Operational Resource Theory of Coherence

TLDR

We establish an operational theory of coherence in quantum systems by focusing on the optimal rate of performance of certain tasks. We introduce coherence distillation and coherence cost as basic concepts under incoherent operations, and show that in the asymptotic limit their rates are given by the relative entropy of coherence and the coherence of formation, respectively. We find that no bound coherent state exists and that the coherence theory is generically irreversible, with a simple criterion characterizing all reversible states. Rev.

Abstract

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.