Geometric Bloch Vector Solution to Minimum Error Discriminations of Mixed Qubit States

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.