Characterizing Symmetry-Protected Thermal Equilibrium by Work Extraction*

The second law of thermodynamics prohibits work extraction from thermal equilibrium, which is formulated as complete passivity in the quantum setup; a state is called completely passive if one cannot extract work from any number of copies of the state by any unitary operations. It has been established that the necessary and sufficient condition for complete passivity is being a Gibbs ensemble. In many physical situations, however, the class of possible operations is restricted by fundamental constraints such as the symmetries of a system. In this talk, we explain the concept of complete passivity under symmetry constraints. Specifically, we have proved 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 including the conserved charges associated with the symmetry. Remarkably, our result holds for noncommutative symmetry such as SU(2) symmetry, suggesting an unconventional extension of the notion of the generalized Gibbs ensemble. Furthermore, we have investigated the case where a quantum work storage is explicitly introduced, and have proved that the condition for complete passivity remains unchanged. Our result extends the notion of thermal equilibrium to symmetric systems, and would lead to flexible design principles of quantum heat engines and batteries. Moreover, our result lays the foundation for the resource theory of thermodynamics in the presence of symmetries.