Entropic Fluctuations in Quantum Statistical Mechanics: A Review

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.