Unified entropy production in finite quantum systems

TL;DR

The paper addresses the ambiguity of temperature in finite quantum systems and introduces a unified entropy production $\Delta\Sigma(\beta_0,\beta_\tau)=D(\sigma_{SE}\|\sigma_S\otimes\gamma_E(\beta_\tau))-D(\rho_{SE}\|\rho_S\otimes\gamma_E(\beta_0))$, which decomposes into a Clausius-type term and a contribution from the time dependence of the effective temperature. It demonstrates that, under a fixed reference framework, this EP reduces to the conventional Clausius form when the effective temperature is constant and relates to a geometric projection with $\beta^*_t$ defined by energy matching; the EP rate becomes the sum of entropy change and heat flow only if $\beta_t$ is constant or satisfies $\beta_t=\beta^*_t$. The authors derive lower bounds for EP in general initial states using mutual information, relative entropy, and trace-distance tools, and provide a sufficient condition for non-negativity based on the trace distance, illustrated with a two-dimensional environment example. Overall, the work unifies finite-environment thermodynamics with a robust mathematical framework, enabling nontrivial bounds and nonnegativity guarantees for quantum entropy production in nonequilibrium settings.

Abstract

In finite-dimensional quantum systems, temperature cannot be uniquely defined. This, in turn, implies that there are several ways to define entropy production in finite-dimensional quantum systems, because the classical entropy production depends on temperature. We propose a unified definition of entropy production based on the difference in quantum relative entropy with respect to reference states characterized by effective temperatures. We demonstrate that the proposed definition naturally decomposes into a Clausius-type entropy production and an additional contribution arising from the time dependence of the effective temperature. Furthermore, we show that requiring the entropy production rate to take the conventional form as the sum of the entropy change and the heat flow constrains the effective temperature to be either constant or equal to a specific energy-matching effective temperature. For general initial states, entropy production can become negative, in which case we derive lower bounds on entropy production and establish sufficient conditions for its non-negativity using the trace distance.