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arXiv:1804.05154 · Journal of Physics A Mathematical and Theoretical

Is coherence catalytic?

Joan A. Vaccaro, Sarah Croke, Stephen M Barnett

Published Apr 30, 2018• 30 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/aac112

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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.
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.
In the presence of conservation laws, superpositions of eigenstates of the corresponding conserved quantities cannot be generated by quantum dynamics. Thus, any such coherence represents a potentially valuable resource of asymmetry, which can be used, for example, to enhance the precision of quantum metrology or to enable state transitions in quantum thermodynamics. Here we ask if such superpositions, already present in a reference system, can be broadcast to other systems, thereby distributing asymmetry indefinitely at the expense of creating correlations. We prove a no-go theorem showing that this is forbidden by quantum mechanics in every finite-dimensional system. In doing so, we also answer some open questions in the quantum information literature concerning the sharing of timing information of a clock and the possibility of catalysis in quantum thermodynamics. We also prove that even weaker forms of broadcasting, of which Åberg's "catalytic coherence" is a particular example, can only occur in the presence of infinite-dimensional reference systems. Our results set fundamental limits on the creation and manipulation of quantum coherence and shed light on the possibilities and limitations of quantum reference frames to act catalytically without being degraded.
Abstract There has been substantial interest of late on the issue of coherence as a resource in quantum thermodynamics. To date, however, analyses have focused on somewhat artificial theoretical models. We seek to bring these ideas closer to experimental investigation by examining the ‘catalytic’ nature of quantum optical coherence. Here the interaction of a coherent state cavity field with a sequence of two-level atoms is considered, a state ubiquitous in quantum optics as a model of a stable, classical source of light. The Jaynes–Cummings interaction Hamiltonian is used, so that an exact solution for the dynamics can be formed, and the evolution of the atomic and cavity states with each atom-field interaction analysed. In this way, the degradation of the coherent state is examined as coherence is transferred to the sequence of atoms. The associated degradation of the coherence in the cavity mode is significant in the context of the use of coherence as a thermodynamic resource.
In the classical regime, thermodynamic state transformations are governed by the free energy. This is also called as the second law of thermodynamics. Previous works showed that, access to a catalytic system allows us to restore the second law in the quantum regime when we ignore coherence. However, in the quantum regime, coherence and free energy are two independent resources. Therefore, coherence places additional nontrivial restrictions on the state transformations that remain elusive. In order to close this gap, we isolate and study the nature of coherence, i.e., we assume access to a source of free energy. We show that allowing catalysis along with a source of free energy allows us to amplify any quantum coherence present in the quantum state arbitrarily. Additionally, any correlations between the system and the catalyst can be suppressed arbitrarily. Therefore, our results provide a key step in formulating a fully general law of quantum thermodynamics.
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.