Geometric bloch vector solution to minimum-error discriminations of mixed qubit states

Abstract

We investigate minimum-error (ME) discrimination for mixed qubit states using a geometric approach. By analyzing positive operator-valued measure (POVM) solutions and introducing Lagrange operator Gamma , we develop a four-step structured instruction to find Gamma for N mixed qubit states. Our method covers optimal solutions for two, three, and four mixed qubit states, including a novel result for four qubit states. We introduce geometric-based POVM classes and non-decomposable subsets for constructing optimal solutions, enabling us to find all possible answers for the general problem of minimum-error discrimination for N mixed qubit states with arbitrary a priori probabilities.