Geometric coherence and quantum state discrimination

One of the important features of quantum mechanics is that non-orthogonal quantum states cannot be perfectly discriminated. Therefore, a fundamental problem in quantum mechanics is to design a optimal measurements to discriminate a collection of non-orthogonal quantum states. We prove that the geometric coherence of a quantum state is the minimal error probability to discrimination a set of linear independent pure states, which provides an operational interpretation for geometric coherence. Moreover, the closest incoherent states are given in terms of the corresponding optimal von Neumann measurements. Based on this idea, the explicitly expression of geometric coherence are given for a class of states. On the converse, we show that, any discrimination task for a collection of linear independent pure states can be also regarded as the problem of calculating the geometric coherence for a quantum state, and the optimal measurement can be obtain through the corresponding closest incoherent state.