The second law of quantum thermodynamics as an equality

Abstract

We investigate the connection between recent results in quantum thermodynamics and fluctuation relations by adopting a fully quantum mechanical description of thermodynamics. By including a work system whose energy is allowed to fluctuate, we derive a set of equalities which all thermodynamical transitions have to satisfy. This extends the condition for maps to be Gibbs-preserving to the case of fluctuating work, providing a more general characterisation of maps commonly used in the information theoretic approach to thermodynamics. For final states, block diagonal in the energy basis, this set of equalities are necessary and sufficient conditions for a thermodynamical state transition to be possible. The conditions serves as a parent equation which can be used to derive a number of results. These include writing the second law of thermodynamics as an equality featuring a fine-grained notion of the free energy. It also yields a generalisation of the Jarzynski fluctuation theorem which holds for arbitrary initial states, and under the most general manipulations allowed by the laws of quantum mechanics. Furthermore, we show that each of these relations can be seen as the quasi-classical limit of three fully quantum identities. This allows us to consider the free energy as an operator, and allows one to obtain more general and fully quantum fluctuation relations from the information theoretic approach to quantum thermodynamics.

11+3 pages. V4: Updated to match published version. Discussion of thermo-majorization and implementing arbitary unitaries added. V3: Added funding information. V2: Expanded discussion on relation to fluctuation theorems

References in corpus (9)

• Description of quantum coherence in thermodynamic processes requires constraints beyond free energy
• Quantum coherence, time-translation symmetry and thermodynamics
• The thermodynamic meaning of negative entropy
• Fluctuation Theorem for Arbitrary Open Quantum Systems
• Path-integral analysis of fluctuation theorems for general Langevin processes
• Microcanonical and resource-theoretic derivations of the thermal state of a quantum system with noncommuting charges
• Fluctuation theorems in driven open quantum systems
• Gibbs-Preserving Maps outperform Thermal Operations in the quantum regime
• Uncertainty relations and approximate quantum error correction

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