Quantum superposition of thermodynamic evolutions with opposing time’s arrows

ARTICLE Quantum superposition of thermodynamic evolutions with opposing time’s arrows Giulia Rubino1,2✉, Gonzalo Manzano 3,4 &Časlav Brukner1,4 Microscopic physical laws are time-symmetric, hence, a priori there exists no preferential temporal direction. However, the second law of thermodynamics allows one to associate the “forward”temporal direction to a positive variation of the total entropy produced in a thermodynamic process, and a negative variation with its “time-reversal”counterpart. This definition of a temporal axis is normally considered to apply in both classical and quantum contexts. Yet, quantum physics admits also superpositions between forward and timereversal processes, whereby the thermodynamic arrow of time becomes quantummechanically undefined. In this work, we demonstrate that a definite thermodynamic time’s arrow can be restored by a quantum measurement of entropy production, which effectively projects such superpositions onto the forward (time-reversal) time-direction when large positive (negative) values are measured. Finally, for small values (of the order of plus or minus one), the amplitudes of forward and time-reversal processes can interfere, giving rise to entropy-production distributions featuring a more or less reversible process than either of the two components individually, or any classical mixture thereof. https://doi.org/10.1038/s42005-021-00759-1 OPEN 1Vienna Center for Quantum Science and Technology (VCQ), Faculty of Physics, University of Vienna, Boltzmanngasse 5, Vienna 1090, Austria. 2Quantum Engineering Technology Labs, H. H. Wills Physics Laboratory and Department of Electrical & Electronic Engineering, University of Bristol, Bristol BS8 1FD, UK. 3Institute for Cross-Disciplinary Physics and Complex Systems (IFISC) UIB-CSIC, Campus Universitat Illes Balears, Palma de Mallorca E-07122, Spain. 4Institute for Quantum Optics and Quantum Information (IQOQI), Austrian Academy of Sciences, Boltzmanngasse 3, Vienna 1090, Austria. ✉email: [email protected] COMMUNICATIONS PHYSICS | (2021) 4:251 | https://doi.org/10.1038/s42005-021-00759-1 | www.nature.com/commsphys 1 1234567890():,; In spite of it being seemingly straightforward, physics is still nowadays seeking to provide a comprehensive understanding of the apparent passage of time1. The concept of time flow is intimately related to the observation of a change in physical systems. However, the recognition that, at their most fundamental level, physical systems generally obey time-reversible laws led to the realisation that systems’evolutions do not intrinsically differentiate between forward and backward time directions. Attempts to uphold with physical arguments the evidence of the time flow are being made on multiple fronts, mainly on the basis of empirical observations: we see that entropy in the universe increases (thermodynamic time’s arrow), that the universe expands (cosmological time’s arrow), that causes always precede their effects (causal time’s arrow). Likewise, there have been several proposals as to the explanation of the time’s arrow in a quantum-mechanical contexts2–6. The peculiarity of the quantum framework is that it enables processes to be placed in quantum superposition. Applied to the notion of thermodynamic time’s arrow, this implies that quantum mechanics can allow the superposition of thermodynamic processes (namely, dynamic processes wherein a system of interest exchanges either heat, work, or both with other systems, the environment and/or external agents) producing opposite variations in the entropy. This raises the question of how a well-defined thermodynamic arrow of time can be established in the quantum framework when such superpositions are in place. To address this question, in this work we show that a measurement of the entropy production has a decisive role in restoring a definite thermodynamic time’s arrow, and we investigate interference effects in such superpositions. Our investigations bear a conceptual similarity to the field of indefinite quantum causality, wherein the order of operations is placed in a quantum superposition7–9. Note, however, that there is a crucial difference between these two types of studies. In indefinite quantum causality, operations are performed in the same temporal direction (here referred to as “forward”)in each amplitude of the superposition. In contrast, in the present case, we analyse superpositions of thermodynamic processes with opposing thermodynamic arrows of time. In thermodynamics, the time’s arrow is introduced by the second law of thermodynamics, according to which the total entropy of the universe can only either increase or remain constant. Consequently, one might think that observations of entropy changes are all we need to distinguish the past from the future: an overall increase in entropy shall be identified with the direction of time “forward”, while an overall decrease in entropy with its “time-reversal”counterpart. Yet, for a microscopic system, fluctuations blur the direction of the time’s arrow, and the time flow is only defined on average. More specifically, in this regime, the time’s arrow cannot be inferred, as both positive and negative entropy changes can be observed with comparable probability in a single experimental run. As a consequence, for such systems, the two opposite time’s arrows become classically indistinguishable. The extension of this indistinguishability to the quantum domain gives rise to quantum superpositions between opposite time’s arrows, whose investigation is the focus of the present work. In what follows, we will explore how a definite time’s arrow arises in quantum superpositions between “forward”and “timereversal”processes (i.e., thermodynamic processes whose quenches are related by time-inversion symmetry). In particular, we will show that quantum measurements of the dissipative work W diss (or, equivalently, entropy production ΔS tot ) can restore the time directionality of the process. The dissipative work W diss = W−ΔFis the amount of work Winvested in a thermodynamic transformation between equilibrium states having a free energy difference ΔF, which cannot be recovered by reversing the process. Furthermore, the relation between the dissipative work and the entropy production (or total entropy) ΔS tot in the process is established through the relation: ΔS tot =βW diss , where β¼ ðkBTÞ1is the inverse temperature, with k B being the Boltzmann constant and Tthe temperature of the bath10–12. We will then show that, when the measured dissipative work equals βW diss ≫ 1, the superposition is effectively projected onto the forward process, whereas when βW diss ≪−1, it is effectively projected onto the time-reversal one, hence recovering a definite thermodynamic arrow of time (albeit, in each individual execution of the experiment, the outcome “forward" or “time reversal" is random). Conversely, when β∣W diss ∣is of the order of one, the forward and the time-reversal thermodynamic processes can quantum mechanically interfere under certain conditions, resulting in a work probability distribution describing work fluctuations that have no classical counterpart. More precisely, in the case of interference, the probabilities take on values that cannot be obtained by any classical (convex) mixture of the forward and the time-reversal processes. Results Superposition of forward and time-reversal dynamics. We start by defining the framework used to characterise thermodynamic processes and work fluctuations. First, we will introduce all the necessary elements to formally construct a state representing the quantum superposition of a thermodynamic process evolving in the forward temporal direction, and one evolving in the opposite (time-reversal) direction. Then, we will discuss how to characterise work and entropy-production fluctuations in such superposition states using an extended two-point-measurement (TPM) scheme, and we illustrate how the outcomes achieved through processes with well-defined time directions can be recovered inside our framework. We consider a thermodynamic system Sbeing, in both forward and time-reversal processes, initially in equilibrium with a thermal reservoir at inverse temperature β. The process occurring in the forward direction will be realised by a quench U(t,0) induced by the time-dependent Hamiltonian HλðtÞðÞexecuting a controlled protocol Λ≡{λ(t); 0 ≤t≤τ} in the time-frame t∈[0, τ], followed by a final thermalisation in contact with the reservoir at β. Here, Uðt1;t2Þ¼~ Texp i _Rt2 t1dνHλðνÞðÞ hi , where ~ Tis the so-called “time-ordering”operator resulting from the Dyson decomposition. Its time-reversal twin will be described by a quench ~ Uðτt;0Þassociated to the implementation of the operational time-reversal protocol ~ ΛfλðτtÞ;0≤t≤τg, where in both cases λis a control parameter, and again the quench is followed by a final thermalisation step. The micro-reversibility principle for non-autonomous systems establishes a strong relation between forward and time-reversal quenches lying at the core of fluctuation theorems13,14: ~ Uðτt;0Þ¼ΘUyðτ;tÞΘy;ð1Þ where Θdenotes the (anti-unitary) time-reversal operator acting on the system’s Hilbert space, which flips the sign of observables with odd parity under time-reversal. This operator verifies the relations Θ1i¼1iΘ, and ΘΘ y¼ΘyΘ¼1. In order to describe superpositions of forward and timereversal processes, the initial equilibrium states of the system S can be purified by including some environmental degrees of freedom Ewith a generic Hamiltonian H E in the description. These purifications are not unique, and they can be represented ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00759-1 2COMMUNICATIONS PHYSICS | (2021) 4:251 | https://doi.org/10.1038/s42005-021-00759-1 | www.nature.com/commsphys by joint states of the system and the environment of the form ψ0 S;E¼∑ kffiffiffiffiffiffiffiffiffiffiffiffi eβEð0Þ k Z0 sEð0Þ k ESεð0Þ k EE;ð2aÞ ~ ψ0 S;E¼∑ kffiffiffiffiffiffiffiffiffiffiffiffi eβEðτÞ k Zτ sΘEðτÞ k ESεðτÞ k EE;ð2bÞ where Eð0Þ kand EðτÞ kare the eigenvalues of the Hamiltonian at times t={0, τ}, i.e., H[λ(0)] and H[λ(τ)], whereas jEð0Þ kiSand jEðτÞ kiSare the corresponding eigenvectors (for the sake of brevity, we will henceforth omit the subscript Sin the system’s energy eigenvectors). Furthermore, jεð0Þ kiE,jεðτÞ kiErepresent the corresponding sets of states of the environmental degree of freedom, which can always be chosen as sets of orthogonal states. Notice that the environment may possess further degrees of freedom that are not entangled with the system under consideration, and which we will thus not explicitly account for. The state jψ0iS;Eabove corresponds to the initial state of the process evolving in the forward direction as defined by Λ,whereas j~ ψ0iS;Eis the initial state of the time-reversed process as defined by ~ Λ. Notice that, by tracing out the environmental degrees of freedom, we recover the corresponding Gibbs thermal states for the system ρth 0TrEðjψ0ihψ0jS;EÞ¼eβH½λð0Þ=Z0and ~ ρth 0 TrEðj~ ψ0ih~ ψ0jS;EÞ¼ΘeβH½λðτÞΘy=Zτ,beingZ0¼Tr eβH½λð0Þ  , and Zτ¼Tr eβH½λðτÞ  the partition functions. Moreover, we introduce an auxiliary system Awhose two orthogonal states f0 ji A;1 ji Aggovern the evolution of the process in the two temporal directions. This is a quantum analogue of the coin tossed to decide classically which process to run (forward or time reversal). With this in place, the global Hamiltonian of the system, the environment, and the auxiliary qubit reads HðtÞ 0 ji0 hj AH½λðtÞþ 1 ji1 hj AΘH½λðτtÞΘy  1Eþ1S;A HE. We then entangle each orthogonal auxiliary state to one of the initial states in Eq. (2). The overall initial state of thermodynamic system, environment and auxiliary system reads therefore: Ψ0 S;E;A¼α0ψ0 S;E0 ji Aþα1~ ψ0 S;E1 ji A;ð3Þ with arbitrary coefficients α0;α12C,∣α 0 ∣2+∣α 1 ∣2=1. If, subsequently, in each branch of the superposition in Eq. (3)theforward and time-reversal quenches are respectively applied, the evolved state at some arbitrary instant of time t∈[0, τ] is given by jΨðtÞiS;E;A¼ α0½Uðt;0Þ1E;Ajψ0iS;Ej0iAþα1j½~ Uðt;0Þ1E;Aj~ ψ0iS;Ej1iA: In this expression, the first and the second amplitudes correspond to the forward and the time-reversal processes, respectively. Furthermore, we assume that the system does not interact with the environment during the timescale of the quenches (however, after the quench, the system thermalises through the interaction with the thermal reservoir). This is verified whenever the quenches are implemented in a fast timescale as compared to the characteristic relaxation time of the system in interaction with the environment15, or when the system is artificially disconnected from the environment during the quench implementation and reconnected after it. Furthermore, we will consider the quenches U(t,0) and ~ Uðt;0Þin the superposition to be implemented by some external (classical) control. As we discuss in the Supplementary Note 116–20, this limit is adequate in our setup, and it corresponds to the case in which the control mechanism acts approximately as an ideal reservoir of energy and coherence21–24, as is the case, for instance, with lasers or radiofrequency pulses. Taking a gas enclosed in a vessel as a pictorial example, the aforementioned state can be constructed by entangling the position of the piston with a further auxiliary quantum system, thereby establishing a quantum superposition of the following two processes: (i) a process wherein the gas particles are initially in thermal equilibrium confined in one half of the vessel by a piston, and the piston is pulled outwards, and (ii) the reverse process, in which the piston is pushed towards the gas, starting from an initial state where the gas occupies the entire vessel in thermal equilibrium. Extended two-point measurement scheme. We will now measure the work of the system undergoing the above-mentioned superposition of forward and time-reversal dynamics. In order to implement such a measurement, we formally construct a procedure described by a set of measurement operators forming a completely positive and trace-preserving (CPTP) map. In this regard, we will refer to a standard TPM procedure to measure work in quantum thermodynamic processes13. Implementations of the TPM in quantum setups25–29, as well as suitable extensions30–33, have recently received increasing attention. Our procedure can be seen as a generalisation of the TPM scheme to situations where different thermodynamic processes are allowed to be superposed, and can consequently interfere. In the TPM scheme, work is defined as the energy difference between the initial and final states of the system, which are measured through ideal projective measurements of the system Hamiltonian implemented before and after the thermodynamic process associated with the protocol Λ34,35. This measurement scheme can be performed, individually, both for the forward and the time-reversal processes, enabling the construction of the work probability distributions P(W) and ~ PðWÞ, respectively. As far as the forward process is concerned, the probability to observe a transition Eð0Þ n !EðτÞ m is given by pn;m¼pmjnpð0Þ n, where pð0Þ n¼eβEð0Þ n=Z0is the probability of observing the energy Eð0Þ nat t=0, and pmjn¼hEðτÞ mjUðτ;0ÞjEð0Þ ni  2is the conditional probability of measuring EðτÞ mat t=τafter having measured Eð0Þ n at the beginning of the process. Similarly, for the time-reversal process one has ~ pm;n¼~ pnjm~ pð0Þ m, where ~ pð0Þ m¼eβEðτÞ m=Zτis the probability to obtain the energy EðτÞ mat the beginning of the timereversal process, and ~ pnjm¼hEð0Þ njΘy~ Uðτ;0ÞΘjEðτÞ mi  2is the corresponding conditional probability for observing the inverse transition ΘEðτÞ m !ΘEð0Þ n given that one obtained EðτÞ min the first measurement. The micro-reversibility principle in Eq. (1) relates the conditional probabilities in the forward and timereversal processes as ~ pnjm¼pmjn13,14. The TPM scheme allows one to compute the stochastic work invested by the external driver in a single realisation of the protocol Λ,Wn;mEðτÞ mEð0Þ n, associated to the outcomes of initial and final energy measurements. Its probability distribution reads: PðWÞ¼∑ n;mpn;mδðWWn;mÞ:ð4Þ Analogously, the probability distribution associated to the work invested in the time-reversal protocol, ~ Wn;m¼Eð0Þ nEðτÞ m¼ Wn;m,isgivenby: ~ PðWÞ¼∑ n;m ~ pn;mδðW~ Wn;mÞ:ð5Þ Hereafter, we consider an extension of the TPM scheme in which we include energy measurements at t=0 and t=τin both COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00759-1 ARTICLE COMMUNICATIONS PHYSICS | (2021) 4:251 | https://doi.org/10.1038/s42005-021-00759-1 | www.nature.com/commsphys 3 branches of the superposition between a forward and a timereversal processes, as illustrated in Fig. 1. More precisely, starting with the initial state in Eq. (3), and conditionally on the auxiliary state, we consider the application of the projectors Eð0Þ n Eð0Þ n  and ΘEðτÞ m EðτÞ m Θyto the initial states ψ0 S;Eand ~ ψ0 S;E, respectively. Subsequently, the unitary quenches U(τ, 0) and ~ Uðτ;0Þare implemented in each branch, after which the projectors EðτÞ m EðτÞ m and ΘEð0Þ n Eð0Þ n Θyare respectively applied. Consequently, given the outcomes Eð0Þ nand EðτÞ m, a work W n,m is invested in the forward-dynamics branch by applying the protocol Λ, whereas the work invested in its time-reversal counterpart ~ Λis ~ Wn;m¼Wn;m(i.e., the same amount of work as in the forward dynamics is here extracted). The operator representing the application of the scheme through which the work W n,m is obtained can be written as: Mn;m¼EðτÞ m EðτÞ m Uðτ;0ÞEð0Þ n Eð0Þ n 1E0 ji0 hj A þΘEð0Þ n Eð0Þ n Θy~ Uðτ;0ÞΘEðτÞ m EðτÞ m Θy1E1ji1hj Að6Þ The set of operators {M n,m }formsaCPTPmap,EðρÞ ∑n;mMn;mρMy n;m, acting on the composite system S,E,Aand fulfilling ∑n;mMy n;mMn;m¼1S;E;A.ThemapEdescribes the average effect of the measurement scheme on an arbitrary initial state of the composite system ρ, while the operations En;mðρÞMn;mρMy n;m provide the probability PðWÞ∑n;mTr ½En;mðρÞδðWWn;mÞto measure the work W. It is important to stress that the operations EWpreserve the coherence between the forward and time-reversal thermodynamic processes. Indeed, performing a standard quantum measurement on the process would destroy the coherence, as it would reveal the time at which the measurement has been performed, and, from this, also whether the outcome E m was observed before (in the forward process) or after the outcome E n (in the time-reversal process). In other words, such a measurement would reveal the time direction, and it would be equivalent to the measurement of the auxiliary qubit in the basis f0 ji A;1 ji Ag. However, there exist also measurement schemes in which the result is encoded in an auxiliary system through its entanglement with the measured system, and the result is then read-only at the end of the whole evolution, thereby preserving its coherence. (Such a measurement scheme was recently used to measure the system undergoing superposition of causal orders36) In such a scheme, the system on which the thermodynamic quenches act and the auxiliary system can be encoded on two different degrees of freedom of the same quantum system. If the auxiliary degree of freedom is of sufficient dimension, it is possible to encode the results of each measurement taking place within the process in a state of this system. More precisely, suppose that the auxiliary system has two additional registers A0which can store the results of the two energy measurements. When the auxiliary system is in the 0 ji A (1 ji A) state, the thermodynamic system is subject to an unitary U1 (~ U1) that couples the energy of the system to the first (second) register of the auxiliary system. This results in an overall unitary that entangles the thermodynamic system with the auxiliary system: 0 ji A0 hj U1þ1 ji A1 hj ~ U1;ð7Þ where U1Eð0Þ n Sx;y A0¼Eð0Þ n Sxn;y A0;ð8Þ ~ U1ΘEðτÞ m Sx;y A0¼ΘEðτÞ m Sx;ym A0;ð9Þ for any basis state x;y A0of the two registers. Here, the symbol ⊕means the sum modulo the total number of different energy values. Subsequently, the thermodynamic system is subject to a quench, followed by another entangling unitary 0ji A0hjU2þ1ji A1hj~ U2;ð10Þ with U2EðτÞ m Sx;y A0¼EðτÞ m Sx;ym A0;ð11Þ ~ U2ΘEð0Þ n Sx;y A0¼ΘEð0Þ n Sxn;y A0;ð12Þ which now couples the energy of the thermodynamic system after the quench in the second (first) register when the auxiliary system is in the state 0 ji A(1 ji A). If the two registers are initially prepared in the state 0;0 ji A0, their final state n;m ji A0will encode both energy values. The coherence of the overall state has to be maintained until the end of the entire thermodynamic process when the auxiliary system is measured in the basis fð 0 ji A±1 ji AÞ=ffiffiffi 2 pgto erase any information as to whether the system has gone through the “forward”or “time-reversal”process (which might be encoded, for instance, in the temporal or directional mode of the auxiliary system). A sketch of a possible experimental realisation of the extended TPM scheme is shown in Fig. 2in the case of two measurements outcomes for Eð0Þ n,EðτÞ m. For simplicity, in this study we consider only two states of the auxiliary system (Fig. 1). Nevertheless, all the conclusions drawn herein can be extended to the case of more than two states. In order to evaluate the work probability distribution in the extended TMP scheme, it is also crucial to take into account the mutual phases between the conditional probabilities. We thus write, in general hEðτÞ mjUðτ;0ÞjEð0Þ ni:¼ffiffiffiffiffiffiffiffi pmjn peiΦn;m;ð13aÞ hEð0Þ njUyðτ;0ÞjEðτÞ mi:¼ffiffiffiffiffiffiffiffi ~ pnjm qei~ Φm;n;ð13bÞ Fig. 1 Superposition of a forward thermodynamic quench with its timereversal counterpart. A thermodynamic system Sand its environment are coupled to an auxiliary system Ain a suitable entangled state. Depending on the state of the auxiliary system, 0 ji Aor 1 ji A, when the state of the environment is traced out, the system Sis initially prepared in a thermal state of the initial or final Hamiltonians, H(0) and H(τ), respectively. This is then sent through a thermodynamic quench U(t, 0) or its time reversal ~ Uðt;0Þin the time-frame t∈[0, τ]. Before and after each quench, the system’s energy is measured. The measurement outcomes Eð0Þ nand EðτÞ mare found when the auxiliary system is in 0 ji A, whereas the outcomes Eð0Þ mand EðτÞ nare obtained when the auxiliary system is in 1 ji A. After these measurements, the system may eventually undergo a second thermalisation with the environment. Note that the first (second) measurement when the auxiliary system is in 0 ji A, and the second (first) measurement when it is in 1 ji Aare physically one and the same measurement. A possible implementation of this scheme is reported in Fig. 2. ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00759-1 4COMMUNICATIONS PHYSICS | (2021) 4:251 | https://doi.org/10.1038/s42005-021-00759-1 | www.nature.com/commsphys and we notice that ffiffiffiffiffiffiffiffi ~ pnjm qei~ Φm;n¼DEð0Þ njUyðτ;0ÞjEðτÞ mE ¼ffiffiffiffiffiffiffiffi pmjn peiΦn;m  ¼ffiffiffiffiffiffiffiffi pmjn peiΦn;m; from which we get Φn;m¼~ Φm;n, since ~ pnjm¼pmjn. We now consider the concatenation of the operation M n,m with a projection of the auxiliary qubit onto an arbitrary state ξ ji A.By applying this sequence of operations to the initial state in Eq. (3), we derive the (unnormalized) state of the composite system associated to the work outcome W n,m and projection of the auxiliary qubit onto ξ ji A: Ψξ n;m ES;E;A1S;Eξ jiξ hj A  Mn;mΨ0 S;E;A ¼jΞξ 0iþjΞξ 1i;ð14Þ where we identified the two branches of the superposition corresponding to the forward ðjΞξ 0iÞ and the time-reversal dynamics ðjΞξ 1iÞ. They read, respectively: jΞξ 0i¼α0hξj0iffiffiffiffiffiffiffiffi pn;m peiΦn;mEðτÞ m εð0Þ n Eξ ji Að15aÞ jΞξ 1i¼α1hξj1iffiffiffiffiffiffiffiffi pn;m peβ 2ðWn;mΔFÞiΦn;m ΘEð0Þ n εðτÞ m Eξ ji Að15bÞ where, in the second equation, we made use of ~ pn;m¼ pn;meβðWn;mΔFÞ(see the Supplementary Note 2), and of the relation between the forward and time-reversal phases ~ Φm;n¼Φn;m. The final thermalisation step, which effectively leads to the irreversible dissipation of work W diss , occurs only after the projection onto the auxiliary qubit, and is thus not included within the (extended) TPM scheme. The joint probability of measuring the work Wand projecting the auxiliary state onto ξ ji Ais given by Pðξ;WÞ¼∑n;mjjjΨξ n;miS;E;Ajj2 δðWWn;mÞ.Furthermore,fromthejointprobabilitiesPðξ;WÞ, one can obtain the conditional ones PξðWÞ:¼PðWjξÞ¼ Pðξ;WÞ=PðξÞ, which we will hereafter refer to as “post-selected work probability distributions”,andwherePðξÞ¼RdW Pðξ;WÞ. By introducing the notation qξ 0¼jα0j2jhξj0ij2=PðξÞand qξ 1¼ jα1j2jhξj1ij2=PðξÞ,wecanrewritePξðWÞas: PξðWÞ¼qξ 0PðWÞþqξ 1~ PðWÞþ2Re IξðWÞ  ;ð16Þ whereweidentified the probability distributions for the work in the forward process P(W), and in the time-reversal one ~ PðWÞas given in Eqs. (4)–(5), respectively. From this, we obtain the interference term: IξðWÞ¼α 0α1h0jξihξj1i PðξÞ∑ n;mpn;meβ 2ðWn;mΔFÞ e2iΦn;mEðτÞ m ΘEð0Þ n εð0Þ njεðτÞ m  δðWWn;mÞð17Þ The functional dependence of PξðWÞon Wconsists of two parts: (i) an “incoherent”part, reflecting the fact that each work value W obtained in the scheme is compatible with running the process in one or the other temporal direction with a given probability (i.e., investing the work Wwhen running the protocol Λ, and extracting the same amount of work −Wwhen executing its time-reversal counterpart ~ Λ), and (ii) a “coherent”part, which is a genuinely quantum feature arising from the superposition of the two temporal directions of the quench. In the case jα0j¼jα1j¼1=ffiffiffi 2 p, the forward state jΞξ 0iand the time-reversal one jΞξ 1iin Eq. (15) have the same amplitudes in the superposition. Nevertheless, as in the standard scenario of welldefined temporal directions, one may use the properties of the work probability distribution PξðWÞtogether with Bayesian reasoning to infer the time’s arrow of the thermodynamic process. As we will see shortly, in some cases, the thermodynamic time’s arrow can be determined even in a single realisation of the process, which effectively projects the state jΨξ n;miS;E;Aonto either its forward or its time-reversal component. Fig. 2 Possible implementation of the extended two-point measurement scheme. A beam splitter (BS) creates a quantum superposition of the auxiliary state in j0iA,j1iAas represented by the upper (solid) and lower (dashed) paths in the left part of the figure, over which the initial state in Eq. (3) is prepared. Unitary operators U1;2and ~ U1;2couple the system with two additional internal registers A0, initially prepared in the state 0;0 ji A0 (for simplicity, unitaries U1;2and ~ U1;2are here imagined to produce binary results). Encoding pairs of system energy eigenstates (n,m) onto the registers A0leads to further subdivisions into different paths (middle part of the figure), which are recombined and measured only at the final stage of the interferometer. The unitaries U1;2and ~ U1;2and the final measurement replace the initial and final projective measurements of Eð0Þ nand EðτÞ min the TPM scheme in Fig. 1. When the auxiliary system is in the state j0iA(j1iA), the thermodynamic system is first subjected to a unitary U1(~ U1), then to the thermodynamic process U(τ,0)[~ Uðτ;0Þ] within the time interval [0, τ], and finally to a second unitary U2(~ U2), see solid (dashed) paths in the figure. Unitary U1encodes the energy of the eigenstates Eð0Þ n of the thermodynamic system into the first register n;0ji A0(n=0, 1) of the auxiliary system, while unitary U2encodes the energy of the eigenstates EðτÞ m into the states n;mji A0(m=0, 1) of the second register. The four possible outcomes are indicated as four solid paths (bottom part of the figure), each labelled as j0iAjn;miA0. Similarly, the unitaries ~ U1and ~ U2 encode the energies of the thermodynamic system before and after the quench in the second and the first register respectively, when the auxiliary system is in 1 ji A(four dashed paths labelled as 1 ji Am;n ji A0in the top part of the figure). This enables to maintain the coherence of the auxiliary system’s states until the end of the interferometer. There, the states 0 ji An;m ji A0and 1 jim;n ji A0are interfered pairwise through further BSs, and measured. The results of the final measurements over the system Ain the diagonal basis f± ji Agare indicated by the symbols E± n;m. COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00759-1 ARTICLE COMMUNICATIONS PHYSICS | (2021) 4:251 | https://doi.org/10.1038/s42005-021-00759-1 | www.nature.com/commsphys 5 Effective projection onto a definite time’s arrow. In the following, we demonstrate that measuring work values such that W−ΔF≫β−1,orW−ΔF≪−β−1, in single realisations of the extended TPM scheme effectively results in projecting the state jΨξ m;niS;E;Ain Eq. (14) onto either the forward or the time-reversal components in Eq. (15) (i.e., jΞξ 0ior jΞξ 1i, respectively). In order to show this, we consider the probabilities for the superposition state jΨξ n;miS;E;Ato be found in either jjjΞξ 0ijj2or jjjΞξ 1ijj2, respectively. In particular, we notice that the term jjjΞξ 1ijj2is upper bounded by jjjΞξ 1ijj2¼jα1j2jhξj1ij2pn;meβðWn;mΔFÞ ⩽eβWdiss ∑ n;mpn;m¼eβWdiss ;ð18Þ where we used the fact that ∣α 1 ∣2∣〈ξ∣1〉∣2⩽1, and ∑ n,m p n,m =1. Consequently, in the limit βW diss ≫1, we have jjjΞξ 1ijj20, and hence jjjΞξ 0ijj21, that is, jΨξ n;miS;E;A’jΞξ 0i. Indeed, applying the detailed fluctuation theorem in Eq. (21) to Eq. (16), we obtain: PξðWÞ¼PðWÞqξ 0þqξ 1eβWdiss  þ2Re IξðWÞ  qξ 0PðWÞ;ð19Þ where we made use of the fact that IξðWÞ/eβWdiss=2. Therefore, we obtained that, whenever one performs a measurement of the work in the extended TPM scheme and observes W−ΔF≫β−1 (or, equivalently, ΔS=βW diss ≫1), the state of the system is projected onto the forward component of the quantum superposition without measuring the auxiliary qubit (similarly to what one would obtain, had one projected the joint state ΨðtÞ S;E;A through a projective measurement 0 ji0 hj Aon the auxiliary system, and subsequently observed the work value W). The probability to observe this work value in the extended TPM scheme is given by Eq. (19). Analogously, whenever the result of the extended TPM scheme is such that W−ΔF≪−β−1(or, equivalently, ΔS=βW diss ≪−1), one can neglect the term jjjΞξ 0ijj2≤eβWdiss , and thus obtain the projection jΨξ WiS;E’jΞξ 1i. In this case, we correspondingly achieve: PξðWÞ¼~ PðWÞqξ 0eβWdiss þqξ 1  þ2Re IξðWÞ  qξ 1~ PðWÞ:ð20Þ Hence, here the joint state is projected onto the time-reversal component of the quantum superposition (as if a projective measurement 1 ji1 hj Aon the auxiliary system was performed, followed by the observation of the work value W).Similarlytothe previous case, Eq. (20) provides the probability to get such an outcomeinanestimationofthework. Interference effects in the work distribution. In the previous section, we observed that, for individual runs of the process’ superposition, whenever the observed entropy production is of the order ∣ΔS∣≫1 (or, equivalently, ∣W−ΔF∣≫β−1), the system is effectively projected onto a state with a definite thermodynamic time’s arrow. Conversely, if the measured entropy production is ∣ΔS∣≲1 (or equivalently ∣W−ΔF∣≲β−1), the superposition state Eq. (14) resulting from the application of the extended TPM scheme lacks a definite time’s arrow, exhibiting interference effects. A closer examination of the term I ξ (W) highlights the fact that the second source of loss of interference effects in the extended TPM scheme lies in the presence of environmental decoherence, manifested in a negligible overlap between the environmental degrees of freedom, i.e., εð0Þ njεðτÞ m  0 for all n,m. This is the case in all instances where the environment is large and uncontrollable, thus leading the states εð0Þ n and εðτÞ m to have scarcely any significant overlap. However, for small environments or purposely-engineered environments, such effects can be avoided. For instance, one way to implement this scheme would be keeping a sufficiently small path separation in the interferometer in Fig. 2, such that the particle can be assumed to interact with the same environmental degree of freedom regardless of the path it takes. In this specific case, εð0Þ njεðτÞ m  ¼δn;m. As an illustrative example, we study the effect of interference in the work distribution in the case of a spin-1 2system, as illustrated in Fig. 3. In particular, in the forward quench, the spin system is subjected to a magnetic field whose direction is rotating within the x−zplane at constant angular velocity Ωaround the yaxis (ωbeing the spin’s natural frequency) HðΩtÞ¼_ω 21þcos ΩtðÞσzþ  sin ΩtðÞσx. In the extended TPM scheme, we superpose the forward quench and its time-reversal twin, and we project the auxiliary system onto the diagonal basis ± ji A¼ð0 ji A±  1 ji AÞ=ffiffiffi 2 pg. This leads to the work probability distributions P±ðWÞ, which illustrates the role played by the interference term. In the limit of a rapid quench (ω≪Ω) (and hence of a large degree of irreversibility), the distributions are presented in Fig. 4(yellow and blue bars), together with the one corresponding to a classical mixture of the forward and time-reversal processes (turquoise bars), where here PðWÞ¼~ PðWÞ.Whiletheclassicalmixture displays large fluctuations in the work probability distributions, the contribution of the interference term in P±ðWÞcan sharpen [PþðWÞ]orflatten [PðWÞ] the coherent work distribution, effectively increasing or decreasing the degree of reversibility, respectively. Specifically, the probability that the process will occur in a reversible fashion (i.e., that W=0) is higher for P + (W=0) [lower for P − (W=0)] than for a classical mixture (see the “Case study: a spin-1 2system”subsection in “Methods”). In this example, reversibility and adiabaticity coincide, being both reached for slow modulations. In the post-selected case, we can obtain a probability distribution PþðWÞcorresponding to that of a slower realisation of the quench. In this sense, through our protocol, one can achieve a net “speed-up”of the realisation of an adiabatic quench. Discussion Viewed in isolation, a thermodynamic system coupled to a reservoir undergoes a dynamic which is generally non-unitary, Fig. 3 Two-point measurement scheme in the forward process for our spin-1 2system. A spin-1 2particle in the thermal state of the initial Hamiltonian is measured in its eigenbasis fz± gat time t=0. After the action of the quench described by the time-dependent Hamiltonian in Eq. (25), it is measured in the eigenbasis fx± gof the final Hamiltonian at time t=τ. Depending on the measured states at the two times, the thermodynamic quench causes an energy change ΔE=0, ± ℏω, with ω being the spin’s natural frequency, and ℏthe reduced Planck constant. ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00759-1 6COMMUNICATIONS PHYSICS | (2021) 4:251 | https://doi.org/10.1038/s42005-021-00759-1 | www.nature.com/commsphys even though the joint state of the system and the environment evolves in a unitary, reversible fashion. Depending on whether this dynamics favours events involving a positive or a negative change in the total entropy, it is possible to establish the temporal direction of the quench to which the system has been subjected to, that is, the time’s arrow is aligned along the direction where the total entropy increases. (Notice that, under our sign convention, this means that the time’s arrow matches a positive entropy change in the case of the forward process, and a negative entropy change for the time-reversal process.) However, it can be expected that, under some circumstances, the joint state of the system and the environment may as well evolve in an arbitrary superposition of the two, whereby the direction of evolution is controlled by a further quantum system. We note that this superposition of thermodynamic processes does evolve according to an external dynamical time (e.g., the time as shown by the laboratory clock). However, from a quantum-mechanical perspective, there is a priori no preferential thermodynamic time’s arrow (i.e., the forward protocol Λand the time-reversal one ~ Λ occur in a quantum superposition), and this peculiarity is what this work has explored. In particular, the core questions behind this work are (i) how a definite (thermodynamic) arrow of time can emerge in such a picture, and (ii) what the signature of quantum interference among the forward-in-time and backwardin-time thermodynamic processes is. We showed that the coherence between the two temporal directions is effectively lost when the entropy production in the process is measured: the observation of a large increase (decrease) of dissipative work effectively projects the system in the forward (time-reversal) temporal direction. It is conceivable to imagine that such a projection could also result from the interaction of the system with the environment, which decoheres the system in a well-defined thermodynamic time’s arrow. Furthermore, when considering the total-entropy production in our process, one could consider adding the contribution arising from the irreversibility of the measurement itself. In Supplementary Note 3, we clarify that the entropy production linked to such a measurement, however, does not contribute to the definition of the orientation of the time’s axis associated to the quantum superposition of forward and time-reversal processes. Finally, for small values of the observed dissipative work (of the order of β−1), the system and the auxiliary state may display interference effects. This aspect bears important implications, insofar as, by measuring the state of the control, the system can exhibit a work (entropy-production) distribution which is classically impossible with the protocols at hand. This feature can be best observed when both the forward and the time-reversal processes are, to a high degree, irreversible (i.e., the probability of zero entropy production is low). In this case, indeed, the quantum superposition between the two irreversible processes can result in a dynamics that is no longer such (i.e., the above probability can be significantly increased due to constructive interference). Formally, this means that when the distribution of the work P±ðWÞ is affected by interference effects, this can result in a probability distribution radically different from any classic mixture of PðWÞ and ~ PðWÞ. As a consequence, P±ðWÞdoes not generally satisfy the fluctuation theorem (21). This is not extremely surprising given that the process generating P±ðWÞdoes not verify the requirements needed for the work fluctuation theorems. In particular, the initial state in Eq. (3) is not a thermal state neither of the system alone, nor of the system together with the control, and the work performed is defined differently in the two quenches of the superposition. Nevertheless, this violation has a crucial implication: it entails that the distribution P±ðWÞcannot be generated by any thermodynamic process starting in equilibrium with the environment, and being subsequently driven out of it by means of any given protocol Λ. Consequently, our procedure provides a recipe to generate thermodynamic processes with a work probability distribution that cannot be reproduced within the standard framework of fluctuation theorems. Methods Fluctuation theorems and the thermodynamic time’s arrow. The link between work fluctuations and the thermodynamic time’s arrow can be illustrated in terms of a “guessing the time directionality game”which was introduced by Jarzynski37. There, the author supposes to record the motion of a non-equilibrium thermodynamic process, and then to toss a coin. Depending on the outcome of the coin, he either plays the movie in the order in which it took place, or in the time-reversal one. In order to determine in which order the movie is being shown, the optimal guessing strategy for a macroscopic system follows from the second law of thermodynamics: if 〈W〉>ΔF, the movie proceeds in the correct order, while if 〈W〉<ΔF, the movie is being run backwards. Here, 〈W〉is the average work performed on the system by the external driving mechanism, and ΔFthe difference in free energies of the thermodynamic states at the beginning and at the end of the movie. Conversely, for a microscopic system, the optimal guessing strategy exploits the so-called “fluctuation theorems”13,38–40, together with Bayesian probabilistic reasoning41,42. We review this study briefly in Supplementary Note 4. In one of its most famous versions43–46, the fluctuation theorem describes the fluctuations of the dissipative work W diss associated to the observation of a particular value of Win a single realisation of a non-equilibrium driving protocol (i.e., a single shot of the movie): PðþWÞ ~ PðWÞ¼eβWdiss ;ð21Þ where P(+W) represents the probability that a work Wis invested along the forward thermodynamic evolution, whereas ~ PðWÞis the probability linked to recovering the same amount of work along the time-reversal evolution, both of which start in equilibrium with a thermal bath. From this equation, it follows that both the probability of total-entropy-decreasing events (βW diss < 0) in the forward evolution, and that of total-entropy-increasing ones (βW diss > 0) using the timereversal dynamics vanish exponentially with the size of the total-entropy variation: PðβWdiss<ξÞ≤eξ;ð22aÞ ~ PðβWdiss>þξÞ≤eξ;ð22bÞ for any ξ≥0, and where the second inequality (22b) arises from the fact that, in the time-reversal process, the dissipative work equals −W diss . In other words, large reductions in the total entropy are unlikely in the forward evolution, while events leading to a large entropy production are unlikely in the time-reversal one. (Notice that the sign of the entropy change is defined to match that of the dissipative work in the forward process.) On the other hand, it is evidenced that, when βW diss is of the order of one, it is inherently impossible to tell in which of the two orders the process has occurred. In this region, the directionality of time flow cannot be Fig. 4 Work probability distribution for a spin-1/2 system in a superposition of forward and time-reversal processes. The coherent work probabilities P±ðWÞand the work probabilities of a classical mixture PðWÞþ~ PðWÞ  =2 are compared in the limit of the rapid quench ω≪Ω for φ=π. The results are temperature-independent. COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00759-1 ARTICLE COMMUNICATIONS PHYSICS | (2021) 4:251 | https://doi.org/10.1038/s42005-021-00759-1 | www.nature.com/commsphys 7 inferred, and the time’s arrow is, so to say, blurred. A clear temporal directionality is then reestablished for β∣W diss ∣≫1. We remark that, here, “forward”and “time-reversal”are interchangeable labels since each process represents the time-inverted version of the other. Moreover, it is worth noticing that considerations on time-inversion only take on relevance in the absence of complete time-symmetry, as this latter may lead to ΔS tot equal to zero in every single realisation. In order to exhibit time-asymmetry, in the present study the two conjugated processes are assumed to start from equilibrium states, a standard procedure in the derivation of fluctuations theorems13,39. This introduces afinal (implicit) thermalisation step which enables irreversibility to emerge11,47,48. Case study: a spin-1 2system. In this section, we detail on the interference effects between forward and time-reversal thermodynamic evolution of a spin-1 2system. To this end, we further develop the general expression of Eq. (16). Specifically, we project the auxiliary system onto the diagonal basis ξ ji A¼± ji A¼ð0 ji A±1 ji AÞ=ffiffiffi 2 p .This leads to the joint state of the system and the environment jΨ± n;miS;E;Að1S;Ej±ih±jAÞMn;mjΨ0iS;E;A. The corresponding post-selected work probability distribution, conditioned on the projection of the auxiliary system onto ± ji A, reads: P±ðWÞ¼q± 0PðWÞþq± 1~ PðWÞþ2Re I±ðWÞ  ;ð23Þ where the interference term I ± (W) is given by Eq. (17) with h0j±iA¼1=ffiffiffi 2 pand h±j1iA¼±1=ffiffiffi 2 p. We recall that the states ΘEð0Þ n in the above expressions are the eigenstates of the Hamiltonian ΘH½λð0ÞΘy¼H½~ λð0Þ. Moreover, we notice that the distribution P±ðWÞin Eq. (23) differs by the term I ± (W)≠0 from what one would have obtained by applying the extended TPM scheme to a (classical) convex mixture jα0j20 ji0 hj Aρth 0þjα1j21 ji1 hj A~ ρth 0of the initial states. For the outcome W=0, the interference term in Eq. (17) can be simplified when ΔF=0, and the sets of eigenvalues of the initial and final Hamiltonians coincide, i.e., Eð0Þ n¼EðτÞ n. In that case: I±ðW¼0Þ¼±α 0α1 2PðξÞ∑ npn;ne2iΦn;nhεð0Þ njεðτÞ niEðτÞ n ΘEð0Þ n :ð24Þ As a result, it emerges that the interference effects can increase (decrease) the probability of observing the work value W=0. This yields to a work probability distribution P±ðWÞanalogous to the one potentially generated by a more reversible (irreversible) process than the forward and time-reversal processes themselves, or any classical mixture therefrom. We remark that the interference term I ± (W) may show non-zero values for W≠0 in general, as we will see below. We conclude by evaluating Eq. (23) in the concrete example of a spin-1 2system introduced in the “Interference effects in the work distribution”subsection in the Results. We consider a spin system with natural frequency ωin a magnetic field λ !ðtÞwhose direction is rotating within the x−zplane at a constant angular velocity around the yaxis: Hλ !ðtÞ hi ¼_ω 21þλ !ðtÞ σ ! hi ¼_ω 21þcos ΩtðÞσzþsin ΩtðÞσx  ;ð25Þ where λ !tðÞ¼ λ0sin ΩtðÞ;0;λ0cos ΩtðÞ  and λ 0 =1 is the dimensionless magnetic field, and where the protocol reads Λ¼fλ !ðtÞ;0≤t≤π=ð2ΩÞg.We notice that ΘHλ !ðtÞ hi Θy¼H½ λ !ðtÞ, implying that the time-reversal of the control parameter corresponds to a flip of the magnetic field. At the initial and final times of the protocol, the Hamiltonian is diagonal in the z±  and x±  bases, respectively. Therefore, Eð0Þ n ¼fz± Sg, with corresponding eigenvalues Eð0Þ n¼f0;_ωg, and EðτÞ m ¼fx± S¼1 ffiffi2 pz S±zþ S  g, with eigenvalues EðτÞ m¼f0;_ωg(we shifted the lower energy level by ℏω/2 to avoid negative energy eigenvalues). As a result, F0¼Fτ¼log 1 þeβ_ω  and W n,m ={ℏω,0,−ℏω}. In the frame rotating around the yaxis at frequency Ω, the Hamiltonian becomes time-independent, and the unitary governing the evolution can be obtained straightforwardly. Turning back to the Schrödinger picture, the applied unitary U(t, 0) reads: Uðt;0Þ¼ei 2Ωσytei 2½ωð1þσzÞΩσyt:ð26Þ This is used below to compute the work distribution. Effect of interference on reversibility. In this subsection, we will represent the environment as a spin-1 2system which is left unaffected during the quench. For instance, we can assume that the purification of the thermal states in Eq. (2a)–2b read ψ0 S;E¼ffiffiffiffiffi 1 Z0 sz Sz Eþffiffiffiffiffiffiffiffiffiffiffi eβ_ω Z0 szþ Szþ E;ð27aÞ ~ ψ0 S;E¼ffiffiffiffiffi 1 Z0 sx Sz Eþffiffiffiffiffiffiffiffiffiffiffi eβ_ω Z0 sxþ Szþ E:ð27bÞ Furthermore, we will assume to begin the protocol in the state in Eq. (3) with α0¼1=ffiffiffi 2 p,α1¼eiφ=ffiffiffi 2 p, with φbeing a controllable phase between the forward and the time-reversal processes. Next, we compute P±ðWÞ: P±ðW¼0Þ¼ 1 2Pð±Þp0;0þp1;1  1 2ffiffiffi 2 pPð±Þp0;0cos 2Φ0;0þφ  þp1;1cos 2Φ1;1þφ hi ; ð28Þ where we used the fact that hEðτÞ njΘjEð0Þ ni¼1=ffiffiffi 2 pfor all n, whereas hεð0Þ njεðτÞ niE¼1, and where the marginal probability of the auxiliary system reads Pð±Þ¼1 2±1 2ffiffi2 pp0;0cos 2Φ0;0þφ  þp1;1cos 2Φ1;1þφ hi , with p0;0¼hxjUðτ;0Þjzi jj 2 1þeβ_ω,eiΦ0;0¼hxjUðτ;0Þjzi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hxjUðτ;0Þjzi jj p, and p1;1¼hxþjUðτ;0Þjzþi jj 2 1þeβ_ωeβ_ω, eiΦ1;1¼hxþjUðτ;0Þjzþi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hxþjUðτ;0Þjzþi jj p. (To evaluate hEðτÞ njΘjEð0Þ niS, we made use of the fact that the time-reversal operator Θfor a spin-1 2system acts as Θ=iσ y K, where Kis the complex conjugation operator. Thus, Θjz±i¼jzi). From this result, we deduce that it is possible to observe interference between thermodynamic processes occurring in the forward and time-reversal temporal directions. Following the same procedure for the cases W=±ℏω, we get P±ðW¼_ωÞ¼ p0;1 4Pð±Þ1þeβ_ω  ;ð29aÞ P±ðW¼_ωÞ¼ p1;0 4Pð±Þ1þeβ_ω  ;ð29bÞ which do not feature interference. In the last expressions, p0;1¼hxþjUðτ;0Þjzi jj 2 1þeβ_ω, eiΦ0;1¼hxþjUðτ;0Þjzi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hxþjUðτ;0Þjzi jj p, and p1;0¼hxjUðτ;0Þjzþi jj 2 1þeβ_ωeβ_ω,eiΦ1;0¼hxjUðτ;0Þjzþi ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hxjUðτ;0Þjzþi jj p. We illustrate the probability distribution in Eq. (28)–(29) in Fig. 4. Interference terms for varying ±ℏω. In the previous case study, we represented the environment as a spin-1 2system which is left unmodified by the thermodynamic quench. This caused the cancellation of all interference terms in P±ðW¼±_ωÞ.In this subsection, on the contrary, we suppose that the environment undergoes a spin-flip during the quench: ψ0 S;E¼ffiffiffiffiffi 1 Z0 sz Sz Eþffiffiffiffiffiffiffiffiffiffiffi eβ_ω Z0 szþ Szþ E;ð30aÞ ~ ψ0 S;E¼ffiffiffiffiffi 1 Z0 sx Szþ Eþffiffiffiffiffiffiffiffiffiffiffi eβ_ω Z0 sxþ Sz E:ð30bÞ This change results in hεð0Þ njεðτÞ miE¼0, for n=m. For the sake of simplicity, below we will also set φ=π. The three probabilities discussed in the previous section become therefore: P±ðW¼0Þ¼ 1 2Pð±Þp0;0þp1;1  ; P±ðW¼_ωÞ¼ p0;1 4Pð±Þ1þeβ_ω±ffiffiffi 2 peβ_ω 2cos 2Φ0;1 hi ; P±ðW¼_ωÞ¼ p1;0 4Pð±Þ1þeβ_ωffiffiffi 2 peβ_ω 2cos 2Φ1;0 hi ; ð31Þ where the marginal probability of the auxiliary system is now Pð±Þ¼1 2±1 2ffiffi2 pp0;1eβ_ω 2cosð2Φ0;1Þp1;0eβ_ω 2cosð2Φ1;0Þ hi , and where p 0,0 ,Φ 0,0 , p 1,1 , and Φ 1,1 are the same as in case study “Effect of interference on reversibility”. In Fig. 5, we show the work probability distributions for varying ℏω. For work values ℏωsmaller than, or of the order of β−1, we observe strong interference effect, as shown by the difference between PþðW¼_ωÞand PðW¼_ωÞ. For work values ℏω≫β−1, this difference vanishes, and the probability PðW¼_ωÞ:¼ PþðW¼_ωÞþPðW¼_ωÞto obtain the work value ℏωtends to the probability p 0,1 of first projecting the auxiliary system onto the forward direction, and then ARTICLE COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00759-1 8COMMUNICATIONS PHYSICS | (2021) 4:251 | https://doi.org/10.1038/s42005-021-00759-1 | www.nature.com/commsphys obtaining the work value ℏω. This trend shows that the observation of large work values effectively projects the system into a well-defined temporal direction. Data availability All data needed to evaluate the conclusions of the paper are present in the paper and/or the Supplementary Information. 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Equilibrium free energies from nonequilibrium measurements using maximum-likelihood methods. Phys. Rev. Lett. 91, 140601 (2003). 42. P., M., Ritort, F., Bustamante, C., Karplus, M. & Crooks, G. Equilibrium free energies from nonequilibrium measurements using maximum-likelihood methods. J. Chem. Phys. 129, 024102 (2008). 43. Bochkov, G. N. & Kuzovlev, I. E. General theory of thermal fluctuations in nonlinear systems. Zh . Eksperimentalnoi i Teoreticheskoi Fiz. 72, 238–247 (1977). Fig. 5 Work probabilities of a spin-1/2 system under the time-dependent Hamiltonian with varying the work invested ℏω.For values of ℏωsmaller or of the order of β−1=k B T=1/2 (k B =ℏ=1), the work probabilities PþðW¼_ωÞand PðW¼_ωÞ(see Eq. (31); turquoise and purple curves) strongly depend on the interference terms. For values ℏω≫β−1, PþðW¼_ωÞþPðW¼_ωÞ(green curve) tends to the value p 0,1 (yellow curve), which is obtained by projecting the process to the forward direction and obtaining the work difference ℏω. This illustrates that observing large work values ℏω≫β−1(ℏω≪−β−1) effectively projects the process onto the forward (time-reversal) direction. COMMUNICATIONS PHYSICS | https://doi.org/10.1038/s42005-021-00759-1 ARTICLE COMMUNICATIONS PHYSICS | (2021) 4:251 | https://doi.org/10.1038/s42005-021-00759-1 | www.nature.com/commsphys 9