Categorical Coherence, Central Extensions, and the Limits of Unitarity: Triangle-Level Stability and Higher-Diagram Anomalies in Quantum Mechanics and Gravity
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Categorical Coherence, Central Extensions, and the Limits of Unitarity: Triangle-Level Stability and Higher-Diagram Anomalies in Quantum Mechanics and Gravity We develop a categorical framework in which observer-dependent choices form a groupoid and coherence conditions on simplices govern the emergence of physical laws. At the triangle level, coherence breakdown corresponds to nontrivial classes in H2 ( G, U (1)), yielding central extensions such as the canonical commutation relations in quantum mechanics and curvature identities in general relativity. We prove that all triangle-level obstructions reduce to central extensions, thereby preserving unitarity and linearity no matter how coherence fails at this order. At the pentagon and hexagon level, however, higher coherence anomalies are governed by H3 data and associators. These can reproduce central extensions but also generate non-central obstructions, opening the possibility of nonlinear dynamics and unitarity breaking. We formalize these effects using groupoid cohomology, weak monoidal structures, and higher-gauge holonomy, and derive scaling laws for interferometric tests such as nested Mach–Zehnder setups, Berry-phase experiments, and weak-gravity interferometry. Our results identify a sharp boundary between structural robustness (triangle-level unitarity) and genuinely new physics (pentagon/hexagon-level anomalies), offering a unified categorical perspective on quantum mechanics, general relativity, and their potential extensions. Part I Foundations Section 1. Coherence as the Common Skeleton of Quantum Mechanics and General Relativity 1.1. Why coherence? From invariance to consistency of composition From its birth, modern physics has treated invariance under change of viewpoint as a first principle. Noether taught us that continuous symmetries constrain dynamics through conserved quantities [ 1 ]; Wigner and the postwar school expressed physical content in terms of (projective) unitary representations of symmetry groups [ 2 – 6 ]. In differential geometry, parallelogram laws for parallel transport and their higher analogues—codified by Ehresmann connections and the Ambrose–Singer theorem—turn consistency of composition around small loops into curvature and holonomy [7, 8]. These familiar stories have a unifying core: coherence. Whether one composes symmetry transformations (in quantum mechanics) or infinitesimal transports (in general relativity), there are diagrams whose commutativity expresses that two (or more) ways of getting from here to there agree. When such diagrams only commute up to a controlled defect, the defect is usually highly structured: phases (central multipliers) for quantum symmetries; holonomies for geometric connections; or, in more elaborate settings, associators and braidings in monoidal and braided categories [14–16]. Thesis of this paper. Observer-dependent physics can be organized as a monoidal (indeed, higher) functor of “context changes” into a (higher) linear setting. The key data are the coherence maps that compare different compositions of context changes. We show that: (T) Triangle-level (2-simplex) coherence breakdown is necessarily central: it is classified by a U (1)- valued 2-cocycle (equivalently, a central extension), and thus preserves linearity and unitarity. (P/H) Pentagon and hexagon (3-simplex and braided 3-simplex) anomalies are governed by 3-cocycles and braided coherence; these can still be central (hence unitary) but, in general, admit non-central realizations whose effective action can induce unitarity breaking and controlled non-linearity. Item (T) explains why the ubiquitous phase phenomena (Aharonov–Bohm, Berry, Pancharatnam– Uhlmann) appear as triangle-level defects while leaving unitary quantum theory intact [ 9 – 12 ]. Item (P/H) points to where qualitatively new physics may consistently enter: at the level of associators and braidings (pentagon/hexagon), as in higher gauge/gerbe theory and quasi-Hopf structures [ 16 , 17 ]. We will show that our framework not only recovers standard QM and GR in appropriate limits, but also clarifies which deformations are a priori allowed and which are excluded by coherence itself. 1.2. Contexts, changes, and their compositions: a categorical kinematics We take contexts (frames, gauges, choices of foliation, instrument settings, etc.) as objects of a small groupoid C . Morphisms g : A→B encode context changes. Sequential composition in C captures doing one change after another; the monoidal product ⊗represents composition of independent systems/contexts.
2 Physical content is assigned by a (weak) monoidal pseudofunctor S: (C,⊗)−→ (Hilb,), sending a context A to a Hilbert space S( A )and a change g : A→B to a linear map (typically unitary) S(g) : S(A)→S(B). “Weak” here means we admit nontrivial coherence isomorphisms: φg,h :S(g)◦S(h)⇒S(g◦h), αX,Y,Z : (XY)Z=⇒X(YZ), βX,Y :XY=⇒YX, as well as the usual unitors. In a strict target these are identities; in physics they need not be. Coherence axioms demand that the obvious diagrams relating φ , α , β commute. At the triangle level (unitors with α ) and pentagon level (associator alone) this is Mac Lane’s coherence [ 14 ]; with a braiding β , the hexagon enters [ 15 ]. Our goal is to interpret controlled failures of these laws and to classify their physical effect. 1.3. Triangle-level defects are phases: central extensions and unitarity Consider composable context changes h:A→Band g:B→C. In general we may have S(g)S(h) = ω(g, h)S(g◦h), ω(g, h)∈U(1). Associativity of composition forces ωto satisfy the 2-cocycle law ω(h, k)ω(g, hk) = ω(gh, k)ω(g, h), so that ω∈Z2 ( MorC, U (1)). Two choices ω, ω0 differing by a coboundary ω0 = δχ ·ω are physically equivalent (a rephasing of the functor by χ : MorC → U (1)). Equivalence classes are thus [ ω ] ∈ H2(MorC, U(1)), and the defect is represented by a central extension 1−→ U(1) −→ b C −→ C −→ 1, on which Slifts to a strictly monoidal functor. This is precisely the classical classification of projective unitary representations (Bargmann multipliers) [ 3 ] and of the Stone–von Neumann uniqueness in Weyl form [4, 5]. Triangle Stability Theorem (informal). If the only coherence defect in Soccurs at the triangle level, then (i) the defect is central and classified by H2 ( −, U (1)); (ii) after passing to the central extension b C , the theory is equivalent to a strictly coherent, unitary and linear functor. Hence triangle-level anomalies cannot break unitarity. Physical meaning. Phases acquired upon circling small triangular loops in context space are AB/Berry/Pancharatnam–Uhlmann phases and their mixed-state generalizations [ 9 – 12 ]. They are central, commute with all observables, and leave the probability calculus untouched. In spacetime, the same mathematics manifests through holonomy of Ehresmann connections; by Ambrose–Singer, curvature is recovered from small loops [ 7 , 8 ]. The COW experiment’s gravitationally induced neutron phase [ 13 ] fits exactly here. 1.4. Beyond triangles: associators, braidings and higher coherence At the next layer, the associator α (and, in braided settings, the braiding β ) enters. Algebraically, α carries a 3-cocycle class [Φ] ∈H3 ( −, U (1)) that must satisfy the pentagon identity to strictify; in braided monoidal categories, the hexagon governs compatibility of αwith β[14–16]. Two qualitatively distinct regimes arise: • Central (unitary) 3-cocycles. When α and β act by central phases, one obtains higher geometric phases (2-holonomies) akin to the surface holonomies of higher gauge/gerbe theory [ 17 ]. These can be absorbed by a coherent choice of 2-connection and preserve unitarity.
3 • Non-central associator/braiding. In general, a quasi-monoidal (or quasi-Hopf) structure allows α to take values in non-central automorphisms (stateor subsystem-dependent). Then different parenthesizations of a triple composition implement distinct physical maps related by a non-scalar operator AX,Y,Z . If AX,Y,Z fails to commute with all observables on the relevant space, effective dynamics can become non-associative at the level of channels and, after reduction to state evolution, non-unitary or even non-linear. This is precisely the arena where higher-categorical anomalies and quasi-Hopf associators live [16, 18]. A geometric parallel. Ordinary gauge theory encodes curvature as a 2-form with loop holonomy. Higher gauge theory adds a 2-connection with 3-form curvature (“fake curvature” constraints), whose surface holonomies provide the natural home for H3 -classes [ 17 ]. Our observer-centric setting mirrors this: triangles detect central phases ( H2 ), while pentagons/hexagons detect genuine higher obstructions ( H3 ) that need not be absorbable by a phase and can affect compositional structure. 1.5. What is new here? 1. We derive a sharp separation principle: triangle-level defects are forced to be central extensions (hence unitary), whereas pentagon/hexagon defects admit both central and non-central realizations. The latter provide the only coherent gateway in our framework to unitarity breaking and controlled non-linearity, consistent with the structure of quasi-monoidal/ braided coherence. 2. We unify familiar phase phenomena (AB/Berry/Pancharatnam/Uhlmann) and GR holonomy as the same categorical H2 -class, while placing higher-gauge surface effects and quasi-Hopf associators at H3. 3. We provide operational tests: bracket-dependence (associator tomography) and braid-dependence (hexagon tomography) separate central from non-central higher defects. Their low-energy limits connect to well-known bounds on non-linearity and no-signalling constraints in quantum mechanics [19–21]. 1.6. Intuition in one picture Think of C as a mesh of admissible context changes. A triangle is the smallest 2-simplex: compose three edges to return to where you started. Any deviation from exact commutativity can only be a scalar on the state space—exactly a phase. A pentagon compares two ways of rebracketing four edges; now the comparison lives in operators acting on a tensor product of spaces, and there is room for non-central structure. Physics that lives at triangle order must be unitary; physics that first appears at pentagon/hexagon order can be exotic. 1.7. Roadmap Part I formalizes the setting and proves the Triangle Stability Theorem. Part II develops the higher (2-)gauge viewpoint and classifies H3 anomalies, giving explicit associator/hexagon forms and their experimental signatures. Part III treats examples: AB/Berry holonomy, gravitational phases (COW), Thomas precession as transport, and surface phases in higher-gauge backgrounds. Part IV analyzes when and how non-central pentagon/hexagon data can induce effective non-unitarity/non-linearity without violating foundational constraints, and states bounds compatible with existing precision tests [19–21]. Notation. All Hilbert spaces are separable; Hilb denotes the (symmetric) monoidal category with the completed tensor product. Cohomology H• ( −, U (1)) is groupoid cohomology when C is not a single-object group. “Central” means lying in the image of the canonical embedding U(1) ,→AutHilb(1)∼ =C×.
4 Part I Foundations Section 2. Triangle Stability: Central Extensions, Unitarity, and First-Order Coherence 2.1. Setting and hypotheses We recall from Section 1 that contexts (frames, gauges, instrument settings) form the objects of a small groupoid C , with morphisms g : A→B encoding admissible context changes. Physical content is assigned by a weak monoidal, dagger pseudofunctor S: (C,⊗,1)−→ (Hilb,,C), that sends objects A to Hilbert spaces S( A )and morphisms g to linear isometries (typically unitaries) S(g) : S(A)→S(B). Weakness means there are coherent comparison isomorphisms for composition φg,h :S(g)◦S(h)⇒S(g◦h)and φA: idS(A)⇒S(idA), natural in g, h and satisfying the usual triangle/pentagon coherence (Mac Lane) identities. Hypotheses for triangle analysis. We impose three mild structural conditions: (H1) (Local irreducibility/Schur property) For each connected component of C and each object A therein, the endomorphism algebra EndS(A) := {T∈ B(S(A)) |TS(g) = S(g)T∀g∈AutC(A)}is CI.[91] (H2) (Dagger naturality) The composition isomorphisms are unitary (dagger) natural transformations: φ† g,h =φ−1 g,h. (H3) (Normalization)φidB,g =φg,idA=Ifor all g:A→B(this is the usual “normalized” choice). 2.2. Projective functoriality and 2-cocycles By naturality and (H1), each φg,h must lie in the commutant of S( g◦h )and hence is a scalar multiple of the identity on S(dom h)transported to S(codg): φg,h =ω(g, h)Iwith ω(g, h)∈U(1).(1) Associativity of composition in C , together with Mac Lane’s triangle identity for the composition constraint of a pseudofunctor, forces ω to satisfy the groupoid 2-cocycle identity (Eilenberg–Mac Lane; see [ 22 – 24 ]) ω(h, k)ω(g, hk) = ω(gh, k)ω(g, h)for all composable g, h, k, (2) with the normalization ω ( id, g ) = ω ( g, id )=1by (H3). Rephasing the pseudofunctor by a U (1)-valued 1-cochain χ(g), eS(g) := χ(g)S(g), changes ω by a coboundary ω7→ ω0 =( δχ ) ω (i.e. ω0 ( g, h ) = χ ( g ) χ ( h ) χ ( gh ) −1ω ( g, h )). Thus the defect class [ω]lives in the groupoid cohomology H2(C, U(1)). Remark .1 (Projective representations).If we freeze an object A and restrict to its isotropy group GA := AutC ( A ), (1) – (2) say precisely that S GA is a projective unitary representation with Bargmann multiplier ω∈Z2 ( GA, U (1)) [ 3 , 25 ]. For locally compact GA , the measurable multiplier classification is due to Moore and Kleppner [26, 27]. 2.3. Central extensions and strictification at triangle level Given ω∈Z2 ( C, U (1)), define the central U (1)-extension b C with the same objects as C and morphisms Homb C(A, B) = {(g, z)|g∈HomC(A, B), z ∈U(1)}, and composition (g, z)◦(h, w) := (g◦h, zw ω(g, h)). The canonical projection π : b C → C is identity on objects and ( g, z ) 7→ g on morphisms. The cohomology class of ωcontrols the equivalence class of the extension [22–24].
5 Lemma .2 (Triangle strictification).The assignment bS(A) := S(A),bS(g, z) := zS(g), extends (uniquely) to a strict (composition-preserving) dagger functor bS : b C → Hilb . In particular, all triangle coherence is absorbed by the central U(1) and disappears in bS. Proof. Compute: bS(g, z)bS(h, w) = zw S(g)S(h) = zw ω(g, h)S(gh) = bS((g, z)◦(h, w)). Dagger compatibility follows from |z|= 1 and (H2). Theorem .3 (Triangle Stability Theorem) . Under (H1)–(H3), any triangle-level coherence defect of S is classified by a U (1)-valued 2-cocycle [ ω ] ∈H2 ( C, U (1)); it is central, and after passing to the central extension b C the theory is equivalent to a strictly coherent dagger functor bS . In particular, triangle defects preserve unitarity and linearity. Proof. Scalarity of φg,h is from (H1). The 2-cocycle law is (2) , a restatement of pseudofunctor associativity; normalization gives a normalized cocycle. The extension and strictification are as above. Since U (1) lies in the center of U (S( A )), conjugation by bS ( g, z )preserves transition probabilities and the Born rule (hence unitarity persists). Linearity is unchanged because rephasing by scalars commutes with superpositions. Corollary .4 (Wigner consistency at triangle order) . If transitions probabilities are preserved by Sand only triangle-level defects occur, then by Wigner’s theorem the induced transformations are unitary/antiunitary on each S( A )[ 28 ]. The triangle defect merely contributes a central phase and cannot induce nonlinearity. 2.4. Quantum kinematics: CCR as an H2-class Fix an object A and consider the subgroup GA∼ = ( R2, +) that integrates translations in position and momentum (Heisenberg–Weyl kinematics). A measurable multiplier ω∈Z2 ( R2, U (1)) is necessarily of the form ω(x, p),(x0, p0)= expni 2~xp0−px0o, up to coboundary (Kleppner’s classification on abelian groups [ 27 ]; see also [ 29 , Ch. 2]). The associated central extension U (1) →b GA→R2 is the (Stone–von Neumann) Heisenberg group. The corresponding Weyl relations read W(x, p)W(x0, p0) = ei 2~(xp0−px0)W(x+x0, p +p0), and differentiating (Stone’s theorem) gives the CCR [ X, P ] = i~I on the common domain [ 4 , 5 ]. Uniqueness of the regular irreducible representation is Stone–von Neumann (see [ 29 , Ch. XIII.1]). By Theorem .3, these are precisely the triangle-level defects in kinematics: central and unitary. 2.5. Geometric kinematics: curvature and holonomy as first-order obstructions Let Π 1 ( M )be the path groupoid of a smooth manifold M . A principal G -bundle with connection ∇ determines a holonomy functor Hol∇: Π1(M)−→ G, [γ]7−→ Pexp−Zγ A, such that Hol∇ ( γ2◦γ1 ) = Hol∇ ( γ2 ) Hol∇ ( γ1 ). On a small contractible triangle ∂ Σ, Stokes’ theorem yields Hol∇(∂Σ) = exp−ZΣ F,
6 where Fis the curvature 2-form. Infinitesimally, this is the commutator of covariant derivatives [∇µ,∇ν]Vρ=Rρσµν Vσ, as recalled in Section 1 (Ehresmann; Ambrose–Singer [ 7 , 8 ]). In our language, first-order (triangle) coherence breakdown expresses precisely that parallel transport around a small loop need not be trivial; its effect is geometric holonomy, not a breakdown of the orthonormal/metric structure (unitarity analogue in GR). No non-associative or non-linear phenomena can arise at this order: composition in Π 1 ( M )is strictly associative; defects are controlled entirely by F. 2.6. No route to nonlinearity at triangle order There are two independent reasons why triangle defects cannot generate nonlinearity: 1. Centrality: By Theorem .3, all triangle defects are central U (1) phases; central scalars cannot lead to nonlinear superposition rules or trace non-preserving channels. 2. Probability preservation: Wigner’s theorem shows that any symmetry of transition probabilities acts unitarily/antiunitarily [ 28 ]. Triangle rephasings preserve all Born probabilities (they cancel from hψ|ψiand transition amplitudes), hence cannot induce nonlinear expectation functionals. Consequently, any controlled departures from unitarity or linearity (if present) must first appear at the level of higher coherence (pentagon/hexagon), where non-central associators may occur. This is exactly the door we open in Part II. 2.7. Normal forms and gauge of multipliers For later use we recall two standard reductions: • (Normalization) Any 2-cocycle is cohomologous to a normalized one satisfying ω ( id, g ) = ω ( g, id ) = 1. • (Skew form on abelian groups) On a finite-dimensional real vector group V , multipliers are classified (up to coboundary) by alternating bilinear forms σ∈ ∧2V∗ . In particular for V = R2n , one may choose Darboux coordinates so that σ=Pn j=1 dxj∧dpjand ω(u, v) = ei 2σ(u,v). These gauges will be used in Section 3 to extract clean lowest-order phenomenology from nested interferometer loops. 2.8. Summary of Part I, Section 2 Triangle coherence breakdown is completely exhausted by U (1)-central 2-cocycles. In the quantum sector this is the familiar realm of projective representations/Heisenberg CCR; in the geometric sector it is curvature/holonomy. Both are structurally “unitarity-preserving” at first order: no nonlinearity or non-associativity can arise. Hence any genuine departures—including controlled unitarity breaking or nonlinear effective dynamics—must be tied to higher coherence anomalies, analyzed next via associators and hexagon identities. Part I Foundations Section 3. Higher Coherence: Associators, 3–Cocycles, and Hexagon Anomalies 3.1. From first-order coherence to higher coherence Section 2 showed that all triangle-level defects are exhausted by central U (1)-valued 2-cocycles and can be absorbed by a central extension without disturbing unitarity or linearity. We now pass to higher coherence data. Let S: (C,⊗,1)−→ (Hilb,,C)
7 be a weak monoidal dagger pseudofunctor as before. The associator and (when present) braiding are unitary natural isomorphisms αX,Y,Z : (XY)Z∼ = ==⇒X(YZ), βX,Y :XY∼ = ==⇒YX(3) subject to Mac Lane’s pentagon and the hexagon coherence laws [ 14 – 16 ]. We allow nontrivial α, β as controlled departures from strict associativity/commutativity. Our guiding questions are: (Q1) When is higher coherence still central (and hence unitary-preserving), and how is it classified? (Q2) When can higher coherence be non-central, and what are its operational consequences (bracket dependence, effective nonlinearity, unitarity breaking)? 3.2. Central associators and 3–cocycles We say the associator is central if, for all objects X, Y, Z , the map αX,Y,Z is proportional to the identity on XYZ: αX,Y,Z = ei φ(X,Y,Z)IXYZ, φ(X, Y, Z)∈R/2πZ.(4) The pentagon identity αW,X,Y Z◦αWX,Y,Z =IWαX,Y,Z ◦αW,XY,Z ◦αW,X,Y IZ then reduces to the 3-cocycle condition (δφ)(W, X, Y, Z) = φ(X, Y, Z)−φ(WX, Y, Z)+φ(W, X Y, Z)−φ(W, X, Y Z)+φ(W, X, Y ) = 0, (5) so [ φ ] ∈H3 ( C, U (1)) [ 16 , 22 , 23 ]. A change of monoidal gauge by a U (1)-valued 2-cochain θ ( X, Y ) modifies φby δθ, so only the cohomology class matters. Theorem .5 (Central higher coherence ⇔U (1) 3–cocycles) . Under the hypotheses of Section 2 and centrality (4) , higher coherence of Sis classified by a normalized class [ φ ] ∈H3 ( C, U (1)). Conversely, any such 3-cocycle defines a monoidal structure with associator (4) satisfying the pentagon. If β is present and central, the hexagon imposes the standard abelian compatibility constraints between [ φ ]and the braiding bicharacter [15, 16, 30]. Sketch. Pentagon ⇒ cocycle is (5) . Cocycle ⇒ associator: pick representatives and define α as in (4) ; the pentagon holds by δφ = 0. Braided case: central β corresponds to a symmetric bicharacter; the two hexagon identities force the familiar abelian compatibility (see [15, 16, 30]). Unitarity and linearity. Because α acts by scalars, it lies in the center of U ( XYZ ); all amplitudes and the Born rule are unchanged. Thus: Corollary .6 (Triangle+ 3–cocycle centrality preserves unitarity) . If all coherence defects up through the pentagon/hexagon level are central ( H2 and central H3 ), then Sis monoidally equivalent to a unitary theory; no effective nonlinearity can be induced by higher coherence. 3.3. 2–group central extensions and higher holonomy When C restricts to a single-object group G (contexts with a fixed background), a central 3-cocycle [ φ ] ∈H3 ( G, U (1)) defines a central extension by U (1) as a 2–group (a.k.a. a string extension) [ 31 , 32 ]. Physically, this is the correct home for surface (2-)holonomies: parallel transport of lines along surfaces in higher gauge theory [ 17 ]. Our framework identifies these higher holonomies with central higher coherence—again unitary-preserving.
8 3.4. Non-central associators: bracket dependence, anomalies, and effective dynamics We now allow non-central α: αX,Y,Z ∈Aut(XY)Z, X (YZ), αX,Y,Z 6∈ CI. (6) Pentagon still holds (so the data define a genuine monoidal structure), but different parenthesizations need not coincide as operators. In operational terms, two circuits that differ only by bracketing, (UgUh)Ukα ==⇒UgUhUk, transform by αX,Y,Z . If αis not scalar, observable bracket dependence can occur. Proposition .7 (Associator anomaly observable) . Fix X, Y, Z and unit vectors ψX, ψY, ψZ . Prepare ΨL:= ((ψXψY)ψZ)and ΨR:= (ψX(ψYψZ)). For any probe Macting on XYZ, ∆M:= ΨR, M ΨR−ΨL, α† X,Y,Z M αX,Y,Z ΨL vanishes for all M iff αX,Y,Z is proportional to the identity on the support of the state. In particular, if αX,Y,Z is non-central, there exist choices of probes/states with ∆M6= 0. Proof. The “only if” direction is immediate by choosing M as a spectral projector for the polar decomposition of αX,Y,Z; the converse is trivial. Effective nonlinearity and (non-)unitarity. Two regimes must be distinguished: (i) Fixed unitary associator. If each αX,Y,Z is a fixed unitary natural isomorphism (independent of state) satisfying the pentagon, then the global theory remains linear and unitary on XYZ . Bracket dependence is physical but does not by itself induce nonlinearity or loss of trace. (One may still observe nontrivial scattering phases controlled by α.) (ii) State/context-dependent associator or non-unitary coherence. If αX,Y,Z depends on the intermediate state (or measurement branch) or fails to be unitary as a natural transformation, then the effective map on subsystems after discarding/conditioning can become nonlinear or non-CP. Such nonlinearities are tightly constrained by no-signalling considerations [ 19 – 21 , 34 ]. Our framework pinpoints their origin: they cannot occur at triangle order, but can first appear through non-central higher coherence. 3.5. Hexagon anomalies with braiding In braided settings, a unitary braiding β yields additional constraints. Central ( α, β )pairs are classified by abelian data (a 3-cocycle together with a bicharacter), and remain unitary-preserving [ 15 , 16 , 30 ]. Non-central braidings satisfy hexagon but can act nontrivially on XY ; the two hexagon diagrams quantify a braid-sensitive anomaly. Operationally, one may compare circuits that differ by exchanging middle factors; a non-central β leads to exchange-dependent amplitudes even when the underlying morphisms commute. 3.6. “Associator tomography”: an operational protocol To diagnose central vs. non-central higher coherence we propose a simple interferometric/tomographic routine. Protocol. 1. Prepare a reference state on XYZ and a path superposition that implements the two bracketings: |Φi=1 √2|Li⊗((UgUh)Uk)|ψi+|Ri⊗Ug(UhUk)|ψi. 2. Insert αX,Y,Z coherently on the left arm. 3. Close the interferometer and measure the relative phase/visibility. Central α = e iφI yields a pure phase shift; non-central α reduces visibility in a stateand probe-dependent way via Proposition .7. A nested Mach–Zehnder implementation realizes step (1)–(3) physically; Section III will provide concrete scaling laws.
9 3.7. Examples and normal forms Group-graded vector spaces. For a discrete group G , the category VectG of G -graded vector spaces admits monoidal structures whose associator is determined by ω∈Z3(G, U(1)): αX,Y,Z =X g,h,k∈G ω(g, h, k)projXg⊗Yh⊗Zk. This is central; pentagon is precisely δω = 0 [16, 30]. Quasi-Hopf (non-central) associators. In a quasi-Hopf algebra ( H, ∆ , , Φ), the associator Φ ∈H⊗3 need not be central [ 18 , 33 ]. Representations inherit a non-central α acting on XYZ . Our framework identifies precisely how such αenters operationally and when it can (and cannot) be gauged away. 3.8. Summary of Part I, Section 3 Higher coherence splits sharply: • Central associators/braidings ⇒H3 (and abelian braided) classes: unitary-preserving, physically higher geometric phases and 2-holonomy (safe). • Non-central associators/braidings ⇒ bracket/exchange anomalies: globally unitary if fixed and unitary, but capable of inducing effective nonlinearity or non-CP behaviour upon conditioning/discarding. This is the first diagrammatic locus where such effects can consistently arise in our framework. The next section formalizes braided constraints and quantifies interferometric signatures in concrete models. Part I Foundations Section 4. Braided (Hexagon) Coherence: Abelian vs. Non-Abelian Structure and Operational Diagnostics 4.1. Why a separate braided analysis? Sections 2–3 established that (i) triangle coherence defects are forced to be central ( H2 ) and thus preserve unitarity and linearity, and (ii) higher (pentagon) coherence can be central or non - central, with central cases classified by H3 and non - central cases allowing bracket anomalies. In many physical settings—from exchange statistics to interferometry with path labels that act like species—one also needs abraiding βX,Y : XY∼ = ==⇒YX , constrained with the associator α by the hexagon coherence laws [14–16]. The braided layer distinguishes two qualitatively different situations: •Abelian (diagonal) braiding: β acts as a phase times the canonical swap on each homogeneous component (“phase exchange”). Hexagon reduces to bilinearity constraints on the phase and compatibility with the 3-cocycle φof α(abelian data). This regime is unitary-preserving. •Non-abelian (mixing) braiding: β moves amplitude between different fusion channels (or subspaces) and cannot be written as a scalar times the swap. Hexagon still holds, but exchange processes become operator - valued. This is the first braided locus where exchange anomalies can be operationally distinguished from mere phases; combined with conditioning/discarding, such non-centrality can seed effective nonlinearity or non-CP behaviour. We now formalize these statements and connect them to standard structures: quasi - (triangular) Hopf algebras, the Drinfel’d center, and non-degeneracy / Müger center.
16 Sagnac (rotation) phase. For counter-propagating beams around a closed contour C with vector area A, the Sagnac time delay in a stationary spacetime is ∆tSag =−2 c2IC g0idxirot.frame =4Ω·A c2,(15) hence the optical phase shift is ∆φ(light) Sag =ω∆tSag =4ω c2Ω·A.(16) If one prefers the Sec. 5 parametrization δφ = ( ωT ) ξ , then ξ(light) Sag = 4Ω·A c2T (with T the interrogation time). In ring lasers, (??) reproduces the textbook ∆φ= 8πA Ω/(λc)[52]. 6.3. Matter-wave interferometry: COW and Sagnac/dragging Let mbe the particle mass and λdB =h/(mv)its de Broglie wavelength at speed v. COW (Colella–Overhauser–Werner) phase. For a rectangular loop of area A oriented vertically, with horizontal speed v, the gravitational potential contributes ∆φCOW =m g A ~v,(17) as measured in neutron interferometry [13]. (Units: mgA/(~v)is dimensionless.) Sagnac/rotation for matter waves. For a loop with vector area A, ∆φ(matter) Sag =2m ~Ω·A.(18) Frame-dragging acts as an effective angular velocity ΩLT (see below), giving the additive correction 2m ~ΩLT ·A[51, 55]. 6.4. Frame dragging (Lense–Thirring) and gravitomagnetic Sagnac In the field of a rotating body with angular momentum J , the stationary metric includes a “gravitomagnetic” term gLT 0i∼(J×r)i/r3. The corresponding Sagnac time delay is[92] ∆tLT =−2 c2IC gLT 0idxi≃4 c2ΩLT(x0)·A,ΩLT(x)∼G c2r3h3(J·ˆ r)ˆ r−Ji,(19) evaluated near a representative point x0of the loop. Thus ∆φ(light) LT =4ω c2ΩLT ·A,∆φ(matter) LT =2m ~ΩLT ·A. Near Earth’s surface |ΩLT| ∼ 10 −14 s −1 , about nine orders of magnitude below the kinematic Sagnac term from Earth’s rotation |Ω⊕| ≃ 7 . 292 × 10 −5 s −1 . Hence differential protocols (Sec. 5.8) and long baselines are essential to isolate ΩLT [51, 52]. 6.5. Tidal (curvature) phases in the local lab frame Let Xi ( t )denote the arm separation in Fermi coordinates. The lab clock measures the optical phase ωRdt ; the redshift factor is √−g00 ≃ 1+ 1 2R0i0jXiXj . To leading order, the curvature-induced differential phase is therefore ∆φ(light) tidal =ω 2ZT 0 R0i0j(τ)Xi(t)Xj(t)dt, (20) which is suppressed by the small dimensionless factor R0i0jL2 (with L a characteristic arm separation). For stationary arms with mean separation moments XiXj, (20) reduces to ∆φ(light) tidal ≃ωT 2R0i0jXiXj= (ωT)ξR, ξR:= 1 2R0i0jXiXj.(21)
17 Equivalently, using the area tensor Aij := Hx[idxj] = RRd Σ ij (units of length 2 ) and Stokes’ theorem on a stationary loop, ∆φ(light) tidal ∼ω 2R0i0jAij,(22) up to geometry-dependent order - unity factors. Since R0i0j has units 1 /length2 , the combinations R0i0jXiXjand R0i0jAij are dimensionless (consistency check). For matter waves, an analogous expansion of the proper-time action S = −mc2Rdτ yields (in the same approximations) ∆φ(matter) tidal ≃mc2 2~ZT 0 R0i0jXiXjdt =mc2 ~(ωT)R0i0jXiXj 2ω,(23) which reduces to the familiar gravity - gradient terms in atom interferometers when R0i0j≃∂i∂j Φ /c2 [46, 55]. 6.6. Mapping to the Sec. 5 parameter ξ The central higher-coherence scaling in Sec. 5 was δφHC =εΓ (ωT)ξ. For weak-gravity backgrounds: ξ= ξΦ= ∆Φ/c2,(static redshift, Eq.(14)), ξ(light) Sag =4Ω·A c2T,(optical Sagnac, Eq.(??)), ξR=1 2R0i0jXiXj∼1 2R0i0jAij/T, (tidal curvature, Eqs. (21)–(22)), and similarly for matter waves after replacing ω with mc2/~ where appropriate (cf. Eqs. (17) , (18) , (23) ). These forms ensure ξis always dimensionless and vanish in the flat/zero-potential limit. A. Numbers at Earth (order of magnitude) For an interferometer of area A = 0 . 1 m2 at latitude ϑ , the Earth’s rotation vector projects as Ω ⊕·A = Ω ⊕Acosϑ with Ω ⊕ = 7 . 292 × 10 −5s−1 . Using Eq. (16) , the Sagnac phase shift for light of wavelength λ= 780 nm is ∆φ(light) Sag ≈4ω c2Ω⊕Acosϑ∼7.8×10−7cosϑ. This is of the same order of magnitude as the gravitational redshift phase for a 1m vertical separation, ∆φredshift ∼ωgh c2T∼2.6×10−7, and both are many orders larger than the tidal curvature contribution, which for a baseline h = 1 m and separation L= 1m yields ∆φtidal ∼10−12. Thus, in realistic laboratory scales the Sagnac (light) and redshift phases are comparable in size (10 −7 ), while tidal terms are negligible by comparison. All three remain comfortably above the 10 −12 level where technical noise would mask them, and our higher-coherence toggles (triangle lock, associator flip) allow these central backgrounds to be cleanly distinguished from non-central effects.
18 Effect Typical magnitude (lab scale) Comments Sagnac (light, A = 0.1m2,λ= 780 nm) ∆φ(light) Sag ∼7.8×10−7cos ϑ Comparable to redshift; central (phaseonly). Redshift (h= 1 m) ∆φredshift ∼2.6×10−7 Same order as Sagnac; central (phaseonly). Tidal curvature ( h = L= 1 m) ∆φtidal ∼10−12 Negligible compared to Sagnac/redshift. TABLE I: Baseline general-relativistic phases at Earth for laboratory scales. Sagnac and redshift are both ∼ 10 −7 , while tidal curvature is suppressed to ∼10−12. 6.8. Assumptions, approximations, and consistency checks •Approximations. Stationary background; first post - Newtonian order O ( c−2 )in g00 and O ( c−2 ) Sagnac formula via Hg0idxi ; small loop/arm separations so that Fermi expansions truncate at quadratic order in x. No torsion; Levi-Civita connection. •Unit/limit checks. All ξ are dimensionless; phases vanish in the limits Φ → 0, Ω→ 0, R→ 0. The Sagnac expressions reduce to textbook formulas in rotating frames and to the Lense–Thirring corrections for gLT 0i. •Separation from higher coherence. The terms in this section are baseline GR/QM phases (central, triangle - level or geometric); they superpose linearly with the higher - coherence signals of Sec.5. Protocols in Sec. 5.8 cancel the large kinematic Sagnac/redshift backgrounds to isolate associator/braiding effects. Part II Phenomenology and Operational Tests Section 7. Berry, Pancharatnam–Uhlmann Holonomy, and Visibility Budgeting under Noise 7.1. Aim and context Section 5 established interferometric signatures of higher coherence (associator/braiding) in ideal, unitary settings; Section 6 quantified weak-gravity baselines. Real platforms face decoherence and dissipation. Here we (i) unify pure-state Berry/Pancharatnam phases and mixed-state Uhlmann holonomy within our coherence framework; (ii) present interferometric measurement protocols; and (iii) derive a visibility budget under completely positive (CP) evolutions that isolates central ( H3 -phase) effects from non-central associator/braiding, even with noise. 7.2. Pure states: Pancharatnam and Berry revisited For a differentiable path t7→ |ψ(t)iwith |ψ(0)i=|ψ(T)i, the Pancharatnam phase is φP= arghψ(0)|ψ(T)|.i(24) Imposing the parallel-transport gauge hψ(t)|˙ ψ(t)|=i0, Berry’s phase along a parameter loop λ(t)is φB=IλA,A:= ihψ(λ)|∇λψ(λ)|·idλ, (25) with curvature F = dA [ 10 , 11 ]. Non-adiabatic cycles (Aharonov–Anandan/Samuel–Bhandari) are obtained by rephasing the dynamical contribution out of (24) [ 56 , 57 ]. In our language, these are triangle-level central defects: they shift the interferometric phase but preserve visibility (Sec. 5.2).
19 7.3. Mixed states: Uhlmann bundle and holonomy Let D ( H )be the manifold of full-rank density operators on H . An Uhlmann amplitude is W∈ B2 ( H ) with WW† = ρ (Hilbert–Schmidt bundle); W and WU , U∈ U ( H ), represent the same ρ . Uhlmann’s parallelism condition along a curve t7→ ρ(t)reads W†˙ W=˙ W†W⇐⇒ W†˙ Wis Hermitian,(26) which fixes the horizontal lift t7→ W ( t )up to a constant unitary on the right [ 12 ]. The Uhlmann holonomy around a loop is the phase φU= argTrW(0)†W(T),(27) a mixed-state generalization of (24) that reduces to Berry in the pure, adiabatic gauge. Operational meaning. With a purification |Ψ(t)i ∈ HK , W acts as the partial isometry from K to H , and (26) enforces that the ancilla “absorbs” the dynamical phase so that only geometric content remains in (27). 7.4. Measuring phases interferometrically with noise Any physical arm experiencing noise is modelled by a CP, trace-preserving map E with Kraus operators Kj, E(ρ) = X j KjρK† j,X j K† jKj=I. (28) In a Mach–Zehnder with a unitary U on the reference arm and E on the probe arm, the complex fringe amplitude is A= TrU†E(ρ)=X j TrU†KjρK† j= Tre U†ρ,e U:= X j K† jUKj,(29) with visibility V=|A| and phase arg A. Equation (29) makes two facts manifest: •Central higher coherence ( U7→ eiφU ): multiplies A by eiφ , shifting only the phase (no visibility loss). •Non-central associator/braiding (unitary entanglers on system+ancilla or branch-dependent maps): changes e U non-scalar, thereby reducing V after tracing out environment/ancilla (Sec. 5.3, Eq. (5.3)). Stinespring picture. Any E admits a unitary dilation V on system S + environment E (Stinespring) with E ( ρ ) = TrE [ V ( ρ⊗|0ih0| ) V† ][ 60 – 62 ]. If a higher-coherence associator on S acts as eiφI (central) on S and trivially on E , then A 7→ eiφA . If it acts non-centrally on S or correlates S with E , V picks up nontrivial entangling terms and Vreduces. 7.5. Two mixed-state protocols (A) Interferometric mixed-state phase (Sjöqvist et al.). Given a unitary loop ρ7→ UρU† with parallel transport in the Pancharatnam sense on each eigen-subspace, the interferometric mixed-state phase is [ 58 ] φint = arg X k wkhψk|U|ψk|,i(30) for ρ=Pkwk|ψkihψk|. This is directly accessible as arg Ain (29) for unital Eand unitary U. (B) Uhlmann phase via purification. Choose a fixed ancilla K and prepare a purification |Ψi = Pk√wk|ψki|kiK . Impose the Uhlmann parallelism (26) by actively rotating the ancilla along the path (feedback), then close the interferometer between the initial and final purifications; the phase shift at the output equals φU in (27) . For nonunitary evolutions, adopt the kinematic construction à la Tong et al. [ 59 ] to define geometric phase for general CP dynamics and measure it from arg A with suitable reference.
20 7.6. Visibility budgeting: central vs. non-central, plus noise Let V0be the triangle-calibrated visibility (Sec. 5.2). Under noise and higher coherence, V=V0×1−1 2ε2Varψ(Λ) | {z } non-central associator, Sec. 5.3 ×ηnoise |{z} CP map (decoherence) +O(ε3),(31) where ε is the higher-coherence strength, Λthe (Hermitian) non-central generator in the associator block, and 0 ≤ηnoise ≤ 1is the noise visibility factor determined by E and ρ through (29) . For common channels: ηdepol(p) = |1−4 3p|, ηdeph(γ) = |1−2γ|, ηAD(γ) = p(1 −γ)(on off-diagonals), with p the depolarizing probability, γ the dephasing or amplitude damping parameter (single-qubit illustration; generalizations are immediate). Central higher coherence multiplies the complex amplitude by a phase and leaves each ηunchanged. Difference-of-differences under noise. Protocol of Sec. 5.8 still isolates higher-coherence signals: (i) swap inner bracketing L↔R at fixed E ; (ii) flip α7→ α† (changes sign of the phase term but not ηnoise ); (iii) for exchange tomography, alternate (Σ , β )on the arms (abelian β changes phase only; non-abelian β reduces Vat O(ε2)). 7.7. Worked examples (one qubit + one qubit ancilla) (i) Dephasing + central associator. Let Eγ ( ρ ) = (1 −γ ) ρ + γZρZ on the probe arm and U = I on the reference. With input ρ=|+ih+|,A= (1 −2γ)eiφ, so arg A=φ(central H3phase) and V=|1−2γ|. (ii) Dephasing + non-central associator. Insert α = eiεΛ on the left arm with Λ = 1 2 ( σxσx )acting between polarization and time-bin. Gauge tr ( ρ Λ) = 0. Then V≈ | 1 − 2 γ| (1 −1 2ε2Varψ (Λ)), while the phase shift is φ+O(ε3): central vs. non-central parts separate cleanly. (iii) Amplitude damping + Uhlmann protocol. Let EAD ( ρ ) = E0ρE† 0 + E1ρE† 1 with E0 = |0ih0| + √1−γ|1ih1| , E1 = √γ|0ih1| . Purify on an ancilla and enforce (26) by rotating the ancilla so that W†˙ W is Hermitian. The output interferometric phase equals φU of (27) , while the contrast follows (31) with ηAD(γ). 7.8. Assumptions, limits, and consistency checks •Assumptions. Narrowband interferometer; weak higher coherence ( ε 1); CP noise model with stationary parameters during a run; perfect path balance. •Limits. ε→ 0reduces to standard mixed-state phases (§7.4); γ, p→ 0restores the unitary case of Sec. 5; central H3phases never change V. •No-signalling. Even with post-selection/conditioning, triangle-level effects cannot induce signalling or nonlinearity. Any effective nonlinearity must be traced to state/context dependence of non-central higher coherence (Part I, §3–§4), not to central phases. Part II Phenomenology and Operational Tests Section 8. Entanglement Tests: CHSH, MABK, and Higher-Coherence Tomography 8.1. Aim and scope Sections 5–7 examined single– and two–arm interferometers. We now use entanglement tests to probe higher coherence. The guiding principles from Part I are: •Triangle (H2) and central (H3) effects are phases: they preserve unitarity and linearity, thus cannot alter Bell bounds. Operationally they shift interferometric phase but not visibility.
21 •Non-central (H3) effects: fixed, unitary associators/braidings can reshape correlations but still obey Tsirelson’s bound. State/branch–dependent or non - unitary higher coherence can, upon conditioning/discarding, induce effective nonlinearity/non - CP behaviour; such effects are tightly constrained by no-signalling and precision Bell tests (cf. Part I, Secs. 3–4). We formalize these statements for CHSH (bipartite) and MABK/Mermin (tripartite) scenarios, give tomography protocols that separate central from non - central higher coherence, and supply worked examples and bounds. 8.2. CHSH preliminaries and LU invariance For a bipartite state ρ on HAHB , dichotomic observables A, A0 on A and B, B0 on B with operator norms ≤1, define E(a, b) := Tr(A⊗B)ρ, S := E(a, b) + E(a, b0) + E(a0, b)−E(a0, b0). Local unitaries (LUs) UA⊗UB leave the supremum of S invariant, since they merely rotate measurement axes; in the two - qubit case this is encoded by SO (3) rotations of the correlation matrix Tij = Tr [ ρ σi⊗σj ] [66]. For any quantum state and observables, Tsirelson’s bound holds: |S| ≤ 2√2. Thus any operation whose unconditional action yields a valid density operator and linear Born probabilities cannot produce |S|>2√2[65]. Triangle & central higher coherence are LU–equivalent. Triangle - level phases (H 2 ) and central H 3 associators/braidings act as global U (1) phases or LU gauge on each arm (Part I, Secs. 2–3). Hence they cannot change optimal S; they only shift interferometric phases (already seen in Secs. 5–7). 8.3. Non-central associators in CHSH: contrast and harmonics, not super-Tsirelson Insert a (small) non - central associator acting across A and an internal degree of freedom Z (measured/discarded later), or directly across Aand B: αAB =eiεΛAB ,ΛAB = Λ† AB,0< ε 1, with its scalar part gauged away on the prepared support (so Tr [ ρ Λ AB ] = 0). The measured correlator becomes Eε(a, b) = Trh(A⊗B)αAB ρ α† ABi= Trhe−iεΛAB (A⊗B)eiεΛAB ρi. Baker–Campbell–Hausdorff gives Eε(a, b) = E0(a, b) + iε Tr[A⊗B, ΛAB]ρ−ε2 2Tr[[A⊗B, ΛAB],ΛAB]ρ+O(ε3). With the “ α↔α† flip” (Sec. 5.8) the O ( ε )term cancels, leaving a contrast reduction and angular–harmonic distortions at O(ε2): Eε(a, b) = E0(a, b)−ε2 2Ξab(ρ, ΛAB) + O(ε3),Ξab := Tr[[A⊗B, ΛAB],ΛAB]ρ.(32) Consequently Sε=S0−ε2 2Ξab + Ξab0+ Ξa0b−Ξa0b0+O(ε3), so fixed, unitary non - central αAB reduces the attainable |S| from its quantum optimum but cannot exceed 2 √2 (Tsirelson). Any experimental |S|> 2 √2 would therefore certify either (i) state/branch dependence or non - CP conditioning (effective nonlinearity), or (ii) a systematic error; cf. [ 63 – 65 , 67 ] and Part I’s discussion [21, 34].
22 8.4. Worked two-qubit example via the Horodecki criterion For a pure two-qubit state |ψθi= cos θ|00i+ sin θ|11i, the maximal CHSH value is [66] Smax(θ)=2p1 + sin22θ. Let a weak non - central associator act across AB as above. To O ( ε2 )the effect is captured by a contraction of the correlation matrix T7→ (I−ε2C)T+O(ε3)for a positive semidefinite C=C(ρ, ΛAB). Hence Smax(θ, ε)=2q(1 −ε2c1)+(1 −ε2c2) sin22θ+O(ε3)≤2√2, with c1,2≥ 0the leading eigenvalues of CTC . Central H 3 associators correspond to C = 0 and leave Smax unchanged. 8.5. Braiding (hexagon) in bipartite tests Anon-abelian braiding block βAB (Part I, Sec. 4) inserted coherently before measurement acts as Eex ε(a, b) = Tr(A⊗B)βAB ρ β† AB. If β = eiχ Σ(abelian), Eex differs from E0 only by a phase remapping of settings (LU–equivalent). If β = eiχeiεK Σwith non - central K , the expansion mirrors (32) with Λ AB 7→ K and the same O ( ε2 ) visibility loss. Again, fixed unitary β cannot breach Tsirelson; state/branch dependence is required for effective nonlinearity. 8.6. Tripartite tests: Mermin/MABK and associator tomography Associators naturally live at tripartite level. For three parties and dichotomic observables {A, A0} , {B, B0},{C, C0}, the Mermin polynomial is M=hABC0i+hAB0Ci+hA0BCi−hA0B0C0i. Local realism imposes |M| ≤ 2, while quantum mechanics attains |M| ≤ 4(GHZ with equatorial measurements) [68]. Central vs. non - central associators. Let αABC act between the three subsystems. If αABC = eiφI (central H 3 ), all correlators and hence M are unchanged (up to a global phase that cancels in expectation values). For a small non-central αABC =eiεΛABC with gauge Tr[ρΛABC] = 0, hABCiε=hABCi0−ε2 2Tr[[A⊗B⊗C, ΛABC ],ΛABC ]ρ+O(ε3), and likewise for the other terms, so the Mermin value reduces by O(ε2)but never exceeds the quantum bound 4. The same pattern holds for the n-party MABK polynomials [69]. 8.7. Protocol: higher-coherence tomography with Bell tests We propose a difference-of-differences routine robust to noise (cf. Sec. 5.8 and Sec. 7): 1. Baseline & LU calibration. Maximize S (or M ) on the prepared state without higher - coherence blocks; record the optimal settings. 2. Bracket/exchange toggles. Insert the associator α(or braiding β) in one arm/path and implement the two parenthesizations (for tripartite) or the swap vs. physical exchange (for bipartite). 3. α↔α†(and β↔β†) flips. Average the two runs to cancel all O ( ε )contributions; residual changes at O(ε2)are the non-central signature. Central H3leaves S(and M)unchanged. 4. Noise factorization. Use the visibility budgeting of Sec. 7 to factor out CP - noise ηnoise from true higher-coherence contrast changes.
23 8.8. What would a Tsirelson violation mean here? Within linear quantum mechanics, for any CPTP pre - processing (including any fixed unitary non - central α, β ), Bell correlators obey Tsirelson’s bound [ 65 ]. An observed |S|> 2 √2 or |M| beyond the quantum bound would therefore diagnose either (i) state/branch - dependent higher coherence that induces effective nonlinear dynamics upon conditioning (ruled by no - signalling constraints [ 21 , 34 ]), or (ii) a loophole/systems error. Our framework cleanly identifies the only diagrammatic locus where genuine departures could live: non - central higher coherence beyond fixed unitary natural isomorphisms (Part I, Secs. 3–4). 8.9. Assumptions and limits •Balanced sources and fair sampling; stationary noise during runs (Sec. 7). •Small higher-coherence strength ε1; central H3treated as scalar phases. • All unconditional maps are CPTP unless stated (post - selection and branch - dependent controls are explicitly flagged). • In the limits ε→ 0or central H 3 , S, M reduce to their calibrated quantum values (no change beyond LU reparametrization). Part II Phenomenology and Operational Tests Section 9. Experimental Platforms and Resource Estimates 9.1. Aim and summary Sections 5–8 provided operational tests (nested Mach–Zehnder, exchange/hexagon tomography, Bell tests) and the baseline GR phases (Sec. 6). Here we match these protocols to three families of platforms and derive sensitivity to the higher-coherence strength ε: 1. Integrated photonics (path/polarization/time-bin degrees of freedom). 2. Cold-atom and atom-interferometer platforms (external path, internal hyperfine states). 3. Superconducting circuits and trapped ions (gate-based realization of associator/braiding blocks). Two distinct figures of merit will be used: (Phase) (phase) min ≈k Γ(ωT)ξ√Mand (Contrast) (vis) min ≈ 2k σV V0Varψ(Λ)!1/2 , for a kσ detection in shot-noise-limited conditions. Here M is the total number of detected quanta/runs, ω the carrier angular frequency (photons) or the appropriate kinematic scale, T the interrogation time, ξ the dimensionless coupling defined in Sec. 6 (e.g. ∆Φ /c2 , Sagnac/curvature factors), Γ ∈ (0 , 1] a geometry factor, V0 the calibrated visibility (Sec. 5.2), Λthe Hermitian non-central generator (Sec. 5.3), and σV the rms error of the visibility estimate. In a shot-noise-limited fringe measurement, σV.c/√M with c∼O(1), hence (vis) min ≈2kc V0Varψ(Λ) 1/2M−1/4,(33) so contrast-based detection scales as M−1/4because the signal is quadratic in ε(Sec. 5.3).
24 9.2. Integrated photonics Architecture. Universal linear optics is available via multiport interferometers (Reck–Zeilinger meshes), with on-chip phase shifters and reconfigurable couplers [ 48 , 70 ]. Degrees of freedom ( X, Y, Z )can be encoded in path (dual-rail), polarization, and time-bin. Weak associator blocks αX,Y,Z = eiεΛ are realized with low-voltage electro-optic couplers that entangle polarization and time-bin or two spatial modes, while maintaining path balance. Exchange tomography (Sec. 5.5) is implemented by comparing the canonical swap Σwith a calibrated physical exchange (waveguide crossing + mode mixing). Phase (central H3 ) sensitivity. For purely kinematic central higher coherence ( ξ = 1), bright-beam homodyne or balanced detection can achieve σφ∼ 1 /√M ; with ( ωT ) ∼ 10 9 (optical ω and T∼µ s; Sec. 6), one finds (phase) min ∼k 109√M. As an illustration, M =10 12 detected photons and k =5 give (phase) min ∼ 5 × 10 −15 . For gravity-coupled central effects ( ξ 1; Sec. 6.2) the same formula applies with ξ (e.g. ξΦ∼ 10 −16 for h∼ 1m), so the required Mbecomes unrealistically large unless long Tor cavity enhancement is used. Contrast (non-central) sensitivity. With V0∼ 0 . 95 and a Pauli-like Λgiving Varψ (Λ) ∼ 1, Eq. (33) yields (vis) min ≈3.2M−1/4(k=5, c≈1, V0≈1). Examples: single-photon regime M =10 7⇒min ∼ 5 × 10 −2 (few-percent); bright-beam with true shot-noise performance M =10 12 ⇒min ∼ 3 × 10 −3 . Technical noise (phase drift, detector excess noise) will increase c; the difference-of-differences protocol (Sec. 5.8) suppresses such drifts. Practical notes. Thermal cross-talk and phase shifter 1 /f noise limit V0 ; active path-length stabilization and rapid α↔α† toggling are essential. Universal linear-optics chips demonstrated in [ 70 ] already supply the building blocks needed for L/Rparenthesizations and braiding tests. 9.3. Cold atoms and atom interferometers Architecture. Light-pulse atom interferometers (Kasevich–Chu) implement beamsplitters and mirrors via stimulated Raman/Bragg transitions [ 46 , 71 ]. Associate the three subsystems ( X, Y, Z )with external path (momentum states), internal hyperfine state, and an auxiliary momentum/time-bin. A weak associator between internal and external DoF can be engineered by off-resonant Raman couplings that imprint a small, controllable entangling phase; exchange tomography compares the canonical swap of two momentum classes with a physical exchange sequence. Phase (central) sensitivity. Atom interferometers routinely resolve phases .10−3rad per shot, with effective M set by detected atoms and cycle number [ 46 ]. For central, gravity-coupled higher coherence, use the ξ mapping from Sec. 6 (e.g. ξR or ξ(matter) Sag ). Long interrogation times ( T∼ 0 . 1–1s) and large areas boost ( ωT ) ξ analogues in the relevant sector (Sec. 6.3–6.5). The COW/Sagnac baselines [ 13 , 55 ] provide the natural calibration. Contrast (non-central) sensitivity. Shot-noise-limited contrast estimates with M∼ 10 6 total detected atoms give σV∼10−3, thus (for k=5,V0∼0.8,Var∼1) (vis) min ≈r2k σV V0∼0.11. Improving duty cycle (large-momentum-transfer beamsplitters, concurrent interferometers) and operating at the optimal bias phase (quadrature) reduces σV ; nevertheless, contrast-based non-central detection in atoms typically targets the 10 −1 level unless very long integrations and low technical noise are achieved. Practical notes. Gravity gradients, Coriolis terms, and magnetic systematics can mimic small visibility changes. The Sec. 5.8 “triangle lock” and “associator flip” are essential. Established gravimetry/rotation metrology (e.g. [72]) provides noise models and performance benchmarks.
25 9.4. Superconducting circuits and trapped ions Architecture. In gate platforms the associator α = eiεΛ is realized directly as a weak entangling unitary across three registers ( X, Y, Z ), e.g. a calibrated sequence of ZZ or ZX couplings (superconducting transmons) or Mølmer–Sørensen interactions (ions). The two parenthesizations L vs. R are different gate orderings wrapped by a tunable interferometer (Ramsey/echo framing). Non-abelian braiding analogues are implemented through controlled SWAPs plus entangling phases. Phase (central) sensitivity. Gate-based interferometers (Ramsey) achieve phase resolutions ∼ 10 −3 –10 −2 rad per shot with M∼ 10 4 –10 6 repetitions / s [ 73 – 75 ]. For engineered central higher coherence (ξ=1), Eq.(Phase) yields (phase) min ∼k Γ(ωT)√M. Here ωT is set by the Ramsey evolution window (not an optical frequency); with T∼ 10–100 µ s and ω∼ 1 /T , one has ( ωT ) ∼ 1, so the sensitivity is governed by 1 /√M and the geometry factor Γ. Multi-round phase amplification (echo sequences) can boost Γ. Contrast (non-central) sensitivity. With V0∼ 0 . 9and M∼ 10 6 shots, σV∼ 10 −3 , giving (vis) min ∼ 0 . 05 via Eq. (33) . Because Λcan be chosen as a Pauli product with Var∼ 1on balanced inputs, gate platforms are well suited for proof-of-principle non-central tests at the few% level, with error mitigation and randomized compiling reducing coherent drifts. Practical notes. Leakage, crosstalk, and SPAM errors reduce V0 . Embedding the two bracketings in a single echo circuit cancels low-frequency drift; the α↔α† toggle is implemented by reversing the sign of calibrated couplings. 9.5. Cross-platform comparison and design rules •Central (phase) vs non-central (contrast). Central H3 effects are best sought where ( ωT ) ξ is large and phase noise is low (photonic bright beams, longT atoms). Non-central effects are best sought where precise, weak entanglers are available and M can be made large (integrated photonics; gate platforms). •Scaling. Phase detection improves as M−1/2 ; contrast detection as M−1/4 . Thus, whenever possible, engineer central tests (phase-only) rather than contrast tests. •Maximize Varψ (Λ) . Choose balanced inputs so that Λhas large variance (Pauli-type generators on equally populated subspaces). •Difference-of-differences. Always combine: (i) triangle lock (Sec. 5.8), (ii) associator flip α↔α† , (iii) exchange lock Σ↔β, to cancel drifts and isolate even-in-εcontrast changes. 9.6. Risk register and mitigations Drifts and 1/f noise: Mitigate with rapid toggling, interleaved calibration shots, and echo-style embeddings of L/Rbranches. Mode mismatch (photonic) / wavefront aberrations (atoms): Use active mode matching, adaptive optics, and balanced detection. Systematics mimicking contrast loss: Use the α† symmetry (contrast change is even in ε ) and swap L↔Rto verify the sign/geometry dependence predicted by Sec. 5.3. Finite coherence times (gates): Keep associator depth short; use dynamical decoupling; estimate and subtract SPAM baselines.
32 Section 12. Higher-Categorical Frontiers: Open Problems, Conjectures, and Research Program Motivation and Scope Having established that triangle-level coherence breakdowns are always central and hence compatible with strict unitarity, and that pentagon/hexagon anomalies ( H3 data) are the only locus for effective departures, it is natural to articulate a forward-looking program. This section collects mathematical conjectures, open structural questions, and an experimental roadmap designed to test the boundaries of unitarity in both laboratory and gravitational settings. The aim is to situate our results not as a closed classification, but as the beginning of a systematic research agenda. Mathematical Conjectures and Structural Questions Several outstanding issues arise at the interface of higher category theory, groupoid cohomology, and operational quantum mechanics: •Bicategorical strictification. For monoidal categories with only central H2 anomalies, MacLane’s coherence theorem ensures strictification. In bicategorical or tricategorical settings with non-central H3 , it remains conjectural whether all associators can be realized as coboundary deformations of quasi-Hopf algebras or whether there exist genuinely wild classes. •Classification of non-central associators. A systematic taxonomy of non-central 3-cocycles, particularly those not gauge-equivalent to central representatives, is required to determine which anomalies admit operational detection and which are “silent.” •Gerbe uniqueness and Dixmier–Douady refinements. The correspondence between H3 classes and U (1)-bundle gerbes may admit multiple inequivalent 2-connection realizations. Establishing uniqueness criteria or classifying inequivalences is critical for the geometric dictionary. •Interplay with contextuality sheaves. While we identified diagrammatic failures with contextual obstructions, it remains open whether higher obstructions ( H3 non-centrality) admit a sheaf-theoretic witness analogous to the Mermin–Peres magic square. Proposition III.12.1 Conjectural Strictification Bound Every non-central 3-cocycle class [ φ ] ∈H3 ( C, U (1)) arising from a groupoid of contexts can be realized as the associator of a quasi-Hopf algebra ( H, ∆ , Φ) such that Φencodes φ . Moreover, the associator is strictly centralizable if and only if [ φ ]lies in the image of the inflation map from H3 ( G, U (1)) for some global symmetry group G . This conjecture would, if proven, bound the space of genuinely non-central anomalies and clarify which ones are operationally accessible. Conditions for Effective Nonlinearity Our framework isolates precise conditions under which effective nonlinearity can arise: 1. State/context dependence. If a non-central associator αX,Y,Z acts differently across subspaces, then post-selection or conditioning may yield apparent nonlinear dynamics. 2. Discarding maps. Tracing out subsystems in the presence of non-central coherence defects can convert linear unitary operations into effective non-CPTP maps. 3. Interference visibility limits. When background central phases are canceled by symmetry toggles, residual non-central contrast loss may accumulate, mimicking nonlinear evolution in reduced data. A rigorous classification of which non-central anomalies preserve no-signalling versus those that threaten causal consistency is an open task.
33 Experimental Roadmap and Milestones The protocols introduced earlier suggest an incremental experimental program: Stage I: Optical and Microwave Interferometry. Nested Mach–Zehnder setups with controllable bracketings allow immediate tests of central versus non-central associators. Visibility budgeting protocols provide the discriminators. Stage II: Exchange and Hexagon Tomography. Platforms with anyonic or synthetic braiding (e.g., Rydberg arrays, topological superconductors) enable detection of non-abelian braiding anomalies. Here, the “hexagon tomography” provides the critical observable. Stage III: Gravitational Baselines. Ring-laser gyroscopes and atom interferometers with kilometerscale baselines can use Earth’s Sagnac and redshift effects as calibration signals, enabling differential cancellation and isolation of higher coherence anomalies. Stage IV: Quantum Networks and Bell Tests. Extending toggle protocols to entangled photon or atom networks enables probing whether non-central coherence defects alter visibility in Bell tests. This would be the sharpest test of effective nonlinearity consistent with Tsirelson bounds. Broader Outlook The program articulated here emphasizes a sharp dichotomy: •Triangle-level coherence guarantees stability, unitarity, and linearity. •Pentagon and hexagon anomalies (H3) are the sole locus for effective deviations. By providing both a mathematical roadmap (classification, strictification, gerbes) and an experimental roadmap (interferometry, braiding, gravity, networks), Section A establishes a coherent research program. The central open question is whether nature realizes non-central higher coherence anomalies, and if so, whether they manifest as effective nonlinearities at observable scales. Part III Conceptual Bridges and Implications Section 13. Outlook: Open Problems, Estimation Theory, and Experimental Roadmap 13.1. Synthesis of contributions We have argued and substantiated that: 1. Triangle stability. Any triangle-level (2-simplex) coherence defect is classified by a central U (1)-valued 2-cocycle; after passing to the central extension, the theory is strictly coherent and unitary/linear (Parts I.1–I.2). 2. Higher coherence dichotomy. Pentagon/hexagon (3-simplex) anomalies split into central ( H3 phases; unitary-preserving) and non-central (operator-valued) cases. Only the latter can yield effective nonlinearity/non-CP updates upon conditioning/discarding (Parts I.3–I.4, III.12). 3. Operational fingerprints. Central higher coherence produces phase-only shifts; non-central higher coherence produces contrast changes at O(ε2)and harmonic distortions (Parts II.5–II.8). 4. Weak-gravity baselines. Redshift, (gravitational/kinematic) Sagnac, and tidal curvatures superpose linearly as central (phase-only) backgrounds that can be locked and subtracted (Part II.6). 5. Bounds and dynamics. No-signalling enforces convex linearity of unconditional dynamics; under Markovianity this implies a GKSL generator. In the weak repeated-interaction limit, non-central higher coherence generates a GKSL dissipator at O ( ε2 ), quantitatively matching the observed contrast law (Part III.12). 13.2. Mathematically precise open problems We list problems whose resolution would sharpen—mathematically and operationally—the program advanced here.
34 (A) Central strictification in bicategorical generality. Problem .13 (Bicategorical strictification by central extension).Let S: C → Hilb be a dagger pseudofunctor from a small groupoid C into Hilb with unitary composition constraints and associator α . Prove the following bicategorical refinement: If all triangle coherence constraints are central and α is central up to a U (1) 3-cocycle, then there exists a central 2-group extension b C and a monoidal equivalence turning Sinto a strict dagger 2-functor on b C. Relate the obstruction classes to H2 ( C, U (1)) and H3 ( C, U (1)) via higher nerve/Segal-space models; clarify hypotheses (local Schur property, separability) ensuring centrality forces scalar coherence. (A potential route uses the (∞,1)-categorical viewpoint [86].) (B) Classification of non-central associators under dagger positivity. Problem .14 (Dagger-positivity constraints).Classify unitary non-central αX,Y,Z compatible with: (i) dagger structure, (ii) complete positivity under partial trace on ancillary factors, and (iii) hexagon constraints (if braided). Are there intrinsic positivity-induced obstructions forcing α to be central in certain families (e.g. symmetric monoidal Hilb with superselection rules)? (C) Gerbe quantization of central H3on path groupoids. Problem .15 (Differential refinement).For C = Π 1 ( M ), construct a differential cohomology model realizing central [ φ ] ∈H3 (Π 1 ( M ) , U (1)) as a U (1)-bundle gerbe with connection ( A, B )on M and prove that the associator phase equals the 2-holonomy of B on tetrahedra. Determine integrality conditions and the precise relation to the Dixmier–Douady class (cf. Part III.11). (D) Identifiability and uniqueness of higher-coherence tomography. Problem .16 (Identifiability under toggles).Prove that the pair of toggles (L↔R)and (α↔α†) renders the decomposition into odd-inε (central phase) and even-inε (non-central contrast) parts unique up to O ( ε3 )in nested MZI data, given nondegenerate states and probe POVMs. Extend to exchange tomography (Σ ↔β)with hexagon constraints. (E) Bounds in diamond norm for higher-coherence channels. Problem .17 (Channel distance bounds).Let Φ ε be the channel induced by inserting α = exp ( iε Λ) on system + ancilla and tracing ancilla. Prove second-order bounds kΦε−idk≤C ε2kΛk2+O(ε3) with dimension-independent C , and characterize the cases saturating the bound. Connect the bound to visibility loss constants in Sec. II.5 and to GKSL rates (Part III.12). See [85] for background. (F) Contextual sheaf obstructions vs. groupoid cohomology. Problem .18 (Čech vs. groupoid cohomology).Make precise the comparison map between the obstruction cohomology of a measurement cover (sheaf contextuality) and groupoid cohomology of contexts, identifying when phase-only (triangle) obstructions coincide and when higher (3-simplex) obstructions differ (cf. Part III.11). 13.3. Estimation theory for ε: phase vs. contrast channels We collect compact bounds for parameter estimation of the higher-coherence strength ε. (i) Central (phase-only) case. A nested MZI implementing α = eiεφI yields output intensities P± ( ϕ ) = 1 2 1 ±V0cos ( ϕ + ε Φ) with Φ = Γ( ωT ) ξ (Sec. II.5). The classical Fisher information (per shot) for ε , at the optimal working point (ϕ+εΦ) = π 2, reads Icl(ε) = V2 0Φ2, so for Mindependent shots the Cramér–Rao bound gives Var(bε)≥1 M V 2 0Φ2⇒σbε≥1 V0Γ(ωT)ξ√M,(36) matching the scaling used in Sec. II.9 (cf. [84]).
35 (ii) Non-central (contrast) case. With α = eiεΛ gauged so that tr ( ρ Λ) = 0, the fringe visibility obeys (Sec. II.5) V(ε) = V01−1 2ε2Varψ(Λ)+O(ε3). Assuming shot-noise-limited visibility estimation with per-shot variance σ2 V∝ 1 /M near the working point, standard error propagation yields σbε&2σV V0Varψ(Λ)1/2∝M−1/4,(37) again reproducing Sec. II.9. The M−1/4 law is fundamental for quadratic signals; thus central tests are metrologically superior whenever accessible. (iii) Joint estimation. If both a central phase and a non-central contrast are present, joint estimation with two toggles ( L/R )and ( α/α† )yields a block-diagonal Fisher matrix to O ( ε2 )(odd vs. even in ε ), with variances saturated by alternating operating points (phase quadrature and maximal-visibility bias). 13.4. Channel-norm control and tomography guarantees Let Φ ε be the reduced channel induced by inserting α = exp ( iε Λ) on S⊗E and tracing E (Part III.12). Expanding to O(ε2)gives Φε(ρ) = ρ+iε[Heff, ρ] + ε2D(ρ) + O(ε3), with GKSL dissipator D and Heff as in Prop. III.12.1. Using standard inequalities for completely bounded norms (e.g. [85, Ch. 3]), one obtains kΦε−idk≤C1εk[Heff,·]k+C2ε2kDk+O(ε3),(38) with absolute constants C1, C2 . In phase-locked operation (triangle lock), the commutator part cancels in the interference channel and the O ( ε2 )piece dominates—exactly the visibility-loss mechanism of Sec. II.5. 13.5. Experimental roadmap (milestones and falsifiable targets) Near term (tabletop photonics / gate devices). • Goal: Demonstrate associator tomography with L↔R and α↔α† toggles; resolve even-inε contrast at the few-% level; null test: no change under triangle-only reconfigurations. •Metric: Extract bεvia (37) and report diamond-norm upper bounds using (38). Mid term (long-Tatoms / cavity photonics). • Goal: Measure central higher-coherence phases on top of redshift/Sagnac baselines with triangle lock; reach σbε.10−9in purely kinematic runs using (36). • Metric: Report ( ωT ) ξ and geometry Γ; demonstrate phase-only behavior (no contrast change to within uncertainties). Bell layer (all platforms). • Goal: Verify invariance of Tsirelson-limited optima under central H3 insertions; detect O ( ε2 ) distortions under non-central insertions without any super-Tsirelson effect (Secs. II.8, III.12). • Metric: Confidence intervals for Smax / Mmax ; bounds on non-central generators compatible with GKSL rates.
36 13.6. Falsifiability and risk register •Null tests. (i) Triangle-only rewirings must not change visibility; (ii) central H3 must not change visibility; (iii) non-central effects must be even under α→α† and vanish when [ A, B ]=[ B, C ] = [C, A]=0(Sec. II.5). •Bell/CHSH sanity. No super-Tsirelson outcomes should appear for any fixed (state-independent) coherence blocks; otherwise, flag conditioning/non-CP artifacts (Sec. II.8, III.12). •Systematic mimics. Phase drift, mode mismatch, and detector nonlinearities should be rejected by the symmetry toggles and geometry dependence (Sec. II.10.7). 13.7. Beyond the present scope Three directions suggest themselves: 1. Indefinite causal order. Our analysis assumes a well-defined diagram order. Extending highercoherence tomography to process-matrix/quantum-comb frameworks would clarify whether certain non-central anomalies mimic indefinite causal features (see e.g. [87]). 2. Higher-categorical quantization. Replace Hilb by higher linear categories; study when dagger positivity forces centralization of higher coherence (cf. Problem A; [86]). 3. Topological/condensed-matter realizations. Non-abelian braidings in modular tensor categories furnish natural laboratories for exchange tomography beyond phases; mapping our witnesses to categorical S, T-data is an attractive target. 13.8. Concluding perspective Coherence is the common skeleton uniting quantum mechanics and general relativity: first-order (triangle) coherence enforces the familiar, unitary backbone; higher (pentagon/hexagon) coherence is the unique, structured portal to new phenomena. Our program delineates that boundary sharply, provides the metrology to test it, and integrates the mathematics (cohomology, gerbes, quasi-Hopf) with operational diagnostics. Whatever the outcome of the proposed experiments, the message remains: if new physics hides in plain sight, it must respect triangle stability and reveal itself at higher coherence. Appendix A: Groupoid Cohomology, Central Extensions, and Higher Associators 1. Nerve, cochains, and coboundary Let Cbe a small groupoid. Write Cn:= {(g1, . . . , gn)|gicomposable in order} and let U (1) be a trivial C -module. A (normalized) n -cochain is a map c : Cn→U (1) that equals 1 whenever any gi= id. The coboundary δ:Cn(C, U(1)) →Cn+1(C, U(1)) is (Eilenberg–Mac Lane) (δc)(g1, . . . , gn+1) = c(g2, . . . , gn+1) n Y j=1 c(g1, . . . , gjgj+1, . . . , gn+1)(−1)jc(g1, . . . , gn)(−1)n+1 . Then Zn= ker δ,Bn= im δ, and Hn=Zn/Bn.
37 2. Triangle level: H2and central extensions Given a normalized 2-cocycle ω∈Z2(C, U(1)), ω(h, k)ω(g, hk) = ω(gh, k)ω(g, h), define the central U(1)-extension b Cwith the same objects and morphisms Homb C(A, B) = {(g, z)|g∈HomC(A, B), z ∈U(1)} and composition ( g, z ) ◦ ( h, w ) = ( gh, zw ω ( g, h )). Equivalence classes of extensions correspond bijectively to H2(C, U(1)); changing ω7→ ω δχ by a 1-cochain χgives an isomorphic extension. Strictification at triangle level. If S: C → Hilb is a dagger pseudofunctor with composition constraints φg,h =ω(g, h)I, then bS(g, z) := zS(g) is a strict dagger functor b C → Hilb absorbing all triangle defects; see Part I, Sec. 2. 3. Pentagon level: H3and central associators Acentral associator is αX,Y,Z =eiφ(X,Y,Z)I. Mac Lane’s pentagon reduces to (δφ)(W, X, Y, Z)=0, hence [ φ ] ∈H3 ( C, U (1)). A 2-cochain re - gauge θ modifies φ7→ φ + δθ , so only the class matters. If a braiding is present and central, the two hexagon identities impose the familiar abelian compatibility between [φ]and a bicharacter (Part I, Sec. 4). 4. Examples Vector groups. For a finite-dimensional real vector group V∼ =Rn(continuous cohomology), H2(V, U(1)) ∼ =∧2V , H3(V, U(1)) ∼ =∧3V , via ω(u, v) = ei 2σ(u,v)with σ∈ ∧2Vand central 3-cocycles ei τ(u,v,w)with τ∈ ∧3V. Path groupoids. For C = Π 1 ( M ), smooth U (1)-valued 2-cocycles arise by integrating a closed 2-form with integral periods; H2 (Π 1 ( M ) , U (1)) matches H2 ( M, U (1)). Central H3 corresponds to U (1)-bundle gerbes with Dixmier–Douady class in H3(M, Z)(Part III, Sec. 11). Appendix B: Baker–Campbell–Hausdorff, Magnus Expansion, and the Triple Commutator 1. BCH to cubic order and parenthesization For X, Y in a Lie algebra, write BCH(X, Y ) = X+Y+1 2[X, Y ] + 1 12 [X, [X, Y ]] + 1 12 [Y, [Y, X]] + O(4), where O(4) collects terms of degree ≥4in (X, Y ). Define ZL:= BCH(BCH(A, B), C), ZR:= BCH(A, BCH(B, C)). Although eZL = eAeBeC = eZR , the series ZL and ZR differ; the difference is a Lie series whose first nonzero term is cubic. Lemma B.1 (Associator at cubic order).Up to degree 3, ZL−ZR=1 6[A, [B, C]] + [B, [C, A]] + [C, [A, B]].(B1) Sketch. Compute ZL and ZR using the BCH expansion twice. Keep terms up to degree 3and use the Jacobi identity to simplify. Terms of degree 1and 2cancel; at degree 3the cyclic sum (B1) remains. (See also Dynkin’s form of BCH.)
38 2. Operational consequence: interferometric visibility In Sec. 5 we insert a small non-central associator on the left arm: α=eiεΛ, ε 1,Λ=Λ†, and compare L = (( UgUh ) Uk )vs. R = ( Ug ( UhUk )) with Ug = eiA etc. After gauging the scalar part so that hΛiψ= 0, the interference term I(ϕ) = Reeiϕhψ|R†αL|ψ|) expands to I(ϕ) = Reei(ϕ+φ)hψ|R†L|ψ|E)−ε2 2Varψ(Λ) cos(ϕ+φ) + O(ε3), i.e. a contrast loss at O ( ε2 ). The generator Λcan be engineered perturbatively to scale with the BCH associator (B1), yielding the design rule Λ∝JA, B, CK:= 1 3[A, [B, C]] + [B, [C, A]] + [C, [A, B]]. 3. Magnus viewpoint (time-ordered control) For a piecewise constant control H ( t ) ∈ {A, B, C} , the propagator U ( T ) = Texp ( −iRT 0H ( t ) dt )has log Ugiven by the Magnus series Ω1=ZH, Ω2=1 2Zt1>t2 [H(t1), H(t2)],Ω3=1 6Zt1>t2>t3 [H(t1),[H(t2), H(t3)]] + cyc., so changing the nesting order of ( A, B, C )changes Ω 3 by the cyclic triple commutator—again giving (B1). Appendix C: From Microscopic Insertions to GKSL: Detailed Weak-Coupling Derivation 1. Second-order expansion and complete positivity Let S be the system and E an ancilla/environment initially in state σ . Insert a small non-central block α=eiεΛacting on S⊗Ewith Λ=Λ†. Then Φε(ρ) := TrEα(ρ⊗σ)α†=ρ+iε[Heff , ρ]−ε2 2TrE[Λ,[Λ, ρ ⊗σ]]+O(ε3), where Heff = TrE(σΛ). If Λ = PµAµ⊗Bµ, TrE[Λ,[Λ, ρ ⊗σ]]=X µ,ν GµνA† µAνρ+ρA† µAν−2AνρA† µ, with Kossakowski matrix Gµν = TrE ( σB† µBν ) ≥ 0(Gram matrix), hence the second-order piece is a GKSL dissipator. 2. Repeated interactions and coarse graining Apply ΦεNtimes with time step ∆t, keeping Γ := ε2/∆tfixed as ε→0. Then ρn+1 −ρn ∆t=i ∆tε[Heff, ρn]+ΓD(ρn) + o(1).
39 If we triangle - lock the phase (or remove it by toggling), the commutator term vanishes in the observed channel and the coarse-grained generator is purely dissipative: ˙ρt=L(ρt)≡X µ,ν ΓGµνAνρtA† µ−1 2{A† µAν, ρt}. This is the rigorous underpinning of the O(ε2)visibility law in Sec. 5. Appendix D: Interferometric Amplitude with CPTP Noise: General Formulae and Channels 1. General fringe amplitude In a Mach–Zehnder with reference unitary U on one arm and a CPTP map E ( ρ ) = PjKjρK† j on the other, the complex fringe amplitude is A= TrU†E(ρ)=X j TrU†KjρK† j= Tre U†ρ,e U:= X j K† jUKj.(D1) The phase of Ais the measured shift; the visibility is V=|A|. 2. Canonical single-qubit channels (worked) Let ρ=|ψihψ|with |ψi=1 √2(|0i+|1i). Dephasing (phase-flip) with parameter γ∈[0,1/2].Use Kraus K0=√1−γ I,K1=√γ Z. Then E(ρ) = (1 −γ)ρ+γ ZρZ, e U= (1 −γ)U+γ ZUZ. If U=I, then A= 1 −2γand V=|1−2γ|. Depolarizing with probability p∈[0,1].E(ρ) = (1 −p)ρ+p 3Pi=x,y,z σiρσi, so e U= (1 −p)U+p 3X i σiUσi. If U=Iand ρ=|+ih+|, then V=|1−4 3p|. Amplitude damping with parameter γ∈ [0 , 1]. K0 = |0ih0| + √1−γ|1ih1| , K1 = √γ|0ih1| . For U = I and ρ=|+ih+|, one finds V=√1−γ(off-diagonal suppression). Higher - coherence insertion. If a central associator multiplies the noisy arm by eiφ , then A 7→ eiφA : phase-only. If a small non - central block eiεΛ is inserted and traced over, the visibility picks up the O ( ε2 ) loss derived in App. C. Appendix E: Matrix Models for MZI, Nested MZI, and Exchange Tomography 1. Basic MZI Let B = 1 √21i i1 , a balanced beam splitter on the path qubit {|ui,|`i} . Let Φ( ϕ ) = diag (1 , eiϕ )be a phase shifter on the lower arm. For input |ini=|ui|ψi, |outi=B|ui|ψi+|`ieiϕ |ψi/√2, and the two output port probabilities are P± ( ϕ ) = 1 2 1 ±cos ( ϕ + φ0 ) V0 , where V0 and φ0 absorb triangle-level phases.
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