Max- relative entropy of coherence: an operational coherence measure

Abstract

The operational characterization of quantum coherence is the corner stone in the development of resource theory of coherence. We introduce a new coherence quantifier based on max-relative entropy. We prove that max-relative entropy of coherence is directly related to the maximum overlap with maximally coherent states under a particular class of operations, which provides an operational interpretation of max-relative entropy of coherence. Moreover, we show that, for any coherent state, there are examples of subchannel discrimination problems such that this coherent state allows for a higher probability of successfully discriminating subchannels than that of all incoherent states. This advantage of coherent states in subchannel discrimination can be exactly characterized by the max-relative entropy of coherence. By introducing suitable smooth max-relative entropy of coherence, we prove that the smooth max-relative entropy of coherence provides a lower bound of one-shot coherence cost, and the max-relative entropy of coherence is equivalent to the relative entropy of coherence in asymptotic limit. Similar to max-relative entropy of coherence, min-relative entropy of coherence has also been investigated. We show that the min-relative entropy of coherence provides an upper bound of one-shot coherence distillation, and in asymptotic limit the min-relative entropy of coherence is equivalent to the relative entropy of coherence.

v2. 5+6.5 pages, no figure, close to the published version. v1. 5.5+6 pages, no figure

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• One-Shot Operational Quantum Resource Theory
• Maximum coherence in the optimal basis
• Quantifying quantum coherence based on the generalized $\alpha -z-$relative R$\acute{e}$nyi entropy
• Interpreting quantum coherence through a quantum measurement process