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We investigate minimum-error (ME) discrimination for mixed qubit states usinga geometric approach. By analyzing positive operator-valued measure (POVM)solutions and introducing Lagrange operator $\Gamma$, we develop a four-stepstructured instruction to find $\Gamma$ for $N$ mixed qubit states. Our methodcovers optimal solutions for two, three, and four mixed qubit states, includinga novel result for four qubit states. We introduce geometric-based POVM classesand non-decomposable subsets for constructing optimal solutions, enabling us tofind all possible answers for the general problem of minimum-errordiscrimination for $N$ mixed qubit states with arbitrary a prioriprobabilities.

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