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arXiv:1406.3618 · New Journal of Physics

Gibbs-preserving maps outperform thermal operations in the quantum regime

Philippe Faist, Jonathan Oppenheim, Renato Renner

Published Apr 2, 2015• 111 citations
arXivHTMLPDF

Abstract

In this brief paper, we compare two frameworks for characterizing possible operations in quantum thermodynamics. One framework considers thermal operations—unitaries which conserve energy. The other framework considers all maps which preserve the Gibbs state at a given temperature. Thermal operations preserve the Gibbs state; hence a natural question which arises is whether the two frameworks are equivalent. Classically, this is true—Gibbs-preserving maps are no more powerful than thermal operations. Here, we show that this no longer holds in the quantum regime: a Gibbs-preserving map can generate coherent superpositions of energy levels while thermal operations cannot. This gap has an impact on clarifying a mathematical framework for quantum thermodynamics.

DOI
10.1088/1367-2630/17/4/043003

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In this brief paper, we compare two frameworks for characterizing possible operations in quantum thermodynamics. One framework considers thermal operations—unitaries which conserve energy. The other framework considers all maps which preserve the Gibbs state at a given temperature. Thermal operations preserve the Gibbs state; hence a natural question which arises is whether the two frameworks are equivalent. Classically, this is true—Gibbs-preserving maps are no more powerful than thermal operations. Here, we show that this no longer holds in the quantum regime: a Gibbs-preserving map can generate coherent superpositions of energy levels while thermal operations cannot. This gap has an impact on clarifying a mathematical framework for quantum thermodynamics.
In this brief paper, we compare two frameworks for characterizing possible operations in quantum thermodynamics. One framework considers thermal operations—unitaries which conserve energy. The other framework considers all maps which preserve the Gibbs state at a given temperature. Thermal operations preserve the Gibbs state; hence a natural question which arises is whether the two frameworks are equivalent. Classically, this is true—Gibbs-preserving maps are no more powerful than thermal operations. Here, we show that this no longer holds in the quantum regime: a Gibbs-preserving map can generate coherent superpositions of energy levels while thermal operations cannot. This gap has an impact on clarifying a mathematical framework for quantum thermodynamics.
Thermal operations are an operational model of non-equilibrium quantum thermodynamics. In the absence of coherence between energy levels, exact state transition conditions under thermal operations are known in terms of a mathematical relation called thermo-majorization. But incorporating coherence has turned out to be challenging, even under the relatively tractable model wherein all Gibbs state-preserving quantum channels are included. Here we find a mathematical generalization of thermal operations at low temperatures, 'cooling maps', for which we derive the necessary and sufficient state transition condition. Cooling maps that saturate recently discovered bounds on coherence transfer are realizable as thermal operations, motivating us to conjecture that all cooling maps are thermal operations. Cooling maps, though a less-conservative generalization to thermal operations, are more tractable than Gibbs-preserving operations, suggesting that cooling map-like models at general temperatures could be of use in gaining insight about thermal operations.
Gibbs-preserving operations have been studied as one of the standard free processes in quantum thermodynamics. Although they admit a simple mathematical structure, their operational significance has been unclear due to the potential hidden cost to implement them using an operationally motivated class of operations, such as thermal operations. Here, we show that this hidden cost can be infinite-we present a family of Gibbs-preserving operations that cannot be implemented by thermal operations aided by any finite amount of quantum coherence. Our result implies that there are uncountably many Gibbs-preserving operations that require unbounded thermodynamic resources to implement, raising a question about employing Gibbs-preserving operations as available thermodynamics processes. This finding is a consequence of the general lower bounds we provide for the coherence cost of approximately implementing a certain class of Gibbs-preserving operations with a desired accuracy. We find that our lower bound is almost tight, identifying a quantity-related to the energy change caused by the channel to implement-as a fundamental quantifier characterizing the coherence cost for the approximate implementation of Gibbs-preserving operations.
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).
Chapter 17 delves into quantum thermodynamics, building on the concepts introduced in the resource theory of nonuniformity. This chapter focuses on thermal states and athermality as resources within the quantum domain, emphasizing the significance of Gibbs states and their role in quantum statistical mechanics. It outlines the operational framework for thermal operations, setting the stage for discussions on energy conservation and the second law of thermodynamics in quantum systems. A key aspect of the chapter is the exploration of quasi-classical athermality, illustrating how quantum states deviate from thermal equilibrium when the state of the system commutes with its Hamiltonian. In the fully quantum domain, the chapter introduces closed formulas for quantifying athermality, such as the athermality cost and distillable athermality, both in the single-shot and the asymptotic domains. These measures provide a quantitative understanding of the efficiency of thermal operations and the potential for work extraction or consumption.
Quantum thermodynamics is traditionally formulated as a theory of equilibrium states and state transformations. Recent advances in higher-order quantum transformations, which describe physical scenarios beyond states and channels and provide a systematic framework for causal order, raise the question of how equilibrium should be defined and preserved in this more general setting. Starting from the Gibbs state as the unique equilibrium state, we identify equilibrium preservation as its natural higher-order extension. We show that, while this principle can in general give rise to distinct classes of transformations, all such distinctions disappear when equilibrium preservation is required completely, namely under arbitrary ancillary extensions. Remarkably, every transformation satisfying this condition is causally ordered, making causal order an emergent consequence of thermodynamic equilibrium. We establish this result for arbitrary higher-order maps and use the resulting framework to introduce free-energy-like quantities for quantum channels. Our findings reveal a fundamental connection between thermodynamic equilibrium and causal structure, providing a foundation for a fully fledged theory of higher-order quantum thermodynamics.