Max- relative entropy of coherence: an operational coherence measure

TL;DR

This paper introduces the max-relative entropy of coherence as an operational measure, linking it to subchannel discrimination success and resource quantification in quantum coherence theory.

Contribution

It defines a new coherence quantifier based on max-relative entropy, providing operational interpretation and connections to coherence cost and distillation.

Findings

Max-relative entropy of coherence relates to maximum overlap with maximally coherent states.

It characterizes advantage in subchannel discrimination tasks.

In the asymptotic limit, it equals the relative entropy of coherence.

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.