From R\'enyi Relative Entropic Generalization to Quantum Thermodynamical Universality

TL;DR

This paper demonstrates that thermodynamic principles, including the second law and Carnot efficiency, maintain their form invariance when generalized using quantum Rényi entropies, extending classical thermodynamics to nonequilibrium quantum states.

Contribution

It introduces a generalized thermodynamic framework based on quantum Rényi entropies, establishing the invariance of thermodynamic laws beyond equilibrium and connecting them with quantum information measures.

Findings

Thermodynamic form invariance holds for Rényi entropic generalizations.

The universality of Carnot's second law is derived from this invariance.

Clausius inequality is valid using Rényi relative entropies.

Abstract

It is shown that the structure of thermodynamics is "form invariant", when it is derived using maximum entropy principle for various choices of entropy and even beyond equilibrium. By the form invariance of thermodynamics, it is meant that the form of the free energy (internal energy minus the temperature times entropy) remains unaltered when all the entities entering this relation are suitably defined. The useful ingredients for this are the equilibrium entropy associated with thermal density matrix and the relative entropy between an arbitrary density matrix and the thermal density matrix. To delineate the form invariance, we consider the quantum R\'enyi entropic versions (indexed by a parameter $\alpha$), i.e., R\'enyi entropy with appropriate internal energy and equilibrium state defined for all $\alpha$. These results reduce to the well-known Gibbs-von Neumann results when $\alpha \rightarrow 1$. Moreover, we show that the \textit{universality} of the Carnot statement of the second law is the consequence of the form invariance of the free energy. Further, the Clausius inequality, which is the precursor to the Carnot cycle, is also shown to hold based on the known data processing inequalities for the traditional and the sandwiched R\'enyi relative entropies. Thus, we find the thermodynamics of nonequilibrium state and its deviation from equilibrium together determine the thermodynamic laws.