# Max- relative entropy of coherence: an operational coherence measure. (arXiv:1707.08795v2 [quant-ph] UPDATED)

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.