Characterizing symmetry-protected thermal equilibrium by work extraction

TL;DR

This paper extends the concept of thermal equilibrium in quantum systems to include symmetry constraints, showing that generalized Gibbs ensembles characterize completely passive states under such symmetries, with implications for quantum thermodynamics and device design.

Contribution

It proves that under symmetry constraints, completely passive states are generalized Gibbs ensembles, including conserved charges, even for non-commutative symmetries like SU(2).

Findings

Complete passivity under symmetry constraints characterized by GGEs.

Extension of thermal equilibrium concept to symmetry-protected systems.

Framework applicable to quantum heat engines and resource theories.

Abstract

The second law of thermodynamics states that work cannot be extracted from thermal equilibrium, whose quantum formulation is known as complete passivity; A state is called completely passive if work cannot be extracted from any number of copies of the state by any unitary operations. It has been established that a quantum state is completely passive if and only if it is a Gibbs ensemble. In physically plausible setups, however, the class of possible operations is often restricted by fundamental constraints such as symmetries imposed on the system. In the present work, we investigate the concept of complete passivity under symmetry constraints. Specifically, we prove that a quantum state is completely passive under a symmetry constraint described by a connected compact Lie group, if and only if it is a generalized Gibbs ensemble (GGE) including conserved charges associated with the symmetry. Remarkably, our result applies to non-commutative symmetry such as $SU(2)$ symmetry, suggesting an unconventional extension of the notion of GGE. Furthermore, we consider the setup where a quantum work storage is explicitly included, and prove that the characterization of complete passivity remains unchanged. Our result extends the notion of thermal equilibrium to systems protected by symmetries, and would lead to flexible design principles of quantum heat engines and batteries. Moreover, our approach serves as a foundation of the resource theory of thermodynamics in the presence of symmetries.