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arXiv:2606.14519

Extending Covariant Fluctuation Theorems into Quantum Regime through Quasiprobability Approach

Ji-Hui Pei, Tingzhang Shi, Jin-Fu Chen, H. T. Quan

cond-mat.stat-mechquant-phSubmitted Jun 12, 2026• Announced Jun 15, 2026• 0 citations
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Abstract

The covariant formulation of stochastic thermodynamics requires treating the stochastic work as a 4-vector, posing significant challenges for quantum systems due to the non-commutativity. We introduce a new quasiprobability distribution for the work 4-vector, which combines the Wigner and Margenau-Hill quasiprobabilities. This extends the covariant fluctuation theorems from classical to quantum regime. We illustrate our findings with a scalar field driven by classical particles with a generalized version of trace formula. Our work establishes a quasiprobability approach to studying relativistic quantum thermodynamics in a covariant way.

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The covariant formulation of stochastic thermodynamics requires treating the stochastic work as a 4-vector, posing significant challenges for quantum systems due to the non-commutativity. We introduce a new quasiprobability distribution for the work 4-vector, which combines the Wigner and Margenau-Hill quasiprobabilities. This extends the covariant fluctuation theorems from classical to quantum regime. We illustrate our findings with a scalar field driven by classical particles with a generalized version of trace formula. Our work establishes a quasiprobability approach to studying relativistic quantum thermodynamics in a covariant way.
In quantum thermodynamics, the two-projective-measurement (TPM) scheme provides a successful description of stochastic work only in the absence of initial quantum coherence. Extending the quantum work distribution to quasiprobability is a general way to characterize work fluctuation in the presence of initial coherence. However, among a large number of different definitions, there is no consensus on the most appropriate work quasiprobability. In this article, we list several physically reasonable requirements including the first law of thermodynamics, time-reversal symmetry, positivity of second-order moment, and a support condition for the work distribution. We prove that the only definition that satisfies all these requirements is the Margenau-Hill (MH) quasiprobability of work. In this sense, the MH quasiprobability of work shows its advantages over other definitions. As an illustration, we calculate the MH work distribution of a breathing harmonic oscillator with initial squeezed states and show the convergence to classical work distribution in the classical limit.
We derive a quantum extension of the thermodynamic uncertainty relation where dynamical fluctuations are quantified by the Terletsky-Margenau-Hill quasiprobability, a quantum generalization of the classical joint probability. The obtained inequality plays a complementary role to existing quantum thermodynamic uncertainty relations, focusing on observables' change rather than exchange of charges through jumps and respecting initial coherence. Quasiprobabilities show anomalous behaviors that are forbidden in classical systems, such as negativity; we reveal that negativity or a nonclassically enhanced escape rate is necessary to increase an output-to-dissipation ratio beyond classical limitations and show that the requirements are basis independent and stronger than quantum coherence. To illustrate these statements, we employ a model that can exhibit a dissipationless heat current, which would be prohibited in classical systems; we construct a state that has much coherence but does not lead to a dissipationless current due to the absence of anomalous behaviors in quasiprobabilities.
Of indisputable relevance for nonequilibrium thermodynamics, fluctuations theorems have been generalized to the framework of quantum thermodynamics, with the notion of work playing a key role in such contexts. The typical approach consists of treating work as a stochastic variable and the acting system as an eminently classical device with a deterministic dynamics. Inspired by technological advances in the field of quantum machines, here we look for corrections to work fluctuations theorems when the acting system is allowed to enter the quantum domain. This entails including the acting system in the dynamics and letting it share a nonclassical state with the system acted upon. Moreover, favoring a mechanical perspective to this program, we employ a concept of work observable. For simplicity, we choose as theoretical platform the autonomous dynamics of a two-particle system with an elastic coupling. For some specific processes, we derive several fluctuation theorems within both the quantum and classical statistical arenas. In the quantum results, we find that, along with entanglement and quantum coherence, aspects of inertia also play a significant role since they regulate the route to mechanical equilibrium.
The standard approach to deriving fluctuation theorems fails to capture the effect of quantum correlation and coherence in the initial state of the system.Here, we overcome this difficulty and derive the heat-exchange-fluctuation theorem in the full quantum regime by showing that the energy exchange between two locally thermal states in the presence of initial quantum correlations is faithfully captured by a quasiprobability distribution.Its negativities, being associated with proofs of contextuality, are proxies of nonclassicality.We discuss the thermodynamic interpretation of negative probabilities and provide heat-flow inequalities that can only be violated in their presence.Remarkably, testing these fully quantum inequalities, at an arbitrary dimension, is no more difficult than testing traditional fluctuation theorems.We test these results on data collected in a recent experiment studying the heat transfer between two qubits and give examples for the capability of witnessing negative probabilities at higher dimensions.
Abstract We consider Margenau-Hill quasiprobability that measures a statistical weight assigned to an observation of an energy eigenvalue pair and its utilization in assessing work done by isolated quantum ensembles that generally contain coherence. We point out some physically reasonable and intuitive features of the statistical properties of work determined by Margenau-Hill quasiprobability. To be specific, we show that in sharp contrast to other proposals, the moment-generating function given by Margenau-Hill quasiprobability fulfills the composition and shift properties, and thus guarantees the additivity of works performed by joined work protocols. Work fluctuations accessed by Margenau-Hill distribution also give expected values for initially pure energy ensembles, providing a rationale as to why it serves as a unique measure in characterizing quantum work and its fluctuations.
Fluctuation theorems establish exact relations for nonequilibrium dynamics, profoundly advancing the field of stochastic thermodynamics. In this work, we extend quantum fluctuation theorems beyond the traditional thermodynamic framework to quantum multipartite information dynamics, where both the system and the environment are multipartite without assuming any thermodynamic constraints. Based on the two-point measurement scheme and the classical probability, we establish the fluctuation theorem for the dynamics of classical multipartite mutual information. By extending to quasiprobability, we derive quantum fluctuation theorems for multipartite coherence and quantum correlations, presenting them in both integral and detailed forms. Our theoretical results are illustrated and verified using three-qubit examples, and feasible experimental verification protocols are proposed. These findings uncover the statistical structure underlying the nonequilibrium quantum information dynamics, providing fundamental insights and alternative tools for advancing quantum technologies.