of excitonic states are prepared by tailored sequences of ultrafast laser pulses, and their time evolution is encoded in oscillatory features of the nonlinear optical response. From the perspective of our framework, such pump–probe sequences naturally implement the preparation of phase-controlled superpositions analogous to |ψ(ϕ)i in Eq. (641) . If the excitonic system is coupled to well-characterised source and drain environments (for example, by embedding it in a device geometry where exciton injection and extraction can be monitored electrically or optically), one could in principle reconstruct entropy-production statistics for exciton transport by combining ultrafast coherence control with time-resolved measurements of emission or charge separation. This would provide a direct test of how coherent vibronic pathways reshape the tails of P (Σ) in a realistic excitonic network, although significant experimental development is required to reach this level of combined temporal and thermodynamic resolution. Towards fully biophysical chemo–mechanical tests. A more ambitious prospect is to bring the full chemo–mechanical fluctuation structure into contact with reconstituted biological motors such as ATP synthase. Single-molecule trajectory data already allow one to reconstruct step statistics, dwell-time distributions and, with careful calibration, work and heat flows per step. Embedding such experiments in controlled chemicaland mechanical-affinity protocols—for example, periodic modulation of ATP and ADP+P i concentrations combined with time-dependent external torque—would make it possible to approximate the forward and backward driving protocols required for fluctuation theorems at the trajectory level. Our framework suggests that deviations from classical FT predictions arising from coherent or quasi-coherent substeps would be most visible in finite-time, sub-cycle statistics and in the negative-entropy tail of P(Σ). Given the strong decoherence expected in aqueous environments at room temperature, it is likely that fully biophysical tests of coherence-induced FT modifications will require hybrid platforms: for instance, synthetic molecular rotors that emulate key design principles of ATP synthase, but operate in environments and energy scales where quantum coherence can be sustained and controlled. Nevertheless, the coarse-grained operator picture and interference quotients I± developed here provide a natural language for interpreting any future deviations from classical FT predictions observed in such biomimetic machines. Summary of experimental outlook. In summary, the main experimental prospects can be grouped as follows: • Near-term quantum platforms: trapped ions, NMR and superconducting circuits, where ancillabased interferometric measurements of work have already been demonstrated,[ 152 , 153 ] and where adapting these techniques to entropy production and sector projectors ˆ Π± appears technically feasible. • Chaotic and semiclassical systems: cold-atom kicked rotors and microwave billiards,[ 150 , 151 ] which naturally realise mixed phase space and chaos-assisted tunnelling, and where coarse-grained entropy production could be probed via engineered reservoirs. • Molecular and excitonic machines: reconstituted ATP synthase, artificial molecular motors,[ 155 ] and photosynthetic or synthetic excitonic complexes,[ 156 , 157 ] where our minimal chemo–mechanical and excitonic models can guide the interpretation of trajectory-level experiments as spectroscopic and single-molecule techniques continue to improve. Taken together, these platforms suggest that the central predictions of this paper—modified fluctuation relations with interference quotients, chaos-assisted amplification, and phase-controlled negative-entropy sectors—are not purely theoretical constructs, but lie within the scope of current or near-future experiments across atomic, mesoscopic and molecular physics. Appendix A: Operator derivation of the quantum fluctuation theorem In this Appendix we give a self-contained operator derivation of the quantum fluctuation theorem (FT) used in Secs. III C and IV. We (i) state a precise microreversibility condition, (ii) define the entropy-production operator ˆ Σ and the projectors ˆ Π± , (iii) recall how standard quantum FTs follow from microreversibility when the initial state is diagonal in a suitable basis, and (iv) derive the modified ratio Pqm(+) Pqm(−)= eΣst/kB1 + I+ 1 + I− (A1)
118 in a purely operator-theoretic manner. Throughout we work with finite-dimensional Hilbert spaces for clarity; the extension to separable infinite dimensions requires some functional-analytic care but is conceptually straightforward.[158, 164] A.1. Setup and microreversibility We consider a system S with Hilbert space HS coupled to an environment E with Hilbert space HE . The joint Hilbert space is H=HS⊗HEand the total Hamiltonian is ˆ Hλ(t) = ˆ HS(λt) + ˆ HE+ˆ Hint,(A2) where λt denotes a time-dependent control parameter (or set of parameters) defining the driving protocol in the forward process. The joint unitary propagator for the forward protocol is ˆ UF(τ, 0) = Texp−i ~Zτ 0 ˆ Hλ(t) dt,(A3) where Tis the time-ordering operator. We assume the existence of an antiunitary time-reversal operator ˆ Θacting on Hsuch that: 1. ˆ Θiˆ Θ−1=−i. 2. For a time-reversed protocol λR t=λτ−tand possibly reversed magnetic field, ˆ Θˆ Hλ(t)ˆ Θ−1=ˆ HλR(τ−t).(A4) These conditions imply microreversibility in the sense that the unitary propagator for the reversed protocol satisfies[159, 160] ˆ UR(τ, 0) = ˆ Θˆ U† F(τ, 0) ˆ Θ−1.(A5) We further assume that the environment is initially in a product of grand-canonical Gibbs states for reservoirs labeled by α, ˆµE=O α exp−βα(ˆ Hα−µαˆ Nα) Zα ,(A6) and that ˆµE is invariant under time reversal, ˆ ΘˆµEˆ Θ−1 = ˆµE . The system is prepared in an arbitrary initial state ˆρ0on HS, so the global initial state of the forward protocol is ˆρtot 0= ˆρ0⊗ˆµE.(A7) The corresponding final state at t=τis ˆρtot τ=ˆ UF(τ, 0) ˆρtot 0ˆ U† F(τ, 0).(A8) The reduced system state is ˆρτ= TrE[ˆρtot τ]. The reversed process is defined analogously, with initial global state ˆρtot,R 0= ˆρR 0⊗ˆµE,(A9) where ˆρR 0 is chosen according to the specific FT under consideration (e.g. equilibrium for work relations, steady-state for entropy production).[158, 161] A.2. Trajectory-resolved exchanges and entropy production We now assume that the dynamics admits a trajectory-level description in terms of a quantum-jump unraveling of a Lindblad-type master equation, or more generally in terms of discrete measurement records.[162, 163] For concreteness, consider a Markovian master equation for the reduced system, d dtˆρ(t) = Lt[ˆρ(t)],(A10)
119 with generator of GKLS form Lt[ˆρ] = −i ~[ˆ HS(λt),ˆρ] + X rD[ˆ Lr](ˆρ),(A11) where the dissipators are D[ˆ Lr](ˆρ) = ˆ Lrˆρˆ L† r−1 2nˆ L† rˆ Lr,ˆρo.(A12) Each jump operator ˆ Lr is associated with an energy and particle transfer (∆ Eα r, ∆ Nα r )to reservoir α , and satisfies local detailed-balance conditions of the form[158, 161] ˆ Lr= e1 2Pαβα(∆Eα r−µα∆Nα r)ˆ Θˆ L† ˜rˆ Θ−1,(A13) where ˜rlabels the time-reversed jump. Along a quantum trajectory Γ(a sequence of jump types and jump times) the stochastic entropy production can be written as Σ[Γ] = X r σrNr[Γ], σr:= X α βα(∆Eα r−µα∆Nα r),(A14) where Nr [Γ] counts the number of jumps of type r .[ 162 , 163 ] Equivalently, the probability of a trajectory Γin the forward process, denoted PF [Γ], and the probability of its time-reversed counterpart ˜ Γ in the reversed process, PR[˜ Γ], satisfy a quantum version of the Crooks-Gallavotti–Cohen relation[158, 161] ln PF[Γ] PR[˜ Γ] =1 kB Σ[Γ].(A15) This equation is the trajectory-level starting point of our construction. A.3. Entropy-production operator and sector projectors We now pass from trajectory labels Γto an operator representation. Introduce an abstract “record” Hilbert space Hrec spanned by orthonormal vectors {|Γi} , one for each allowed trajectory over the observation time window [0, τ]. Define the entropy-production operator on Hrec as ˆ Σ = X Γ Σ[Γ] |ΓihΓ|,(A16) where Σ[Γ] is given by Eq. (A14) . By construction, ˆ Σ is Hermitian and has purely real spectrum {σs} indexed by s≡Γ. The forward and backward trajectory probability distributions can be encoded as diagonal operators on Hrec, ˆ PF:= X Γ PF[Γ] |ΓihΓ|,ˆ PR:= X Γ PR[˜ Γ] |ΓihΓ|,(A17) where we have relabelled the reversed trajectories by their forward partners so that both operators act on the same basis. The trajectory-level FT (A15) can then be rewritten as a log-likelihood operator identity, ˆ Σ = kBln ˆ PF−ln ˆ PR,(A18) where ln is understood as the functional calculus on positive operators (here, simply diagonal matrices with strictly positive entries).[158] From the spectral decomposition of ˆ Σ we define the projectors onto coarse-grained positive and negative entropy sectors as ˆ Π+:= X Γ: Σ[Γ]>0|ΓihΓ|,ˆ Π−:= X Γ: Σ[Γ]<0|ΓihΓ|.(A19)
120 These projectors satisfy ˆ Π++ˆ Π−+ˆ Π0=ˆ Irec,(A20) where ˆ Π0projects onto the (typically negligible) subspace of zero-entropy trajectories. Although defined here on the record space, the same spectral projectors appear in the system-space formulation when the record is generated by a concrete measurement scheme (TPM or ancilla-based) as discussed in Sec. III C. In particular, in the ancilla-based scheme the characteristic function of ˆ Σ is measured via controlled couplings of the form exp (i uˆ Σ ), and ˆ Π± correspond to coarse-grained measurements of the final ancilla state. A.4. Classical (diagonal) FT for coarse-grained sectors We define the coarse-grained forward probabilities of positive and negative entropy production as Pdiag(+) := Trrec[ˆ Π+ˆ PF] = X Γ: Σ[Γ]>0 PF[Γ],(A21) and similarly Pdiag(−) := Trrec[ˆ Π−ˆ PF] = X Γ: Σ[Γ]<0 PF[Γ].(A22) These are precisely the classical probabilities of positive and negative entropy production in the forward process, and they coincide with the diagonal-sector probabilities discussed in Sec. II. Using Eq. (A15), we can relate Pdiag(+) and Pdiag(−)as follows: Pdiag(+) = X Γ: Σ[Γ]>0 PF[Γ] = X Γ: Σ[Γ]>0 eΣ[Γ]/kBPR[˜ Γ],(A23) Pdiag(−) = X Γ: Σ[Γ]<0 PF[Γ] = X Γ: Σ[Γ]>0 PF[Γ0] (Γ0≡˜ Γ).(A24) Combining these expressions we can formally write Pdiag(+) Pdiag(−)=P Γ: Σ[Γ]>0 eΣ[Γ]/kBPR[˜ Γ] P Γ: Σ[Γ]>0 PF[Γ0].(A25) In general the ratio depends on the full distribution of Σ[Γ]. However, for stationary nonequilibrium states satisfying a Gallavotti–Cohen symmetry at long times, Pτ(Σ) Pτ(−Σ) −−−−→ τ→∞ eΣ/kB,(A26) one can show that 1 τln Pdiag(+) Pdiag(−)−−−−→ τ→∞ Σst kBτ,(A27) where Σ st is the steady-state entropy production over time τ .[ 158 , 161 ] For finite but fixed τ it is convenient to define Σst := kBln Pdiag(+) Pdiag(−),(A28) which reduces to the steady-state entropy production in the long-time limit when large-deviation principles apply.[164] In the classical limit considered in Sec. II, where the trajectory probabilities are generated by a Markov jump process obeying local detailed balance, the above definitions reproduce the usual steady-state fluctuation theorem for coarse-grained positive and negative sectors.
121 A.5. Quantum probabilities, diagonal/off-diagonal split, and interference quotients We now lift the construction to an operator level in the system Hilbert space HS , following the structure of Sec. IV. Consider a measurement scheme (TPM or ancilla-based) that yields the entropy-production record and hence implements the projectors ˆ Π± as effects on HS at time τ . In the TPM scheme, ˆ Π± act on a doubled energy-eigenbasis space indexed by initial and final energies; in the ancilla scheme, they act on the joint system–ancilla space. For the present derivation, it suffices that they are orthogonal projectors on some Hilbert space carrying the system dynamics. Let ˆ U denote the effective propagator over [0 , τ ]in this enlarged space, and let ˆρ0 denote the initial state of the system (plus ancilla if present). The quantum probabilities of positive and negative entropy production are Pqm(±) := Tr ˆ Π±ˆ Uˆρ0ˆ U†.(A29) We now choose an orthonormal basis {|mi} that diagonalises ˆρ0, ˆρ0=X m pm|mihm|+X m6=n ρ0,mn |mihn|, ρ0,mn =hm|ˆρ0|ni.(A30) Expanding Eq. (A29) in this basis we obtain Pqm(±) = X m,n ρ0,mn hn|ˆ U†ˆ Π±ˆ U|mi =X m pmhm|ˆ U†ˆ Π±ˆ U|mi+X m6=n ρ0,mn hn|ˆ U†ˆ Π±ˆ U|mi.(A31) This suggests defining: •The diagonal contribution Pdiag(±) := X m pmhm|ˆ U†ˆ Π±ˆ U|mi.(A32) This coincides with the classical probability of the corresponding sector when ˆρ0 is diagonal in the basis where ˆ Π±define coarse-grained projectors in trajectory space. •The interference correction ∆Pint(±) := X m6=n ρ0,mn hn|ˆ U†ˆ Π±ˆ U|mi,(A33) which vanishes whenever ˆρ0 is diagonal in the {|mi} basis (no initial coherence), or when ˆ U†ˆ Π±ˆ U is diagonal in that basis (no coherence-to-population conversion). Equation (A31) can then be written as Pqm(±) = Pdiag(±)+∆Pint(±),(A34) which is Eq. (4.4) of the main text in operator form. We now introduce the dimensionless interference quotients I±:= ∆Pint(±) Pdiag(±),(A35) whenever Pdiag ( ± ) > 0. In pathological cases where the diagonal contribution to one sector vanishes, the corresponding I±can be defined by continuity or omitted from the ratio. By construction Pqm(±) = Pdiag(±)1 + I±.(A36) Positivity of Pqm(±)implies 1 + I±≥0,(A37) which provides non-trivial bounds on the allowed interference contributions.[ 162 , 164 ] These bounds can be sharpened using Cauchy–Schwarz inequalities applied to the off-diagonal terms in Eq. (A33) , as discussed in Sec. 4.5 of the main text.
128 C.2. Circuit-level derivation of the characteristic function We denote by ˆρ(k) SA the joint state after step kin the protocol. Step 1: Initial state. The initial state is given by Eq. (C2): ˆρ(1) SA = ˆρ0⊗|0ih0|A.(C7) Step 2: First Hadamard on the ancilla. The Hadamard gate acts as HA|0iA=1 √2(|0iA+|1iA), HA|1iA=1 √2(|0iA−|1iA).(C8) After applying HA⊗ˆ ISwe obtain ˆρ(2) SA = (HA⊗ˆ IS) ˆρ(1) SA (HA⊗ˆ IS)† = ˆρ0⊗1 2(|0ih0|A+|0ih1|A+|1ih0|A+|1ih1|A).(C9) Step 3: First controlled-ˆ Σrotation. Applying the controlled unitary of Eq. (C4) yields ˆρ(3) SA =ˆ U1ˆρ(2) SA ˆ U† 1,ˆ U1:= |0ih0|A⊗ˆ IS+|1ih1|A⊗e−iuˆ Σ/2.(C10) By direct multiplication we find ˆρ(3) SA =1 2|0ih0|A⊗ˆρ0+|1ih1|A⊗e−iuˆ Σ/2ˆρ0e+iuˆ Σ/2 +|0ih1|A⊗ˆρ0e+iuˆ Σ/2+|1ih0|A⊗e−iuˆ Σ/2ˆρ0.(C11) Step 4: System evolution under ˆ U.We now apply the physical evolution ˆ Uto the system only, ˆ Utot =ˆ IA⊗ˆ U, (C12) and obtain ˆρ(4) SA = (ˆ IA⊗ˆ U)ˆρ(3) SA(ˆ IA⊗ˆ U†).(C13) Using Eq. (C11) and the definition ˆρτ:= ˆ Uˆρ0ˆ U†, we find ˆρ(4) SA =1 2|0ih0|A⊗ˆρτ+|1ih1|A⊗ˆ Ue−iuˆ Σ/2ˆρ0e+iuˆ Σ/2ˆ U† +|0ih1|A⊗ˆ Uˆρ0e+iuˆ Σ/2ˆ U†+|1ih0|A⊗ˆ Ue−iuˆ Σ/2ˆρ0ˆ U†.(C14) Step 5: Second controlled-ˆ Σrotation. We now apply the inverse controlled unitary of Eq. (C6), ˆ U2:= |0ih0|A⊗ˆ IS+|1ih1|A⊗e+iuˆ Σ/2,(C15) to obtain ˆρ(5) SA =ˆ U2ˆρ(4) SA ˆ U† 2.(C16) Working term by term in Eq. (C14) and using the projectors on the ancilla, we get ˆρ(5) SA =1 2|0ih0|A⊗ˆρτ+|1ih1|A⊗e+iuˆ Σ/2ˆ Ue−iuˆ Σ/2ˆρ0e+iuˆ Σ/2ˆ U†e−iuˆ Σ/2 +|0ih1|A⊗ˆ Uˆρ0e+iuˆ Σ/2e+iuˆ Σ/2ˆ U†+|1ih0|A⊗e+iuˆ Σ/2ˆ Ue−iuˆ Σ/2ˆρ0ˆ U†.(C17) If ˆ Σ commutes with ˆ U (as in many standard constructions where ˆ Σ is defined in the Heisenberg picture at the final time), this simplifies significantly. Assuming [ˆ U, ˆ Σ] = 0 for concreteness, we obtain ˆρ(5) SA =1 2|0ih0|A⊗ˆρτ+|1ih1|A⊗ˆρτ +|0ih1|A⊗ˆ Uˆρ0e+iuˆ Σˆ U†+|1ih0|A⊗e−iuˆ Σˆ Uˆρ0ˆ U†.(C18) Using ˆρτ=ˆ Uˆρ0ˆ U†and cyclicity of the trace, the off-diagonal ancilla blocks can be written as ˆρ(5) 01 := 1 2ˆ Uˆρ0e+iuˆ Σˆ U†=1 2ˆρτe+iuˆ Σ,ˆρ(5) 10 := 1 2e−iuˆ Σˆρτ.(C19)
129 Step 6: Second Hadamard and ancilla readout. Before readout we apply a second Hadamard HA . The reduced state of the ancilla after this final Hadamard, ˆρout A , can be obtained by tracing out the system. It is convenient to compute the expectation values of ˆσx and ˆσy directly from ˆρ(5) SA before the last Hadamard and then rotate the measurement basis. The Bloch vector components of the ancilla prior to the second Hadamard are hˆσxi(5) = TrSAh(ˆσx⊗ˆ IS) ˆρ(5) SAi= 2<TrSˆρ(5) 01 =<TrShˆρτe+iuˆ Σi,(C20) hˆσyi(5) = TrSAh(ˆσy⊗ˆ IS) ˆρ(5) SAi=−2=TrSˆρ(5) 01 =−=TrShˆρτe+iuˆ Σi.(C21) Hence hˆσxi(5) =<χ(u),hˆσyi(5) =−=χ(u),(C22) with χ ( u )defined by Eq. (C3) . After the second Hadamard, the roles of ˆσx and ˆσz are interchanged, so that by measuring ˆσzand ˆσyon the ancilla one can reconstruct χ(u). In summary, the ancilla Bloch vector after the full Ramsey sequence contains the real and imaginary parts of the characteristic function of the entropy-production operator ˆ Σ in the final state ˆρτ . Importantly, no projective measurement is ever performed on the system at t = 0; the initial coherences of ˆρ0 are preserved throughout the protocol, and their effect is encoded in χ ( u )and the resulting distribution of Σ. C.3. Extraction of the entropy-production distribution Once the characteristic function χ(u) = TrSheiuˆ Σˆρτi(C23) has been reconstructed for a set of values u∈R , the probability distribution P (Σ) of entropy production can be obtained by Fourier transform, P(Σ) = 1 2πZ+∞ −∞ due−iuΣχ(u).(C24) In practice, one measures χ ( uk )on a finite grid of points uk and performs a discrete Fourier transform or an appropriate inversion procedure, taking into account statistical noise and finite sampling.[172, 173] The coarse-grained sector probabilities Pqm ( ± )used in the main text can then be obtained by integrating P (Σ) over Σ ≷ 0, or directly by measuring the expectation values of the projectors ˆ Π± in the final state (e.g. by adding an additional ancilla qubit that couples selectively to positive or negative eigenspaces of ˆ Σ). C.4. Pulse sequences in concrete platforms The abstract controlled unitaries C [e ±iuˆ Σ/2 ]can be implemented in a variety of platforms where ancillaassisted interferometry has already been used to measure work or heat statistics.[168, 169, 172, 173] NMR and trapped ions. In liquid-state NMR, the ancilla is a nuclear spin and the system is encoded in one or several additional spins. Conditional evolutions of the form |1ih1|A⊗ˆ Hint can be engineered via refocusing sequences and controlled rotations; exponentiating these Hamiltonians for time t∝u realises e −iuˆ Σ/2 when ˆ Σ is expressed as a linear combination of spin operators.[ 172 ] Analogous constructions exist in trapped-ion systems, where internal states of the ion serve as qubits and motional modes as bosonic degrees of freedom.[173] Superconducting qubits and circuit QED. In circuit QED architectures, transmon or flux qubits can be coupled dispersively to microwave resonators, realising Hamiltonians of the form χˆσzˆa†ˆa . By choosing the system Hilbert space appropriately (e.g. Fock states of the resonator) and combining such dispersive couplings with single-qubit gates, one can engineer effective controlledˆ Σ operations. The Ramsey sequence is implemented as a pair of π/ 2pulses on the ancilla separated by an evolution interval during which the controlled-ˆ Σis active.
130 Preservation of initial coherences. In all these platforms, the crucial point is that the ancilla couples to ˆ Σ after the system has been prepared in ˆρ0 , and that no projective measurement is performed on S at t = 0. The only measurements occur at the end of the protocol and act on the ancilla only. This ensures that initial coherences in the eigenbasis of ˆ Σ (or of the relevant energy/number operators entering its definition) are preserved, and that the resulting χ ( u )and P (Σ) encode the full quantum interference effects rather than a classicalised TPM statistics. C.5. Relation to TPM schemes and resource-theoretic interpretations For completeness, we comment on the relation between this ancilla protocol and standard TPM schemes. TPM as a limiting case. If one inserts a projective measurement of the eigenbasis of ˆ Σ at t = 0 before the first Hadamard, then ˆρ0 is replaced by its dephased version ∆( ˆρ0 ), diagonal in that basis. In this case, the off-diagonal contributions to χ ( u )vanish, and the Fourier transform (C24) yields the TPM distribution of entropy production. Thus TPM can be viewed as a special case of the present scheme where initial coherences are removed “by hand”. Coherence as a control resource. In the protocol of Sec. VIII, the purification of ˆρ0 on a larger Hilbert space and the control of relative phases in that purification allow one to shape the initial coherences that enter χ ( u )and hence the weights of positive and negative entropy sectors. From a resource-theoretic viewpoint, the ancilla-based protocol is an explicit realisation of how coherence in the eigenbasis of ˆ Σ , together with controlled unitaries, can be used as a thermodynamic control resource to manipulate entropy-production statistics.[174, 175] C.6. Summary To summarise, the ancilla-based Ramsey protocol described in this appendix: • Realises the characteristic function χ ( u ) = Tr [e iuˆ Σˆρτ ]of the entropy-production operator ˆ Σ as coherences of a single-qubit ancilla. • Preserves initial system coherences by avoiding any projective measurement at t = 0, unlike TPM schemes. • Allows the full distribution P (Σ) and the sector probabilities Pqm ( ± )to be reconstructed by Fourier transform and coarse-graining. • Can be implemented with existing pulse sequences in NMR, trapped-ion and superconducting-qubit platforms, where analogous interferometric schemes for work and heat statistics have already been demonstrated experimentally. This provides the operational backbone for the “coherence-preserving ancilla-based measurement protocol” used in the main text, and makes it clear how the interference quotients I± and modified FT ratio emerge from a concrete, experimentally viable measurement scheme. Appendix D: Detailed solution of the finite-dimensional example In this appendix we give a complete solution of the finite-dimensional quantum example introduced in Sec. 5. For concreteness, we work with a three-level system (qutrit) whose dynamics is unitary over a single driving step of duration τ , and for which the entropy-production operator ˆ Σ is a diagonal observable with three distinct eigenvalues. This is sufficient to illustrate, in a fully explicit way, how initial coherence in the eigenbasis of ˆ Σproduces interference corrections to the fluctuation-theorem (FT) ratio. Throughout this appendix we set kB= 1 and ~= 1 for simplicity. D.1. Hamiltonian and propagator We consider a three-level system with orthonormal basis {|0i,|1i,|2i} and Hamiltonian ˆ H= Ω|1ih2|+|2ih1|,(D1)
131 where Ω > 0is a real coupling strength. Level |0i is a spectator (it does not participate in the coherent driving), while |1i and |2i form an effective two-level system undergoing Rabi-like oscillations. The model can be regarded as the restriction of a more general three-level LGKS model to a single coherent manifold; see, e.g., Refs. [176, 177]. The Hamiltonian (D1) has eigenvalues and (unnormalised) eigenvectors ˆ H|0i= 0 ·|0i,ˆ H|1i−|2i=−Ω|1i−|2i,ˆ H|1i+|2i= +Ω|1i+|2i.(D2) The corresponding orthonormal eigenbasis is |0i,|−i =1 √2(|1i−|2i),|+i=1 √2(|1i+|2i),(D3) with ˆ H|±i =±Ω|±i. The time-evolution operator for a driving step of duration τis ˆ U(τ) = exp(−iˆ Hτ) = |0ih0|+ exp(−iΩτ)|+ih+|+ exp(+iΩτ)|−ih−|.(D4) In the original basis {|0i,|1i,|2i} this becomes ˆ U(τ) = |0ih0|+ cos(θ)|1ih1|+|2ih2|−i sin(θ)|1ih2|+|2ih1|, θ := Ωτ, (D5) i.e. an SU(2) rotation in the {|1i,|2i} subspace with angle 2θaround a fixed axis. D.2. Entropy-production operator and spectral decomposition To model entropy production for this finite system, we adopt a simple coarse-grained operator definition consistent with the operator framework of Sec. 4 and Appendix A. We assume that trajectories ending in state |1i correspond to positive entropy production + σ , those ending in |2i to negative entropy production −σ, and those ending in |0ito zero entropy production. The corresponding Hermitian operator is ˆ Σ=0·|0ih0|+σ|1ih1|−σ|2ih2|, σ > 0.(D6) In the {|0i,|1i,|2i} basis, ˆ Σis diagonal with eigenvalues σ0= 0, σ1= +σ, σ2=−σ, (D7) and eigenstates |s0i=|0i,|s1i=|1i,|s2i=|2i.(D8) In the language of Sec. 3.3, ˆ Σ = 2 X s=0 σs|sihs|,(D9) and the projectors onto positive/negative entropy sectors are ˆ Π+=|1ih1|,ˆ Π−=|2ih2|.(D10) If desired, the zero-entropy sector σ0 = 0 can be included explicitly, but in the main text we focus on the ±sectors. D.3. Time-evolved state: populations and coherences We take a general initial state that is diagonal in |0ibut may contain coherence between |1iand |2i: ˆρ0=p0|0ih0|+p1|1ih1|+p2|2ih2|+c12 |1ih2|+c∗ 12 |2ih1|,(D11) with p0, p1, p2≥ 0, p0 + p1 + p2 = 1, and c12 ∈C satisfying |c12|2≤p1p2 . We write c12 = cx + i cy with cx, cy∈R.
132 The state at time τis ˆρτ=ˆ U(τ) ˆρ0ˆ U†(τ),(D12) with ˆ U(τ)given by Eq. (D5). A straightforward calculation yields the final populations p0(τ) := h0|ˆρτ|0i=p0,(D13) p1(τ) := h1|ˆρτ|1i=p1cos2θ+p2sin2θ−cysin(2θ),(D14) p2(τ) := h2|ˆρτ|2i=p1sin2θ+p2cos2θ+cysin(2θ),(D15) while the final coherence between |1iand |2ireads ρ12(τ) := h1|ˆρτ|2i=cx+ icycos(2θ) + 1 2(p1−p2) sin(2θ).(D16) Two key features of Eqs. (D14)–(D16) are: • The diagonal part of the initial state ˆρdiag 0 = p0|0ih0| + p1|1ih1| + p2|2ih2| contributes to p1,2 ( τ ) through the usual classical mixing p1↔p2induced by the rotation in the {|1i,|2i} subspace. • The imaginary part cy = = ( c12 )of the initial coherence yields an additional term ±cysin (2 θ )in Eqs. (D14) – (D15) , which shifts probability between the positive and negative entropy sectors. This is the interference contribution. If the initial state is dephased in the {|0i,|1i,|2i} basis (e.g. by a TPM measurement at t = 0), then cy= 0 and the extra terms vanish. D.4. Entropy-production distribution and characteristic function The entropy-production distribution associated with the operator ˆ Σin Eq. (D6) is purely discrete: Pqm(Σ) = p0(τ)δ(Σ) + p1(τ)δ(Σ −σ) + p2(τ)δ(Σ + σ).(D17) The probability for the positive and negative sectors are Pqm(+) = p1(τ) = Pdiag(+) + ∆Pint(+),(D18) Pqm(−) = p2(τ) = Pdiag(−)+∆Pint(−),(D19) where we have defined Diagonal (classical) contributions: Pdiag(+) := p1cos2θ+p2sin2θ, (D20) Pdiag(−) := p1sin2θ+p2cos2θ, (D21) i.e. the probabilities obtained by evolving the dephased initial state ˆρdiag 0; Interference corrections: ∆Pint(+) := −cysin(2θ),(D22) ∆Pint(−) := +cysin(2θ).(D23) The corresponding interference quotients (cf. Sec. 4.4) are I+:= ∆Pint(+) Pdiag(+) , I−:= ∆Pint(−) Pdiag(−).(D24) The ratio of positive to negative entropy-sector probabilities can then be written in the universal form Pqm(+) Pqm(−)=Pdiag(+) Pdiag(−)·1 + I+ 1 + I− ,(D25)
133 exactly matching the “classical factor × interference factor” structure derived abstractly in Sec. 4.4. If the diagonal sector satisfies a classical FT, Pdiag(+) Pdiag(−)= eσ,(D26) then the full quantum ratio becomes ln Pqm(+) Pqm(−)=σ+ ln(1 + I+)−ln(1 + I−),(D27) providing an explicit illustration of the modified FT relation discussed in the main text. The characteristic function of ˆ Σin the state ˆρτis χ(u) := Treiuˆ Σˆρτ=p0(τ)eiu·0+p1(τ)eiuσ +p2(τ)e−iuσ.(D28) Substituting Eqs. (D13)–(D15) and separating diagonal and interference contributions, we obtain χ(u) = p0+p1cos2θ+p2sin2θeiuσ +p1sin2θ+p2cos2θe−iuσ | {z } χdiag(u) −cysin(2θ) eiuσ +cysin(2θ) e−iuσ | {z } ∆χint(u) .(D29) Thus χdiag(u) = Trheiuˆ Σˆ U(τ)ˆρdiag 0ˆ U†(τ)i,(D30) ∆χint(u) = Trheiuˆ Σˆ U(τ)ˆρ0−ˆρdiag 0ˆ U†(τ)i,(D31) and the entire effect of initial coherence on the entropy-production statistics is encapsulated in the simple interference factor sin(2θ)cy. D.5. Including simple dephasing (optional) To model the effect of decoherence on the interference terms in a minimal way, one may add pure dephasing in the {|1i,|2i} basis at rate γ: ˙ ˆρ=−i[ ˆ H, ˆρ] + γˆσzˆρˆσz−ˆρ,ˆσz:= |1ih1|−|2ih2|.(D32) This leaves populations invariant but attenuates coherences, c12 ( t ) = c12 (0) e −2γt in the interaction picture. Repeating the calculation above with cy→cye−2γτ gives ∆Pint(±;γ) = ±cye−2γτ sin(2θ),(D33) so that I± ( γ ) → 0as γτ → ∞ , and the classical FT is recovered. This mirrors the general decoherenceinduced suppression of interference factors discussed in Sec. 4.5. Appendix E: Numerical methods and parameter choices This appendix collects the numerical details required to reproduce the figures for (i) the chaotic model and (ii) the chemo–mechanical molecular machine of Sec. 9, together with representative parameter values and pseudocode-style summaries of the algorithms. The emphasis is on transparency: we spell out the state spaces, generators, integration schemes, and how entropy-production statistics and FT ratios are extracted.[179–181] Unless otherwise stated, we work in units where kBT = 1 and ~ = 1; energies and entropy productions are therefore dimensionless.
134 E.1 General numerical strategy Across all models we follow a common pattern: 1. Define the state space and dynamics. For the chaotic model: classical phase space ( q, p )on a torus and its quantum counterpart via a Floquet operator acting on a finiteN Hilbert space. For the chemo–mechanical machine: a finite Markov network or Liouvillian on a finite-dimensional Hilbert space. 2. Implement the dynamics numerically. For the chaotic model: symplectic maps and split-operator quantum propagation. For the chemo–mechanical machine: direct exponentiation of the generator or time-stepping of the master equation. 3. Attach a trajectory-level entropy production. Each allowed transition i→j is assigned a medium entropy increment ∆Σ i→j consistent with local detailed balance (for Markov dynamics) or with an operator ˆ Σ(for quantum). 4. Compute entropy-production statistics. Either by (a) direct sampling of trajectories (Gillespie-type algorithms)[182] or (b) full counting statistics via tilted generators and characteristic functions. 5. Extract FT ratios and interference corrections. For the chaotic model we focus on the FT log-ratio as a function of driving parameters. For the chemo–mechanical model we additionally compute the phaseand protocol-dependence of the mean entropy production and efficiency. We now describe the two main numerical backbones in more detail. E.2 Chaotic model: kicked rotor numerics For concreteness, we take the chaotic model of Sec. 7 to be the quantum kicked rotor in the mixedphase-space regime, which is standard in studies of quantum chaos and dynamical tunnelling.[183] E.2.1 Classical standard map and parameter regime The classical kicked rotor is defined on the phase-space torus ( q, p ) ∈ [0 , 2 π ) × [0 , 2 π )with stroboscopic map pn+1 =pn+Ksin qn(mod 2π),(E1) qn+1 =qn+pn+1 (mod 2π),(E2) where K is the kicking strength. For 0 < K . 1the phase space is mostly regular; for K& 2a large chaotic sea coexists with regular islands (mixed phase space), which is the regime we use to illustrate chaos-assisted tunnelling. In the numerics we choose values such as K∈ {2.5,3.0,3.5,4.0}.(E3) E.2.2 Quantum Floquet operator and Hilbert-space truncation The quantum kicked rotor is implemented as a Floquet unitary acting on an N -dimensional Hilbert space with basis {|ni} corresponding to discrete angular-momentum states.[ 183 ] The Floquet operator decomposes as ˆ UF= exp−i 2~eff ˆp2exp−iK ~eff cos ˆq,(E4) with effective Planck constant ~eff = 2π/N. We implement ˆ UFvia a split-operator FFT scheme: 1. Represent the wavefunction in the angle basis ψ ( q )on an N -point grid qj = 2 πj/N , j = 0 , . . . , N − 1.
135 2. Apply the “kick” in angle space: ψ(qj)←exp[−iKcos qj/~eff]ψ(qj). 3. FFT to momentum space, apply the free-rotation phase exp[−ip2 n/(2~eff)], then inverse FFT. We typically use: Symbol Description Typical value NHilbert-space dimension N= 256,512 Kkicking strength 2.5–4.0 nsteps kicks per trajectory 100–300 Mnumber of trajectories / initial states 103–104 Table I. Representative parameters for the kicked-rotor numerics. E.2.3 Coarse-grained entropy production and FT statistics In Sec. 7 we define a coarse-grained entropy production associated with motion between two disjoint phase-space regions RAand RBthat play the role of “forward” and “backward” basins. Operationally: • We choose two angular sectors of width ∆ q centred at angles qA and qB with qB = qA + π , and moderate strips in momentum. • For each quantum trajectory (initial state, evolved for nsteps kicks), we monitor the probability flow between these regions. • We assign a coarse-grained entropy increment ∆Σ = Σ 0 when probability flows predominantly A→B , and − Σ 0 for B→A . The prefactor Σ 0 is fixed by the classical steady-state affinity (e.g. torque plus reservoir bias) entering the corresponding classical Markov model. Numerically we compute Pqm(±) = 1 M M X k=1 Trhˆ Π±ˆ Unsteps Fˆρ(k) 0ˆ Unsteps F†i,(E5) where ˆ Π± project onto the coarse-grained sectors associated with positive/negative entropy production (constructed in direct analogy with the operator projectors of Sec. 4), and ˆρ(k) 0are a set of initial states localised on classical tori inside a regular island. The FT log-ratio plotted in Sec. 7 is then ln Pqm(+) Pqm(−)= Σ0+ ln(1 + I+)−ln(1 + I−),(E6) with interference quotients I± extracted by comparing Pqm ( ± )with the classical probabilities obtained from the corresponding Markov discretisation of the standard map.[184] In practice, to avoid statistical noise in the tails, we average over many initial conditions and, when convenient, over several values of nsteps within a fixed time window so that the asymptotic large-deviation regime of the classical FT is approached. E.3 Chemo–mechanical molecular machine numerics We now describe the numerical implementation of the minimal chemo–mechanical molecular machine of Sec. 9. The model is inspired by ATP synthase and similar rotary motors, but is deliberately coarse-grained and dimensionless to retain analytic control and numerical stability.[185, 186]
136 E.3.1 State space and rotor discretisation We discretise the rotor angle into Nθ= 12 equally spaced positions θj=2π Nθ j, j = 0, . . . , Nθ−1.(E7) At each angle we allow three chemical states: • |B, ji: binding-ready site (empty catalytic site). • |S, ji: substrate-bound or synthesis intermediate. • |R, ji: product-ready / release state. The full state space thus has 3 Nθ = 36 states indexed by a composite index i = ( X, j )with X∈ {B, S, R} and j∈ {0, . . . , Nθ−1}. We treat the model as a classical continuous-time Markov jump process with generator W of dimension 36 ×36; the state probabilities form a vector p(t)obeying d dtp(t) = Wp(t).(E8) E.3.2 Thermodynamically consistent rates Each allowed transition i→k is associated with a free-energy change ∆ Gi→k for the environment (chemical or mechanical work), and we impose local detailed balance Wk←i Wi←k = exp+∆Gi→k.(E9) We distinguish three classes of transitions: Chemical transitions at fixed angle. •Binding/unbinding: (B, j)(S, j), with free-energy change ∆GB→S= ∆µATP −∆EBS. •Synthesis/hydrolysis: (S, j)(R, j),∆GS→R=−∆µATP −∆ESR. •Release/binding of products: (R, j)(B, j),∆GR→B= ∆µADP+Pi−∆ERB. Here ∆ µATP and ∆ µADP+Pi are chemical potentials (in units of kBT ) of ATP and products, and ∆ EXY are internal energy differences between chemical states X and Y . For each such step we choose symmetric Eyring-like rates Wk←i=k0exp+∆Gi→k/2, Wi←k=k0exp−∆Gi→k/2,(E10) which automatically satisfy Eq. (E9). Mechanical stepping of the rotor. Mechanical transitions correspond to forward and backward steps of ∆θ= 2π/Nθ: (X, j)(X, j + 1), j ≡j+Nθ.(E11) The free-energy change combines the proton motive force (PMF) and external load torque τload, ∆G(X,j)→(X,j+1) =−∆µH+ Nθ−τload∆θ+ ∆EX(θj+1)−∆EX(θj),(E12) with ∆ µH+ the PMF in units of kBT , and ∆ EX ( θ )an angle-dependent internal potential which can be used to impose a preferred binding angle. Again we use symmetric rates W(X,j+1)←(X,j)=krot exp+∆G(X,j)→(X,j+1)/2,(E13) and the backward rate fixed by detailed balance.
137 Entropy production increments. Each transition carries a medium entropy-production increment ∆Σi→k= ∆Gi→k,(E14) consistent with the standard identification of entropy production with heat (or chemical work) in units of kBT.[186] The total entropy production along a trajectory is then Σ[Γ] = X jumps i→k∈Γ ∆Σi→k.(E15) Representative parameter choices are summarised in Table II. Symbol Description Typical value (dimensionless) Nθangular discretisation 12 ∆µATP ATP chemical potential 20 ∆µADP+Piproduct potential 0 ∆µH+PMF across membrane 8 τload dimensionless torque 1–4 k0chemical attempt frequency 1 krot mechanical attempt frequency 5 ∆EBS ,∆ESR,∆ERB chemical offsets O(1) Table II. Representative parameters for the chemo–mechanical molecular machine. E.3.3 Time integration and entropy-production statistics The master equation is integrated either by • Direct matrix exponentiation: p ( t ) = exp ( Wt ) p0 , using diagonalisation or Padé approximation with scaling and squaring,[179] • or an adaptive Runge–Kutta method (e.g. Dormand–Prince 4/5), ensuring norm conservation to ∼10−10.[180] To obtain the full entropy-production statistics and FT ratios we use full counting statistics via a tilted generator W(λ),[187] Wk←i(λ) = Wk←iexpλ∆Σi→k, i 6=k, Wi←i(λ) = −X k6=i Wk←i(λ).(E16) The moment-generating function for entropy production is Z(λ, t) = X Γ P[Γ] expλΣ[Γ]=1TexpW(λ)tp0,(E17) where 1 is the vector with all entries equal to 1, and p0 the initial probability vector. The characteristic function used in Sec. 9 is χ(u, t) = Z(iu, t). The probability distribution P (Σ , t )can be recovered by numerical inverse Fourier transform of χ ( u, t ), or—more simply for the two-sector observables we focus on—by computing Pcl(+) = Pr[Σ ≥0], Pcl(−) = Pr[Σ <0],(E18) via discretisation of the Σaxis and binning. For the quantum version of the chemo–mechanical model discussed in Sec. 9, the same structure is implemented with a Lindbladian L instead of W , using a vectorised density-matrix representation and the quantum-trajectory or Liouville-space full counting statistics machinery available in standard libraries.[181] E.4 Pseudocode for figure reproduction We conclude with high-level pseudocode for the two main figure families: FT ratios for the chaotic model and FT/efficiency plots for the chemo–mechanical machine. The actual implementations use standard linear-algebra and FFT libraries (e.g. NumPy/SciPy, FFTW, or QuTiP).
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