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.

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