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 q0, 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)) rho 0. 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.