Characterizing Symmetry-Protected Thermal Equilibrium by Work Extraction

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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 including conserved charges associated with the symmetry. Remarkably, our result applies to noncommutative symmetry such as SU(2) symmetry, suggesting an unconventional extension of the notion of generalized Gibbs ensemble. 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.Received 2 August 2021Revised 7 February 2022Accepted 17 February 2022DOI:https://doi.org/10.1103/PhysRevX.12.021013Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasQuantum statistical mechanicsQuantum thermodynamicsResource theoriesQuantum InformationStatistical Physics