Entropic Fluctuations in Quantum Statistical Mechanics: A Review

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.

This review addresses how transient Evans–Searles and steady-state Gallavotti–Cohen entropy-production fluctuation theorems extend to Hamiltonian quantum systems with infinite reservoirs. It synthesizes recent results comparing two-time measurements (2TM), quantum phase-space contraction (QPSC), and entropic ancilla state tomography (EAST), using modular theory to connect finite-system protocols with algebraic quantum dynamics. The three functionals coincide in the reference state but generally differ for other initial states; naive quantization generally fails the transient fluctuation relation. The central 2TM result is measurement-induced rigidity: for ergodic reservoirs, modular averaging representing the first measurement resets their state to the reference reservoir state. Writing $d=\dim\mathcal K_S$, the entropy-production distributions obey $d\lambda_{\min}(\omega_{T,S})\leq dQ^{2\mathrm{TM}}_{\omega_T,t}/dQ^{2\mathrm{TM}}_{\omega,t}\leq d$. Consequently, transient and steady-state large-deviation rate functions coincide when the steady-state small-system marginal is faithful. For EAST, equality of long-time growth rates holds at every finite waiting time, but extending it to the steady state requires exchanging two limits. A spectral-resonance framework relates these growth rates to deformed Liouvilleans. For a spin coupled to two fermionic reservoirs, analytic form-factor and Fermi-golden-rule assumptions establish equality across all three approaches at sufficiently small nonzero coupling. The common function is analytic, satisfies $F(\alpha)=F(1-\alpha)$, and has an explicit order-$\lambda^2$ expansion. Limitations include potentially nonprobabilistic EAST/QPSC functionals outside the reference state and strong model assumptions. For interacting spin reservoirs, the stated thermodynamic-limit justification uses restricted infinite-volume states; the conventional finite-volume Gibbs construction remains open.