The optimal positive operator-valued measure for two nonorthogonal mixed states

Evaluating the amount of information obtained from nonorthogonal quantum states is an important topic in the field of quantum information. The commonly used evaluation method is Holevo bound, which only provides an upper bound for quantum measurement. In this paper, we provide a theoretical study of the positive operator-valued measure (POVM) for two nonorthogonal mixed states. We construct a generalized POVM measurement operation, and derive the optimal POVM measurement by Lagrange multiplier method. With simulation, we find that the optimal POVM measurement provides a lower bound for quantum state measurement, which is significantly lower than that predicted by Holevo bound. The derivation of optimal POVM measurement will play an important role in the security research of quantum key distribution.