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

Abstract

In this paper we consider the optimal discrimination of two mixed qubit states for a measurement that allows a fixed rate of inconclusive results. Our strategy is to transform the problem 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 behaves as an inconclusive degree. First, we introduce the beginning and the end of practical interval of inconclusive result, $$q_{0}^{(0)}$$ and $$q_{0}^{(1)}$$ , which are key ingredients in investigating our problem. Then we obtain the analytic form of them. Next, we show that our 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 $$\rho _{0}$$ , our problem is completely understood. We provide an analytic solution of our 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.