An optimal discrimination of two mixed qubit states with a fixed rate of inconclusive results

In this paper we consider the optimal discrimination of two mixed qubit states for a measurement that allows a fixed rate of inconclusive results(FRIR). Our strategy for the problem is to transform the FRIR of two qubit states into a minimum error discrimination for three qubit states by adding a specific quantum state $\rho_{0}$ and a prior probability $q_{0}$(which we will call an inconclusive degree), which we name the modified FRIR problem. First, we investigate special inconclusive degrees $q_{0}^{(0)}$ and $q_{0}^{(1)}$, which appear naturally in the modified FRIR problem and are the beginning and the end of practical interval of inconclusive degree, and find the analytic form of them. Next, we show that the modified FRIR problem can be classified into two cases $q_{0}=q_{0}^{(0)}$(or $q_{0}=q_{0}^{(1)}$) and $q_{0}^{(0)}\!<\!q_{0}\!<\!q_{0}^{(1)}$. In fact, by maximum confidences of two qubit states and non-diagonal element of $\rho_{0}$, the modified FRIR problem is completely understood. Then, we provide an analytic solution of the FRIR problem when $q_{0}=q_{0}^{(0)}$(or $q_{0}=q_{0}^{(1)}$). However, when $q_{0}^{(0)}\!<\!q_{0}\!<\!q_{0}^{(1)}$, we rather supply the numerical method to find the solution, because of the complex relation between inconclusive degree and corresponding failure probability. Finally we confirm our results using previously known examples.