Complete analysis to minimum-error discrimination of mixed four qubit states with arbitrary prior probabilities.

In this work, we provide a complete analysis to minimum-error discrimination of arbitrary four qubit states with arbitrary $a~ priori$ probabilities. For the complete anaysis, the most important work to do is to find the necessary and sufficient conditions for the existence of null measurement operator. From the geometric structure of qubit states, we obtain the analytic condition for deciding the existence of a null operator in minimum-error measurement for four qubit states, which also gives the necessary and sufficient conditions for every optimal POVM to have non-zero elements. Using the condition, we completely analyze minimum-error discrimination of four qubit states with arbitrary $a~ priori$ probabilities.