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 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 2021
- Revised 7 February 2022
- Accepted 17 February 2022
DOI:https://doi.org/10.1103/PhysRevX.12.021013
Published 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 Society
Physics Subject Headings (PhySH)
Popular Summary
Symmetries not only set fundamental constraints on physical laws but also enrich them in a variety of fields of physics. In thermodynamics, even the very concept of thermal equilibrium can be altered if the system obeys certain symmetries. Here, we establish a generalized notion of thermal equilibrium in the quantum regime, which we refer to as symmetry-protected thermal equilibrium, based on work extraction.
The second law of thermodynamics, also known as Kelvin’s principle, dictates that a positive amount of work can never be extracted by any cyclic operation from a single heat bath. Ever since its establishment in the nineteenth century, the second law has served as the most fundamental constraint on our ability to harvest energy, prohibiting perpetual motion.
It is known that only the Gibbs ensemble, the commonly used expression of thermal equilibrium where only energy is conserved, satisfies Kelvin’s principle. We reveal that the concept of thermal equilibrium should be replaced by yet another ensemble called the generalized Gibbs ensemble, which incorporates quantum-mechanical conserved charges corresponding to symmetries along with energy. We rigorously prove that, if the system respects continuous symmetries, only this ensemble satisfies Kelvin’s principle.
Our result not only opens a new direction of quantum thermodynamics by incorporating the role of symmetries, but also leads to the flexible design of quantum heat engines and quantum batteries, given the rapid development of quantum technologies in recent years.