Family of coherence measures and duality between quantum coherence and path distinguishability

TL;DR

This paper introduces a new class of quantum coherence measures based on $ ext{α}$-affinity, provides their formulas, and explores their operational meaning and relationship with path distinguishability in quantum systems.

Contribution

It develops a novel family of coherence measures using $ ext{α}$-affinity, derives their formulas, and links them to quantum state discrimination and path distinguishability.

Findings

Analytic formulas for $ ext{α}$-affinity coherence measures.

Operational interpretation of $1/2$-affinity as error probability in state discrimination.

A complementarity relation between $1/2$-affinity of coherence and path distinguishability.

Abstract

Coherence measures and their operational interpretations lay the cornerstone of coherence theory. In this paper, we introduce a class of coherence measures with $\alpha$-affinity, say $\alpha$-affinity of coherence for $\alpha \in (0, 1)$. Furthermore, we obtain the analytic formulae for these coherence measures and study their corresponding convex roof extension. We provide an operational interpretation for $1/2$-affinity of coherence by showing that it is equal to the error probability to discrimination a set of pure states with the least square measurement. Employing this relationship we regain the optimal measurement for equiprobable quantum state discrimination. Moreover, we compare these coherence quantifiers, and establish a complementarity relation between $1/2$-affinity of coherence and path distinguishability for some special cases.