Entropic Fluctuations in Quantum Statistical Mechanics: A Review

Aannalisa Panati; Claude-Alain Pillet; Laurent Bruneau; Tristan Benoist; Vojkan Jak\v{s}i\'c

REVIEW

Entropic Fluctuations in Quantum Statistical Mechanics: A Review

Tristan Benoist, Laurent Bruneau, Vojkan Jakšić, Aannalisa Panati, Claude-Alain Pillet

Mathematical Physics · Submitted 29 Oct 2026 · 36 references

Pith's verdict: A genuinely useful review of the quantum fluctuation-theorem landscape: the headline theorems are quoted from published companion papers, the one load-bearing caveat (the 2TM thermodynamic limit) is disclosed by the authors themselves, and the synthesis deserves refereeing.

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arXiv ID2610.40463 (version 1)

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CitationAannalisa Panati, Claude-Alain Pillet, Laurent Bruneau, Tristan Benoist, Vojkan Jak\v{s}i\'c. "Entropic Fluctuations in Quantum Statistical Mechanics: A Review." arXiv:2610.40463 (2026). https://pith.science/paper/SX32Q3Q5

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Editor's note

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Quick take: this is a review, and a good one. Do not come to it for a new proof—its two headline theorems are quoted from published companion papers [BBJ+24, BBJ+26]—but as a synthesis it does real work. The paper's contribution is a usable map of the four competing quantum entropy-production functionals (naive, 2TM, QPSC, EAST): how each is defined, when they coincide, and why they diverge. Section 5.1, on the algebraic definition of the infinite-system 2TM functional, is the most valuable part; the rigidity result—that the modular averaging representing the first measurement resets the reservoir part of the state to the reference equilibrium state—is genuinely surprising and is presented with care.

The paper is unusually honest about its own soft spots. Section 5.1.2 states plainly that the general-state thermodynamic limit for the conventional finite-volume Gibbs approximation remains open, and that even with the restriction approximation one does not prove convergence of F^{2tm}. That is the load-bearing caveat for the physical reading of the steady-state 2TM branch, exactly as the stress-test note says. But the authors flag it in the section where it matters; it is an acknowledged open problem, not a hidden flaw. The mathematical PREF statement in Theorem 7.2 does not rest on that physical reading—it is a statement about the functionals of Definition 6.1, proved in a published J. Math. Phys. paper.

The heavy reliance on four self-cited companion works is worth noting, but all four are refereed journal articles with DOIs, so in a review by the same group this is not a defect. The classical background in Section 2 and the Gärtner–Ellis route are standard and clean. Section 7 is compressed—readers will need [BBJ+26] for the actual spectral-deformation argument—but it is explicitly labeled a summary, and the F_2(alpha) formula is stated with its hypotheses.

Who gets value: graduate students and researchers entering nonequilibrium quantum statistical mechanics who want the modular-theory approach in one place, plus anyone needing the precise domain conventions of the four functionals. It deserves a serious referee; the referee's job is consistency of the synthesis and accuracy of attribution, and both hold up. I would accept it as a review.

What would settle it

Compute, to second order in $\lambda$, the finite-volume two-time measurement functionals $\mathcal{F}^{2\mathrm{tm}}_{\omega_{T,\Lambda},t}(\alpha)$ for a spin-fermion approximation and compare $\lim_{\Lambda}\frac{1}{t}\log\mathcal{F}^{2\mathrm{tm}}_{\omega_{T,\Lambda},t}(\alpha)$ with the modular definition $\mathcal{F}^{2\mathrm{tm}}_{\omega_T,t}(\alpha)$: a discrepancy at any $\alpha\in(0,1)$ and any waiting time $T>0$ would show the modular definition is not the thermodynamic limit. Alternatively, a small-$\lambda$ numerical simulation of the spin-fermion model can test the explicit prediction $F(\alpha)/\lambda^2\to F_2(\alpha)$ for the scaled cumulant generating function, including the symmetry $F_2(\alpha)=F_2(1-\alpha)$ and the strict convexity away from $\beta_1=\beta_2$.

Referee report

Significance

If the cited results hold, the paper provides the first structured comparison of four distinct noncommutative extensions of classical entropic fluctuation theorems and identifies the precise dynamical statement—the quantum PREF—under which their large-time limits coincide. The strongest concrete content is Theorem 7.2: a paradigmatic open quantum system in which the steady-state 2TM, EAST, and QPSC functionals share one limiting rate function, with a real-analytic symmetry and an explicit, parameter-free second-order coefficient F_2. The manuscript is also careful about its own epistemic status: Section 5.1.2 openly states that the thermodynamic limit for the conventional finite-volume Gibbs approximation of the 2TM protocol is not proved, and Sections 4 and 5 repeatedly flag when QPSC or EAST functionals are not Laplace transforms of probability measures. The main theorems are attributed to published journal papers, which is appropriate for a review, and the definitions and hypotheses are sufficiently explicit that the claims are checkable.

Major objections (2)

[Definition 6.1; Theorem 7.2] There is a domain inconsistency between the definition of PREF and the statement of the main theorem. Definition 6.1 requires the interval I=]theta_-, theta_+[ to be contained in ]-theta, theta[, whereas Theorem 7.2(2) states that the PREF holds on ]-theta, 1+theta[. For every theta>0 the latter interval is not contained in the former, so the theorem's statement does not fall under the definition as printed. The intended domain is presumably centered at alpha=1/2, consistently with the symmetry F(alpha)=F(1-alpha) and with the analytic continuation strips in Proposition 7.1. The definition should be corrected (for example to I subset of ]-theta, 1+theta[, or an equivalent shifted interval), and Theorem 7.2 should be restated in the same convention.
[Section 5.1.2; Theorem 7.2] The physical interpretation of the 2TM branch is conditional on an open thermodynamic-limit problem. The authors correctly state that the general-state TDL for the conventional finite-volume Gibbs approximation remains open, and that even with the restriction approximation one does not prove convergence of F^{2tm}_{omega_{T,Lambda},t} to F^{2tm}_{omega_T,t}. I agree that this is a disclosed limitation rather than a hidden flaw, and it does not affect the mathematical PREF statement for the functionals of Definition 6.1. However, the abstract and the discussion around Theorem 7.2 present the result as an extension of the steady-state Gallavotti-Cohen theorem to two-time measurement entropy production. The paper should carry the caveat more prominently, distinguishing the algebraic modular functional (5.1) from a verified thermodynamic limit of finite-system two-time measurements, so that the physical reading of the steady-state 2TM branch is not overstated. The authors acknowledge this: “The general-state TDL for the conventional finite-volume Gibbs approximation remains open.”

Minor objections (3)

[Abstract and reference list] The companion paper [BBJ+25b] is cited in the abstract with volume 38 and year 2026, while the reference entry reads Rev. Math. Phys. (2025); please reconcile these data.
[Proposition 7.1] The analytic continuation domains are stated separately for the 2TM, QPSC, and EAST representations, and they have different half-widths. Since Theorem 7.2 uses a single interval ]-theta, 1+theta[, a sentence explaining how the three strips are intersected to obtain that interval would help the reader verify the consistency of the theorem with Definition 6.1.
[Section 6.1] The footnote stating that the EAST functional extends to the strip S(2theta), while the other functionals extend to S(theta), is useful but appears only after Definition 6.1; it should be integrated into the definition of the common domain so that the roles of theta and 2theta are transparent.

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The paper

The paper claims

In the spin-fermion model, four quantum entropy-production functionals — transient and steady two-time measurement, ancilla tomography, phase-space contraction — converge to one symmetric rate function.

The reading

Classical non-equilibrium statistical mechanics knows two universal statements about entropy-production fluctuations: the Evans–Searles relation for a transient state and the Gallavotti–Cohen relation for a steady state. When moved to quantum mechanics, each statement splits into several inequivalent definitions — two-time measurement, quantum phase space contraction, and ancilla state tomography — because the entropy-production variable is not an observable that can be simply measured once. The paper reviews this landscape and argues that in the algebraic setting of infinitely extended reservoirs a steady-state version exists, with the first measurement of the two-time protocol represented by an ergodic average over the modular group. Its central reported result, Theorem 7.2, is that in the spin-fermion model all these quantum definitions share one limiting large-deviation rate function $F$ with the symmetry $F(\alpha)=F(1-\alpha)$ — the quantum principle of regular entropic fluctuations — whose leading weak-coupling form is given explicitly. If this is right, a paradigmatic driven open quantum system has a unique fluctuation statistics for entropy production, computable from spectral resonances of quantum transfer operators, and the previously missing quantum Gallavotti–Cohen theorem holds in the two-time measurement approach.

What carries the argument

The machinery has three parts. First, the entropic functionals $\mathcal{F}^{\#}_{\nu,t}(\alpha)=\int e^{-\alpha s}\,dQ^{\#}_{\nu,t}(s)$, Laplace transforms whose large-time growth rate $\lim_{t\to\infty}\frac{1}{t}\log\mathcal{F}^{\#}_{\nu,t}(\alpha)$ feeds the Gärtner–Ellis theorem and produces the large-deviation rate function. Second, the Araki–Connes cocycle $[D\omega_{-t}:D\omega]_\alpha$, the one modular object that belongs to the observable algebra: the QPSC functional is its expectation, the 2TM functional is a modular ergodic average of it, and the EAST functional sandwiches it as $\nu([D\omega_{-t}:D\omega]^*_{\bar\alpha/2}[D\omega_{-t}:D\omega]_{\alpha/2})$. Third, the $\alpha$-Liouvilleans $L_\alpha=L_{\mathrm{fr}}+V-J\varsigma^{-i\bar\alpha}_\omega(V)J$ and their EAST companion $\hat L_\alpha$: the exponential rates of the entropic functionals are $iE(1/2-\alpha)$, where $E$ is the leading spectral resonance of $L_{1/2-\alpha}$, so the equality of the four large-time limits in the PREF becomes an equality of spectral resonances, computable by spectral deformation. The 2TM rigidity is carried by Equation (5.1)'s projection onto $\ker\log\Delta_\omega$, whose rank collapses from reservoir-sized to $(\dim K_S)^2$ in the thermodynamic limit.

Extended reading notes

Core claim

The paper's core claim is that a quantum steady-state fluctuation theorem can be formulated in the algebraic, infinitely extended setting, and that in a canonical model the main quantum protocols converge to the same answer. For two-time measurement on infinitely extended reservoirs, the first measurement is encoded as an ergodic average over the modular group, Equation (5.1): $$\mathcal{F}^{2\mathrm{tm}}_{\omega_T,t}(\$\alpha$)=\lim_{R\to\infty}\frac{1}{R}\int_0^R \omega_T\circ\varsigma^\omega_\$\theta$\big([D\omega_{-t}:D\omega]_\$\alpha$\big)\,d\$\theta$ .$$ A rigidity result (Theorem 5.1) then shows that for ergodic reservoirs this functional is essentially independent of the waiting time $T$: the modular averaging that represents the first measurement resets the reservoir part of the state, a decoherence effect with no classical analogue. The central theorem of the review, Theorem 7.2, attributes to the authors' companion paper the statement that in the spin-fermion model with small nonzero coupling $\lambda$, under assumptions (SFM1)–(SFM4), the quantum PREF holds on $]-\vartheta,1+\vartheta[$: the limits $\mathcal{F}^{2\mathrm{tm}}_\omega=\mathcal{F}^{2\mathrm{tm}}_{\omega_\infty}=\mathcal{F}^{\mathrm{east}}_{\omega_\infty}=\mathcal{F}^{\mathrm{qpsc}}_{\omega_\infty}$ exist and define one real-analytic function $F$ that satisfies $F(\alpha)=F(1-\alpha)$, is strictly convex away from equilibrium, vanishes identically at equilibrium, and has the explicit leading order $F(\alpha)=\lambda^2 F_2(\alpha)+O(\lambda^3)$. The paper is equally explicit about what remains open: the general-state thermodynamic limit of the 2TM functional from finite-volume Gibbs states is not proven, and even with a restriction approximation the finite-volume functionals are not shown to converge to the modular ones.

Load-bearing premise

The steady-state two-time measurement statistics rest on the assumption that the modular ergodic average in Equation (5.1) is the correct thermodynamic limit of the two-time measurement protocol, a limit the paper states is not proven for general initial states.

Editorial extensions

If this is right

If Theorem 7.2 is correct, the spin-fermion model possesses a genuine quantum Gallavotti–Cohen fluctuation relation: steady-state entropy-production statistics follow the same restricted large-deviation principle as the transient ones, with rate function $I$ obeying $I(-s)=I(s)+s$ on the covered interval.
The explicit leading term $F(\alpha)=\lambda^2F_2(\alpha)+O(\lambda^3)$ converts the fluctuation statistics into a quantitative prediction: at weak coupling the mean steady-state entropy production $-F'(0)=\omega_\infty(\sigma)$ and the fluctuation scale are both of order $\lambda^2$, with their relation fixed by the displayed formula in terms of form factors and temperatures.
In the 2TM channel alone, the Evans–Searles relation automatically upgrades to a Gallavotti–Cohen relation under reservoir ergodicity, because the first measurement erases reservoir memory — in contrast with the classical situation, where the same result requires a difficult exchange of limits.
The spectral criterion of Section 7.1 gives a model-independent route to the quantum PREF: any model in which the leading resonances of $L_{1/2-\alpha}$ and $\hat L_{1/2-\alpha}$ can be computed and shown to match will satisfy the PREF with the same symmetry $F(\alpha)=F(1-\alpha)$.
The rate function vanishes identically when $\beta_1=\beta_2$ and is otherwise strictly convex, so the fluctuation theorem is trivial exactly at equilibrium and acquires its full content as the temperature bias is switched on.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable consequence of the rigidity inequality (5.5), which the paper does not spell out experimentally: in mesoscopic realizations of the two-time measurement protocol, the transient and steady 2TM entropy-production distributions should become numerically indistinguishable as the reservoirs are made larger, with residual dependence carried only by the finite-dimensional small-system state $\o
Because $F$ is strictly convex, the weak-coupling fluctuation statistics are approximately Gaussian, and $F(\alpha)=\lambda^2F_2(\alpha)$ fixes the ratio of mean to variance independently of $\lambda$ to leading order — a directly measurable prediction that goes beyond what the paper states explicitly.
The authors prove a restricted LDP window, not a full one; a natural extension is to test whether real-analyticity of $F$ on the full window $]-\vartheta,1+\vartheta[$ combined with the Gärtner–Ellis hypotheses yields a full LDP, and whether the symmetry $F(\alpha)=F(1-\alpha)$ survives beyond that window once higher-order terms in $\lambda$ are included.
The framework suggests that the PREF is the generic behavior in models with sufficient spectral control, with the spin-fermion verification serving as evidence that equality among the three functionals' large-time limits is a dynamical feature of ergodic open quantum systems rather than a coincidence of this model.

Original abstract

The celebrated Evans--Searles, respectively Gallavotti--Cohen, fluctuation theorem concerns certain universal statistical features of the entropy production rate of a classical system in a transient, respectively steady, state. In this paper, we review and compare several possible extensions of these fluctuation theorems to quantum systems. We present, in an introductory way, the approach proposed in a recent series of works [Lett. Math. Phys. 114, 32 (2024): doi:10.1007/s11005-024-01777-0], [Rev. Math. Phys 38, 2461006 (2026): doi:10.1142/S0129055X24610063], [Commun. Math. Phys. 406, 201 (2025): 10.1007/s00220-025-05360-z] and [J. Math. Phys. 67, 022101 (2026): doi:10.1063/5.0311182] including a previously missing extension of the Gallavotti--Cohen (steady state) version of the FT to two-time measurement entropy production in the algebraic setting of quantum dynamical systems. We focus on the open quantum systems, where a small system interacts with several (infinitely extended) reservoirs, and discuss the following approaches: naive, two-time measurement, phase space contraction and ancilla state tomography. We review the connection with modular theory. The latter provides a way to extend the classical notions to the quantum domain. We discuss the quantum version of the principle of regular entropic fluctuations, introduced in the classical context in [Nonlinearity 24, 699 (2011): doi:10.1088/0951-7715/24/3/003]. Finally, we relate the statistical properties of these various notions of entropy production to spectral resonances of quantum transfer operators and apply this approach to a paradigmatic model of non-equilibrium statistical mechanics: the spin-fermion model. The results presented here shed a new light on the nature of entropic fluctuations in quantum statistical mechanics.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-30: 1 request, 10,007 input tokens.

What the abstract itself states, as probabilities: claims first 0.09 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.83 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The celebrated Evans--Searles, respectively Gallavotti--Cohen, fluctuation theorem concerns certain universal statistical features of the entropy production rate of a classical system in a transient, respectively steady, state. In this paper, we review and compare several possible extensions of these fluctuation theorems to quantum systems. We present, in an introductory way, the approach proposed in a recent series of works [Lett. Math. Phys. 114, 32 (2024): doi:10.1007/s11005-024-01777-0], [Rev. Math. Phys 38, 2461006 (2026): doi:10.1142/S0129055X24610063], [Commun. Math. Phys. 406, 201 (2025): 10.1007/s00220-025-05360-z] and [J. Math. Phys. 67, 022101 (2026): doi:10.1063/5.0311182] including a previously missing extension of the Gallavotti--Cohen (steady state) version of the FT to two-time measurement entropy production in the algebraic setting of quantum dynamical systems. We focus on the open quantum systems, where a small system interacts with several (infinitely extended) reservoirs, and discuss the following approaches: naive, two-time measurement, phase space contraction and ancilla state tomography. We review the connection with modular theory. The latter provides a way to extend the classical notions to the quantum domain. We discuss the quantum version of the principle of regular entropic fluctuations, introduced in the classical context in [Nonlinearity 24, 699 (2011): doi:10.1088/0951-7715/24/3/003]. Finally, we relate the statistical properties of these various notions of entropy production to spectral resonances of quantum transfer operators and apply this approach to a paradigmatic model of non-equilibrium statistical mechanics: the spin-fermion model. The results presented here shed a new light on the nature of entropic fluctuations in quantum statistical mechanics.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	The celebrated Evans--Searles, respectively Gallavotti--Cohen, fluctuation theorem concerns certain universal statistical features of the entropy production rate of a classical system in a transient, respectively steady, state.	background 1.00	0.03	0.03	asserted 1.00
s1	In this paper, we review and compare several possible extensions of these fluctuation theorems to quantum systems.	contribution 0.98	0.52	0.04	asserted 0.83
s2	We present, in an introductory way, the approach proposed in a recent series of works [Lett. Math.	contribution 0.94	0.60	0.02	asserted 0.82
s3	Phys. 114, 32 (2024): doi:10.1007/s11005-024-01777-0], [Rev. Math.	contribution 0.57	0.06	0.02	unclear 0.92
s4	Phys 38, 2461006 (2026): doi:10.1142/S0129055X24610063], [Commun. Math.	background 0.54	0.05	0.02	unclear 0.96
s5	Phys. 406, 201 (2025): 10.1007/s00220-025-05360-z] and [J. Math.	background 0.53	0.07	0.02	unclear 0.90
s6	Phys. 67, 022101 (2026): doi:10.1063/5.0311182] including a previously missing extension of the Gallavotti--Cohen (steady state) version of the FT to two-time measurement entropy production in the algebraic setting of quantum dynamical systems.	contribution 0.82	0.79	0.09	asserted 0.98
s7	We focus on the open quantum systems, where a small system interacts with several (infinitely extended) reservoirs, and discuss the following approaches: naive, two-time measurement, phase space contraction and ancilla state tomography.	contribution 0.62	0.48	0.04	asserted 0.94
s8	We review the connection with modular theory.	contribution 0.72	0.33	0.03	asserted 0.87
s9	The latter provides a way to extend the classical notions to the quantum domain.	background 0.50	0.09	0.06	asserted 0.99
s10	We discuss the quantum version of the principle of regular entropic fluctuations, introduced in the classical context in [Nonlinearity 24, 699 (2011): doi:10.1088/0951-7715/24/3/003].	contribution 0.88	0.25	0.05	asserted 0.88
s11	Finally, we relate the statistical properties of these various notions of entropy production to spectral resonances of quantum transfer operators and apply this approach to a paradigmatic model of non-equilibrium statistical mechanics: the spin-fermion model.	contribution 0.74	0.83	0.23	asserted 1.00
s12	The results presented here shed a new light on the nature of entropic fluctuations in quantum statistical mechanics.	interpretation 0.99	0.66	0.55	asserted 0.96

What kind of paper the classifier reports this is: review 1.00 · evidence from literature 0.90.

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sources read

arXiv listing abstract v1

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full text
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

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Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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Str a til a , c. : Modular T heory in O perator A lgebras . Editura Academiei Republicii Socialiste Rom\^ a nia, Bucharest; Abacus Press, Tunbridge Wells, 1981, Translated from the Romanian by the author

View on Pith 1981

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Tasaki, H. : J arzynski relations for quantum systems and some applications. Unpublished, 2000, [DOI:10.48550/arXiv.cond-mat/0009244] https://doi.org/10.48550/arXiv.cond-mat/0009244

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Tasaki, S. Matsui, T. : Fluctuation theorem, nonequilibrium steady states and M ac L ennan- Z ubarev ensembles of a class of large quantum systems. In Fundamental A spects of Q uantum P hysics ( T okyo, 2001) , QP--PQ: Quantum Probab. White Noise Anal., vol. 17, World Sci. Publ., River Edge, NJ, February 2003, p. 100--119, [DOI:10.1142/9789812704412\_0006...

View on Pith doi:10.1142/9789812704412 2001