arxiv: 2603.21281 · v2 · submitted 2026-03-22 · 🪐 quant-ph · cond-mat.other

Gauge-Invariant Non-Hermitian Quantum Theory: Foundation and Applications to Dynamical Phase Transitions

Fei Wang , Guoying Liang , Zecheng Zhao , Bao-Ming Xu This is my paper

Pith reviewed 2026-05-15 07:00 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.other

keywords non-Hermitian quantum theorygauge invariancebiorthogonal formalismdynamical phase transitionsSSH modelopen quantum systemswinding number

0 comments p. Extension

The pith

A gauge-invariant non-Hermitian quantum theory is established by requiring both left and right eigenvectors to satisfy the Schrödinger equation.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper develops a consistent theoretical framework for non-Hermitian quantum systems by starting from two key premises about dynamics and gauge invariance. The resulting theory refines the biorthogonal approach so that physical quantities remain independent of arbitrary phase choices. It includes open-system non-Hermitian evolution as a special case and recovers ordinary quantum mechanics when the Hamiltonian is Hermitian. When applied to dynamical phase transitions in the Su-Schrieffer-Heeger model, the framework produces a generalized transition condition involving the real part of a normalized dot product of the initial and final vectors. It further reveals entirely new transition phenomena that escape detection by the winding number alone.

Core claim

The central discovery is a gauge-invariant formulation of non-Hermitian quantum mechanics obtained by enforcing Schrödinger evolution on both left and right eigenvectors while preserving gauge freedom. This framework naturally generalizes the Hermitian condition d^i_k · d^f_k = 0 for dynamical phase transitions to the non-Hermitian form Re[(d^i_k / d^i_k) · (d^f_k / d^f_k)] = 0 and uncovers additional transitions not captured by topological winding numbers.

What carries the argument

The refined biorthogonal framework with gauge invariance, which ensures that both left and right vectors obey the time-dependent Schrödinger equation independently of phase choices.

If this is right

The open-system effective non-Hermitian evolution emerges as a special case of the general theory.
The theory reduces exactly to standard Hermitian quantum mechanics in the appropriate limit.
Dynamical phase transitions satisfy the condition Re[(d_k^i / d_k^i) · (d_k^f / d_k^f)] = 0.
New dynamical phase transitions appear that cannot be characterized using the winding number.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This approach may help settle ongoing debates about the proper description of states and observables in non-Hermitian physics.
Similar gauge-invariant treatments could be tested in other non-Hermitian models, such as those with gain and loss.
The identification of winding-number-independent transitions suggests that topological invariants may need refinement for non-Hermitian dynamics.

Load-bearing premise

Both the left and right eigenvectors of a non-Hermitian Hamiltonian must satisfy the Schrödinger equation while the overall framework remains invariant under phase redefinitions of these vectors.

What would settle it

An experiment or numerical simulation on a non-Hermitian system showing that measurable quantities depend on the choice of phase for the left or right eigenvectors, or a dynamical phase transition occurring when the real part of the normalized dot product is nonzero.

Figures

Figures reproduced from arXiv: 2603.21281 by Bao-Ming Xu, Fei Wang, Guoying Liang, Zecheng Zhao.

**Figure 3.** Figure 3: FIG. 3. (Color online) Lines of Fisher zeros (a), the time [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. (Color online) Lines of Fisher zeros (a), the time [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

read the original abstract

The description of states and dynamics in non-Hermitian systems is fundamentally linked to the choice of an appropriate theoretical framework -- a point of ongoing debate in the field. This work addresses this issue by proposing a consistent formulation that reconciles existing controversies and establishes a unified theoretical understanding. Our approach rests on two foundational premises: (i) the dynamics of both left and right-vectors of a non-Hermitian system must satisfy the Schr\"{o}dinger equation; (ii) the theoretical framework must preserve gauge invariance, ensuring that physical quantities are independent of unobservable phase choices. Building on these physically motivated assumptions, we refine the biorthogonal framework, leading to a gauge-invariant non-Hermitian quantum theory. Our framework naturally encompasses the open-system effective non-Hermitian evolution as a special case, and can naturally reduce to standard quantum mechanics in the Hermitian limit. As a concrete application, we analyze the dynamical phase transition in a one-dimensional Su-Schrieffer-Heeger (SSH) model within this gauge-invariant non-Hermitian quantum theory. Notably, our formulation naturally generalizes the known condition for such transitions in Hermitian two-band systems, namely, $\mathbf{d}_{k}^i\cdot\mathbf{d}_{k}^f=0$, to the non-Hermitian case, where it takes the form $\mathrm{Re}\Bigl[\frac{\mathbf{d}_{k}^i}{d_{k}^i}\cdot\frac{\mathbf{d}_{k}^f}{d_{k}^f}\Bigr]=0$. Furthermore, we identify entirely new dynamical phase transitions that cannot be characterized by the winding number. We hope that this gauge-invariant non-Hermitian quantum theory will find broad applications in the study of non-Hermitian systems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper builds a gauge-invariant non-Hermitian framework from the premise that left and right vectors obey the same Schrödinger equation, but this premise conflicts with standard evolution rules.

read the letter

The main takeaway is that this work tries to resolve ambiguities in non-Hermitian quantum mechanics by enforcing gauge invariance and requiring that both left and right vectors satisfy the Schrödinger equation i∂tψ = Hψ. Using this, they generalize the Hermitian dynamical phase transition condition d^i · d^f =0 to the non-Hermitian form Re[(d^i / d^i) · (d^f / d^f)] =0 and find new types of transitions in the SSH model that are independent of the winding number. The paper does a solid job of stating its two premises upfront and showing how the framework recovers standard quantum mechanics in the Hermitian limit while treating open-system effective non-Hermitian evolution as a special case. The concrete application to the one-dimensional SSH model produces specific new transition points that the authors say are absent from the literature they cite, which gives the claim some substance. The soft spot is the foundational assumption about the dynamics. Standard non-Hermitian evolution keeps the biorthogonal inner product constant by having the left vectors evolve under the adjoint H† rather than H itself. Imposing the same equation on both sides appears to break that unless the Hamiltonian is Hermitian. The paper presents this as a physically motivated premise, but without seeing how they derive consistent time evolution or verify it against known open-system examples, it is unclear whether the construction holds together. The generalized transition condition is a direct consequence of their setup, but the abstract does not include error analysis or explicit checks, so the strength of the generalization depends on the full derivations. This paper targets theorists working on non-Hermitian systems in condensed matter physics and quantum optics, particularly those interested in dynamical phase transitions. A reader familiar with biorthogonal bases will see the refinements as worth examining. The work is internally consistent given its premises and makes definite predictions about new transitions, so it deserves a serious referee who can verify the math and any supporting calculations. I recommend sending it to peer review.

Referee Report

1 major / 2 minor

Summary. The paper proposes a gauge-invariant non-Hermitian quantum theory resting on two premises: (i) both left and right vectors obey the Schrödinger equation i∂tψ=Hψ, and (ii) the framework preserves gauge invariance. Building on these, it refines the biorthogonal formalism, shows it encompasses open-system non-Hermitian evolution and reduces to standard QM when H is Hermitian, and applies the framework to dynamical phase transitions in a non-Hermitian SSH model, generalizing the Hermitian condition d_k^i · d_k^f =0 to Re[(d_k^i / d_k^i) · (d_k^f / d_k^f)]=0 while identifying new transitions not captured by the winding number.

Significance. If the premises can be shown consistent with standard non-Hermitian evolution, the work would supply a unified framework that treats open-system effective Hamiltonians as a special case and yields falsifiable predictions for new dynamical phase transitions in non-Hermitian two-band models. The absence of an explicit derivation of the generalized transition condition from the stated axioms, however, limits the immediate impact until the foundational consistency is demonstrated.

major comments (1)

[Abstract, premise (i)] Abstract, premise (i): The requirement that both left and right vectors satisfy the identical Schrödinger equation i∂tψ=Hψ for non-Hermitian H is load-bearing for the gauge-invariant refinement and the claimed generalization Re[(d_k^i/d_k^i)·(d_k^f/d_k^f)]=0. Standard non-Hermitian dynamics evolves right states under H and left bras under H† to keep ⟨L(t)|R(t)⟩=1 invariant; the manuscript must derive explicitly how the shared equation preserves biorthogonality and gauge invariance without reducing to the Hermitian case or introducing hidden fitting parameters.

minor comments (2)

[Abstract] The notation d_k^i / d_k^i in the generalized transition condition is ambiguous (vector divided by scalar) and should be defined explicitly, preferably with an equation number, as a normalized vector or equivalent.
[Abstract] The abstract states that the framework 'naturally generalizes' the Hermitian condition and identifies 'entirely new' transitions, but provides no derivation steps or verification; the full manuscript should include these in a dedicated section with explicit algebra from the two premises.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading and constructive feedback on our manuscript. We appreciate the acknowledgment of the potential significance of the gauge-invariant non-Hermitian framework. We address the major comment below and will revise the manuscript accordingly to strengthen the presentation of the foundational premises.

read point-by-point responses

Referee: [Abstract, premise (i)] Abstract, premise (i): The requirement that both left and right vectors satisfy the identical Schrödinger equation i∂tψ=Hψ for non-Hermitian H is load-bearing for the gauge-invariant refinement and the claimed generalization Re[(d_k^i/d_k^i)·(d_k^f/d_k^f)]=0. Standard non-Hermitian dynamics evolves right states under H and left bras under H† to keep ⟨L(t)|R(t)⟩=1 invariant; the manuscript must derive explicitly how the shared equation preserves biorthogonality and gauge invariance without reducing to the Hermitian case or introducing hidden fitting parameters.

Authors: We agree that an explicit derivation of the time evolution under the shared Schrödinger equation is required to fully establish consistency. In the revised manuscript we will add a dedicated subsection that starts from the two premises and derives the preservation of the biorthogonal inner product ⟨L(t)|R(t)⟩=1. The derivation proceeds by imposing a time-dependent gauge choice on the left and right vectors that absorbs the non-Hermitian contribution while keeping the inner product real and unity; this choice is uniquely fixed by the gauge-invariance requirement and introduces no free parameters. We will explicitly show that the resulting dynamics reduces to ordinary unitary evolution when H is Hermitian and recovers the standard open-system effective-Hamiltonian evolution as the special case in which the left-vector evolution is projected onto the adjoint action. The generalized dynamical-phase-transition condition then follows directly from the same gauge-invariant normalization without additional assumptions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation rests on explicit external assumptions

full rationale

The paper states two foundational premises upfront—both left and right vectors obey the Schrödinger equation, and the framework must preserve gauge invariance—then refines the biorthogonal setup and derives the generalized transition condition Re[(d_k^i / d_k^i) · (d_k^f / d_k^f)] = 0 as a consequence when applied to the SSH model. No step reduces by construction to these inputs (the normalized dot-product condition follows from enforcing gauge invariance on the dynamics rather than redefining it), no load-bearing self-citations appear, and no fitted parameters are relabeled as predictions. The framework is self-contained against the stated assumptions and yields new claims (dynamical transitions outside winding-number characterization) that are not tautological.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on two domain assumptions about eigenvector dynamics and gauge invariance; no free parameters or new entities are introduced in the abstract.

axioms (2)

domain assumption The dynamics of both left and right-vectors of a non-Hermitian system must satisfy the Schrödinger equation
Explicitly listed as foundational premise (i) in the abstract.
domain assumption The theoretical framework must preserve gauge invariance, ensuring that physical quantities are independent of unobservable phase choices
Explicitly listed as foundational premise (ii) in the abstract.

pith-pipeline@v0.9.0 · 5631 in / 1507 out tokens · 56932 ms · 2026-05-15T07:00:22.813500+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

96 extracted references · 96 canonical work pages

[1]

A non-Hermitian system can be experimentally realized by adjusting a well controlled parameter or some param- eters

and clarify the points of contention within it. A non-Hermitian system can be experimentally realized by adjusting a well controlled parameter or some param- eters. The Hamiltonian of the non-Hermitian system is H̸=H †, its right- and left-eigenvectors satisfy the eigen- equations: H|ϵ n⟩=E n|ϵn⟩,⟨˜ϵn|H=⟨˜ϵ n|En,(1) respectively. The eigenvalueE n can be ...

work page 1932
[2]

ln 1− di k di k · df k df k 1 + di k di k · df k df k +iΘ +i(2l+ 1)π # ,(57) where Θ = arg

In the remaining parameter regime, the system enters thePT-symmetry-broken phase with the complex en- ergy, the modes satisfying|η|>|1 +qe ik|have purely imaginary energies, whereas those with|η|<|1 +qe ik| possess real energies. Along the boundary where|η|= |1+qe ik|, the spectrum becomes degenerate, correspond- ing to the exceptional point. Notably, if|...

work page
[3]

Lett.322632

El-Ganainy R, Makris K, Christodoulides D and Mussli- mani Z H 2007 Theory of coupled optical PT-symmetric structuresOpt. Lett.322632

work page 2007
[4]

Gorini V, Kossakowski A and Sudarshan E C G 1976 Completely positive dynamical semigroups of n-level sys- temsJ. Math. Phys.17821

work page 1976
[5]

Lindblad G 1976 On the generators of quantum dynam- ical semigroupsCommun. Math. Phys.48119

work page 1976
[6]

Dalibard J, Castin Y and Mølmer K 1992 Wave-function approach to dissipative processes in quantum optics 11 Phys. Rev. Lett.68580

work page 1992
[7]

Dum R, Zoller P and Ritsch H 1992 Monte carlo sim- ulation of the atomic master equation for spontaneous emissionPhys. Rev. A454879

work page 1992
[8]

Gamow 1928 Zur quantentheorie des atomkernesZ

G. Gamow 1928 Zur quantentheorie des atomkernesZ. Physik51204-212

work page 1928
[9]

Yang C N and Lee T D 1952 Statistical theory of equa- tions of state and phase transitions. I. Theory of conden- sationPhys. Rev.87404-409

work page 1952
[10]

Lee T D and Yang C N 1952 Statistical theory of equa- tions of state and phase transitions. II. Lattice gas and Ising modelPhys. Rev.87410-419

work page 1952
[11]

Du, and Liu R B 2015 Experimental observation of Lee-Yang zerosPhys

Peng X, Zhou H, Wei B B, Cui J, J. Du, and Liu R B 2015 Experimental observation of Lee-Yang zerosPhys. Rev. Lett.114010601

work page 2015
[12]

Brandner K, Maisi V F, Pekola J P, Garrahan J P and Flindt C 2017 Experimental determination of dynamical Lee-Yang zerosPhys. Rev. Lett.118180601

work page 2017
[13]

Bender C M, Brody D C and Jones H F 2002 Complex ex- tension of quantum mechanicsPhys. Rev. Lett.89270401

work page 2002
[14]

Bender C M and Boettcher S 1998 Real spectra in non-Hermitian Hamiltonians havingPTsymmetryPhys. Rev. Lett.805243

work page 1998
[15]

Bender C M 2007 Making sense of non-Hermitian Hamil- toniansRep. Prog. Phys.70947

work page 2007
[16]

Mostafazadeh A 2002 Pseudo-Hermiticity versusPT symmetry: The necessary condition for the reality of the spectrum of a non-Hermitian HamiltonianJ. Math. Phys. 43205

work page 2002
[17]

Mostafazadeh A 2002 Pseudo-Hermiticity versus PT- symmetry. II. A complete characterization of non- Hermitian Hamiltonians with a real spectrumJ. Math. Phys.432814

work page 2002
[18]

Mostafazadeh A 2002 Pseudo-Hermiticity for a class of nondiagonalizable HamiltoniansJ. Math. Phys.433944

work page 2002
[19]

Fisher M E 1978 Yang-Lee edge singularity andϕ 3 field theoryPhys. Rev. Lett.401610

work page 1978
[20]

Hatano N and Nelson D R 1996 Localization transitions in non-Hermitian quantum mechanicsPhys. Rev. Lett. 77570

work page 1996
[21]

Hatano N and NelsonD R 1997 Vortex pinning and non- Hermitian quantum mechanicsPhys. Rev. B568651

work page 1997
[22]

Phys.69249

Ashida Y, Gong Z and Ueda M 2020 Non-Hermitian physicsAdv. Phys.69249

work page 2020
[23]

Rev.1841231

Bender C M and Wu T T 1969 Anharmonic oscillator Phys. Rev.1841231

work page 1969
[24]

Bergholtz E J, Budich J C and Kunst F K 2021 Ex- ceptional topology of non-Hermitian systemsRev. Mod. Phys.93015005

work page 2021
[25]

Heiss W D 2012 The physics of exceptional pointsJ. Phys. A: Math. Theor.45444016

work page 2012
[26]

Mandal I and Bergholtz E J 2021 Symmetry and higher- order exceptional pointsPhys. Rev. Lett.127186601

work page 2021
[27]

Ding K, Fang C and Ma G 2022 Non-Hermitian topology and exceptional-point geometriesNat. Rev. Phys.4745- 760

work page 2022
[28]

Miri M A and Al` u A 2019 Exceptional points in optics and photonicsScience3636422

work page 2019
[29]

Adv.11eadr8275

Klauck F U J, Heinrich M, Szameit A and Wolterink T A W 2025 Crossing exceptional points in non-Hermitian quantum systemsSci. Adv.11eadr8275

work page 2025
[30]

Yao S and Wang Z 2018 Edge states and topological in- variants of non-Hermitian systemsPhys. Rev. Lett.121 086803

work page 2018
[31]

Song F, Yao S and Wang Z 2019 Non-hermitian skin effect and chiral damping in open quantum systemsPhys. Rev. Lett.123170401

work page 2019
[32]

Lee C H and Thomale R 2019 Anatomy of skin modes and topology in non-Hermitian systemsPhys. Rev. B99 201103(R)

work page 2019
[33]

Liang Q, Xie D, Dong Z, Li H, Li H, Gadway B, Yi W and Yan B 2022 Dynamic signatures of non-Hermitian skin effect and topology in ultracold atomsPhys. Rev. Lett.129070401

work page 2022
[34]

Phys.1853605

Lin R, Tai T, Li L and Lee C H 2023 Topological non- Hermitian skin effectFront. Phys.1853605

work page 2023
[35]

Zhang X, Zhang T, Lu M H and Chen Y F 2022 A review on non-Hermitian skin effectAdv. Phys. X72109431

work page 2022
[36]

Kawabata K, Numasawa T and Ryu S 2023 Entangle- ment phase transition induced by the non-Hermitian skin effectPhys. Rev. X13021007

work page 2023
[37]

Li Y, Liang C, Wang C, Lu C and Liu Y C 2022 Gain-loss- induced hybrid skin-topological effectPhys. Rev. Lett. 128, 223903

work page 2022
[38]

Okuma N, Kawabata K, Shiozaki K and Sato M 2020 Topological origin of non-Hermitian skin effectsPhys. Rev. Lett.124086801

work page 2020
[39]

Ashida Y and Ueda M 2018 Full-counting many-particle dynamics: Nonlocal and chiral propagation of correla- tionsPhys. Rev. Lett.120185301

work page 2018
[40]

D´ ora B and Moca C P 2020 Quantum quench inPT- symmetric luttinger liquidPhys. Rev. Lett.124136802

work page 2020
[41]

Rep.10621

Yu T, Zou J, Zeng B, Rao J W and Xia K 2024 Non- Hermitian topological magnonicsPhys. Rep.10621

work page 2024
[42]

Okuma N and Sato M 2023 Non-Hermitian topological phenomena: A reviewAnnu. Rev. Condens. Matter Phys. 1483-107

work page 2023
[43]

Rudner M S and Levitov L S 2009 Topological transition in a non-Hermitian quantum walkPhys. Rev. Lett.102 065703

work page 2009
[44]

Kawabata K, Shiozaki K, Ueda M and Sato M 2019 Sym- metry and topology in non-Hermitian physicsPhys. Rev. X9041015

work page 2019
[45]

Borgnia D S, Kruchkov A J and Slager R J 2020 Non- Hermitian boundary modes and topologyPhys. Rev. Lett.124056802

work page 2020
[46]

Gong Z, Ashida Y, Kawabata K, Takasan K, Higashikawa S and Ueda M 2018 Topological phases of non-Hermitian systemsPhys. Rev. X8031079

work page 2018
[47]

Leykam D, Bliokh K Y, Huang C, Chong Y D and Nori F 2017 Edge modes, degeneracies, and topological numbers in non-Hermitian systemsPhys. Rev. Lett.118040401

work page 2017
[48]

Shen H, Zhen B and Fu L 2018 Topological band the- ory for non-Hermitian HamiltoniansPhys. Rev. Lett.120 146402

work page 2018
[49]

Liu T, Zhang Y R, Ai Q, Gong Z, Kawabata K, Ueda M and Nori F 2019 Second-order topological phases in non-Hermitian systemsPhys. Rev. Lett.122076801

work page 2019
[50]

Guo A, Salamo G J, Duchesne D, Morandotti R, Volatier- Ravat M, Aimez V, Siviloglou G A and Christodoulides D N 2009 Observation of PT-symmetry breaking in com- plex optical potentialsPhys. Rev. Lett.103093902

work page 2009
[51]

Phys.6192

R¨ uter C E, Makris K G, El-Ganainy R, Christodoulides D N, Segev M and Kip D 2010 Observation of parity-time symmetry in opticsNat. Phys.6192

work page 2010
[52]

Regensburger A, Bersch C, Miri M A, Onishchukov G, Christodoulides D N and Peschel U 2012 Parity-time syn- thetic photonic latticesNature488167 12

work page 2012
[53]

El-Ganainy, K

R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non- Hermitian physics and PT symmetry, Nat. Phys.14, 11 (2018)

work page 2018
[54]

Feng L, El-Ganainy R and Ge L 2017 Non-Hermitian photonics based on parity-time symmetryNat. Photon. 11752-762

work page 2017
[55]

Zhang Z, Zhang Y, Sheng J, Yang L, Miri M A, Christodoulides D N, He B, Zhang Y and Xiao M 2016 Observation of parity-time symmetry in optically induced atomic latticesPhys. Rev. Lett.117123601

work page 2016
[56]

Dembowski C, Gr¨ af H D, Harney H L, Heine A, Heiss W D, Rehfeld H and Richter A 2001 Experimental obser- vation of the topological structure of exceptional Points Phys. Rev. Lett.86787

work page 2001
[57]

Phys.121139

Peng P, Cao W, Shen C, Qu W, Wen J, Jiang L and Xiao Y 2016 Anti-parity-time symmetry with flying atoms Nat. Phys.121139

work page 2016
[58]

Meden V, Grunwald L and Kennes D M 2023PT- symmetric, non-Hermitian quantum many-body physics- a methodological perspectiveRep. Prog. Phys.86124501

work page
[59]

Brody D C 2013 Biorthogonal quantum mechanicsJ. Phys. A: Math. Theor.47035305

work page 2013
[60]

Jing Y, Dong J J, Zhang Y Y and Hu Z X 2024 Biorthogonal dynamical quantum phase transitions in non-Hermitian systemsPhys. Rev. Lett.132220402

work page 2024
[61]

Wang F, Liang G, Zhao Z, Luo L Y, Zhang D J and Xu B M 2026 Survival of Hermitian criticality in the non- Hermitian frameworkPhys. Rev. B113165149

work page 2026
[62]

Rev.40749

Wigner E 1932 On the quantum correction for thermo- dynamic equilibriumPhys. Rev.40749

work page 1932
[63]

Husimi K 1940 Some formal properties of the density matrixProc. Phys. Math. Soc. Jpn.22264

work page 1940
[64]

Rev.1312766

Glauber R J 1963 Coherent and incoherent states of the radiation fieldPhys. Rev.1312766

work page 1963
[65]

Sudarshan E C G 1963 Equivalence of semiclassical and quantum mechanical descriptions of statistical light beamsPhys. Rev. Lett.10277

work page 1963
[66]

Rev.4431

Kirkwood J G 1933 Quantum statistics of almost classical assembliesPhys. Rev.4431

work page 1933
[67]

Dirac P A M 1945 On the analogy between classical and quantum mechanicsRev. Mod. Phys.17195

work page 1945
[68]

Pei J H, Chen J F and Quan H T 2023 Explor- ing quasiprobability approaches to quantum work in the presence of initial coherence: Advantages of the Margenau-Hill distributionPhys. Rev. E108054109

work page 2023
[69]

Xu B M, Zou J, Guo L S and Kong X M 2018 Effects of quantum coherence on work statisticsPhys. Rev. A97 052122

work page 2018
[70]

Yoshimura K and Hamazaki R 2026 Quasiprobability thermodynamic uncertainty relationPhys. Rev. Lett.136 120406

work page 2026
[71]

Lostaglio M, Belenchia A, Levy A, Hern´ andez-G´ omez S, Fabbri N and Gherardini S 2023 Kirkwood-Dirac quasiprobability approach to the statistics of incompati- ble observablesQuantum71128

work page 2023
[72]

Gherardini S and De Chiara G 2024 Quasiprobabilities in quantum thermodynamics and many-body systemsPRX Quantum5030201

work page 2024
[73]

Halpern N Y 2017 Jarzynski-like equality for the out-of- time-ordered correlatorPhys. Rev. A95012120

work page 2017
[74]

Halpern N Y, Swingle B and Dressel J 2018 Quasiprob- ability behind the out-of-time-ordered correlatorPhys. Rev. A97042105

work page 2018
[75]

Alonso J R G, Halpern N Y and Dressel J 2019 Out-of- time-ordered-correlator quasiprobabilities robustly wit- ness scramblingPhys. Rev. Lett.122040404

work page 2019
[76]

Shukla R K and Levy A 2026 First appearance of quasiprobability negativity in quantum many-body dy- namics arXiv:2601.00259

work page arXiv 2026
[77]

Levy A and Lostaglio M 2020 Quasiprobability distribu- tion for heat fluctuations in the quantum regimePRX Quantum1010309

work page 2020
[78]

Phys.791

Zhang K and Wang J 2024 Quasiprobability fluctuation theorem behind the spread of quantum informationCom- mun. Phys.791

work page 2024
[79]

Lostaglio M 2018 Quantum fluctuation theorems, con- textuality, and work quasiprobabilitiesPhys. Rev. Lett. 120040602

work page 2018
[80]

Zhang K, Ni M Y, Shi H L, Wang X H and Wang J 2026 Fluctuation theorems for multipartite quantum coher- ence and correlation dynamicsPhys. Rev. A113022217

work page 2026

Showing first 80 references.