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arXiv:2406.06234 · Physical Review Letters

Quantum Thermodynamics with Coherence: Covariant Gibbs-Preserving Operation Is Characterized by the Free Energy

Naoto Shiraishi

Published Apr 22, 2025• 3 citations
arXivHTMLPDF

Abstract

The resource theory with covariant Gibbs-preserving operations, also called enhanced thermal operations, is investigated. We prove that with the help of a correlated catalyst, the state convertibility for any coherent state is fully characterized by the free energy defined with the quantum relative entropy. We can extend this result to general resource theories in the form that imposing the covariant condition to a general resource theory does not change the state convertibility, as long as the initial state is coherent and distillable and this resource theory admits the phase estimation and the phase shift. This means that adding a constraint from the law of energy conservation is irrelevant in the correlated-catalytic framework.

DOI
10.1103/physrevlett.134.160402

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The resource theory with covariant Gibbs-preserving operations, also called enhanced thermal operations, is investigated. We prove that with the help of a correlated catalyst, the state convertibility for any coherent state is fully characterized by the free energy defined with the quantum relative entropy. We can extend this result to general resource theories in the form that imposing the covariant condition to a general resource theory does not change the state convertibility, as long as the initial state is coherent and distillable and this resource theory admits the phase estimation and the phase shift. This means that adding a constraint from the law of energy conservation is irrelevant in the correlated-catalytic framework.
In this short review article, we present recent progress in quantum thermodynamics in the framework with a correlated catalyst. We examine two key properties of thermal operations, the Gibbs preserving property and the covariant property. The state convertibility of a Gibbs preserving operation is fully characterized by the second law of thermodynamics with the nonequilibrium free energy. The state convertibility of a covariant operation is shown to be free as long as an initial state has finite coherence. We finally show that these two findings can be combined in the enhanced thermal operation (covariant Gibbs-preserving operation).
Abstract In this short review article, we present recent progress in quantum thermodynamics in the framework with a correlated catalyst. We examine two key properties of thermal operations, the Gibbs-preserving property and the covariant property. The state convertibility of a Gibbs-preserving operation is fully characterized by the second law of thermodynamics with the nonequilibrium free energy. The state convertibility of a covariant operation is shown to be free as long as an initial state has finite coherence. We finally show that these two findings can be combined in the enhanced thermal operation (covariant Gibbs-preserving operation).
The class of possible thermodynamic conversions can be extended by introducing an auxiliary system called catalyst, which assists in state conversion while its own state remains unchanged. We reveal a complete characterization of catalytic state conversion in quantum and single-shot thermodynamics by allowing an infinitesimal correlation between the system and the catalyst. Specifically, we prove that a single thermodynamic potential, which provides the necessary and sufficient condition for the correlated-catalytic state conversion, is given by the standard nonequilibrium free energy defined with the Kullback-Leibler divergence. This resolves the conjecture raised by Wilming, Gallego, and Eisert [Entropy 19, 241 (2017)ENTRFG1099-430010.3390/e19060241] and by Lostaglio and Müller [Phys. Rev. Lett. 123, 020403 (2019)PRLTAO0031-900710.1103/PhysRevLett.123.020403] in the positive. Moreover, we show that, with the aid of the work storage, any quantum state can be converted into another by paying the work cost equal to the nonequilibrium free energy difference. Our result would serve as a step towards establishing resource theories of catalytic state conversion in the fully quantum regime.
Quantum thermodynamics investigates how robust the second law of thermodynamics serves as the unique fundamental law in the small quantum world. To tackle this problem, the quantum coherence constitutes a major difficulty of investigations, which provides severe constraints hindering the recovery of a single thermodynamic potential. Here we solve this long-standing problem of quantum information theory by revealing that the state convertibility under thermal operations is fully characterized by the second law of thermodynamics. Specifically, we prove that whether a quantum state with quantum coherence is convertible to another by a thermal operation with a correlated catalyst is completely determined by the free energy ordering. Unlike previous attempts, our setting does not resort to any additional external coherent assist, providing a faithful operational characterization of thermodynamic state transformation.
The generalized second laws of quantum thermodynamics are usually formulated in terms of Rényi divergences and the associated family of generalized free energies. In catalytic thermal transformations, this framework typically certifies the existence of a suitable catalyst but does not make the catalytic contribution explicit in the resulting system-level inequalities. Here we develop a complementary formulation based on non-additive divergences, whose pseudo-additive structure yields a family of generalized free energies with an explicit catalyst-dependent correction term. For uncorrelated catalytic thermal transformations, we show that this leads to non-additive second-law relations providing nontrivial constraints on admissible catalysts when the catalyst is returned only approximately. We also analyze correlated catalytic thermal transformations and show, through explicit finite-dimensional examples, that reduced-state data are generally insufficient to characterize thermodynamic accessibility: the thermo-majorization behavior of the joint transformation can change while the system and catalyst marginals remain fixed, and even states with identical marginals and the same mutual information can exhibit different thermo-majorization accessibility.
Quantum coherence is one of the fundamental aspects distinguishing classical and quantum theories. Coherence between different energy eigenstates is particularly important, as it serves as a valuable resource under the law of energy conservation. A fundamental question in this setting is how well one can prepare good coherent states from low coherent states and whether a given coherent state is convertible to another one. Here, we show that any low coherent state is convertible to any high coherent state arbitrarily well in two operational settings: asymptotic and catalytic transformations. For a variant of asymptotic coherence manipulation where one aims to prepare desired states in local subsystems, the rate of transformation becomes unbounded regardless of how weak the initial coherence is. In a non-asymptotic transformation with a catalyst, a helper state that locally remains in the original form after the transformation, we show that an arbitrary state can be obtained from any low coherent state. Applying this to the standard asymptotic setting, we find that a catalyst can increase the coherence distillation rate significantly-from zero to infinite rate. We also prove that such anomalous transformation requires small but nonzero coherence in relevant modes, establishing the condition under which a sharp transition of the operational capability occurs. Our results provide a general characterization of the coherence transformability in these operational settings and showcase their peculiar properties compared to other common resource theories such as entanglement and quantum thermodynamics.