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arXiv:1603.00003

Comment on "Catalytic Coherence"

Salil Bedkihal, Joan A. Vaccaro, Stephen Mark Barnett

Published Feb 27, 2016• 1 citation
arXivHTMLPDF

Abstract

Aberg has claimed in a recent Letter," Phys. Rev. Lett. 113, 150402 (2014)", that the coherence of a reservoir can be used repeatedly to perform coherent operations without ever diminishing in power to do so. The claim has particular relevance for quantum thermodynamics because, as shown in "Phys. Rev. Lett. 113, 150402 (2014)", latent energy that is locked by coherence may be extractable without incurring any additional cost. We show here, however, that repeated use of the reservoir gives an overall coherent operation of diminished accuracy and is necessarily accompanied by an increased thermodynamic cost.

DOI
10.48550/arxiv.1603.00003

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Aberg has claimed in a recent Letter," Phys. Rev. Lett. 113, 150402 (2014)", that the coherence of a reservoir can be used repeatedly to perform coherent operations without ever diminishing in power to do so. The claim has particular relevance for quantum thermodynamics because, as shown in "Phys. Rev. Lett. 113, 150402 (2014)", latent energy that is locked by coherence may be extractable without incurring any additional cost. We show here, however, that repeated use of the reservoir gives an overall coherent operation of diminished accuracy and is necessarily accompanied by an increased thermodynamic cost.
Abstract Quantum coherence, the ability to control the phases in superposition states is a resource, and it is of crucial importance, therefore, to understand how it is consumed in use. It has been suggested that catalytic coherence is possible, that is repeated use of the coherence without degradation or reduction in performance. The claim has particular relevance for quantum thermodynamics because, were it true, it would allow free energy that is locked in coherence to be extracted indefinitely . We address this issue directly with a careful analysis of the proposal by Åberg (2014 Phys. Rev. Lett . 113 150402). We find that coherence cannot be used catalytically, or even repeatedly without limit.
Because of conservation of energy we cannot directly turn a quantum system with a definite energy into a superposition of different energies. However, if we have access to an additional resource in terms of a system with a high degree of coherence, as for standard models of laser light, we can overcome this limitation. The question is to what extent coherence gets degraded when utilized. Here it is shown that coherence can be turned into a catalyst, meaning that we can use it repeatedly without ever diminishing its power to enable coherent operations. This finding stands in contrast to the degradation of other quantum resources and has direct consequences for quantum thermodynamics, as it shows that latent energy that may be locked into superpositions of energy eigenstates can be released catalytically.
Coherence, a strictly quantum phenomenon, has found many applications, from quantum information theory and thermodynamics to quantum foundations and biology. When physical constraints are taken into consideration creation of coherence in a system is usually impossible and must therefore be extracted from another system acting as a reservoir. In this article we present two faithful extraction protocols in the sense that the interaction involved between the system and reservoir is strictly coherence preserving. As an example we implement both in the case where the reservoir is a quantum harmonic oscillator in a coherent and a squeezed state, respectively, and study the limits of repeatable extraction. For a single extraction onto qubits it is demonstrated that, perhaps surprisingly, one of the protocols manages to outperform a previous method, known as catalytic coherence, which allows the creation of an extra amount of coherence between degenerate energy eigenstates of the combined system.
A recent Letter by Wadhia et al. reports a realization of a quantum clock using a double quantum dot (DQD) [Phys. Rev. Lett. 135, 200407 (2005)]. This Comment identifies two fundamental issues: (I) the claimed ``quantum clock" exhibits only classical behavior and lacks intrinsic temporal correlations between ticks; it is not sufficient for accurate time as a good clock. (II) the thermodynamic analysis misassigns entropy production and conflates amplification with measurement; the reported combined entropy is an engineering dissipation, not a fundamental cost of quantum timekeeping.
We establish an operational theory of coherence (or of superposition) in quantum systems, by focusing on the optimal rate of performance of certain tasks. Namely, we introduce the two basic concepts-"coherence distillation" and "coherence cost"-in the processing quantum states under so-called incoherent operations [Baumgratz, Cramer, and Plenio, Phys. Rev. Lett. 113, 140401 (2014)]. We, then, show that, in the asymptotic limit of many copies of a state, both are given by simple single-letter formulas: the distillable coherence is given by the relative entropy of coherence (in other words, we give the relative entropy of coherence its operational interpretation), and the coherence cost by the coherence of formation, which is an optimization over convex decompositions of the state. An immediate corollary is that there exists no bound coherent state in the sense that one would need to consume coherence to create the state, but no coherence could be distilled from it. Further, we demonstrate that the coherence theory is generically an irreversible theory by a simple criterion that completely characterizes all reversible states.