Geometric block-coherence and quantum state discrimination

Quantum coherence with respect to orthonormal bases has been studied extensively in the past few years. From the perspective of operational meaning, geometric coherence can be equal to the minimum error probability to discriminate a set of pure states (2018 J. Phys. A: Math. Theor. 51 414005). By regarding coherence as a physical resource, Baumgratz et al (2014 Phys. Rev. Lett. 113 140401) presented a comprehensive framework for coherence. Recently, geometric block-coherence as an effective block-coherence measure has been proposed. In this paper, we reveal an equivalence relationship between mixed quantum state discrimination (QSD) task and geometric block-coherence, which provides an operational interpretation for geometric block-coherence and generalizes the main result in coherence resource theory. Meanwhile, we show that partial coherence is a special case of block-coherence. By linking the relationship between geometric partial coherence and QSD tasks, we show that the value range of the two measures is the same. Finally, we reveal the relationship between geometric positive operator-valued measure-based coherence and QSD task.