In this letter, by treating minimum-error state discrimination as a complementarity problem, we obtain the geometric optimality conditions. These can be used as the necessary and sufficient conditions to determine whether every optimal measurement operator can be nonzero. Using these conditions and an inductive approach, we demonstrate a geometric method and the intrinsic polytope for N -qubit-mixed-state discrimination. When the intrinsic polytope becomes a point, a line segment, or a triangle, the guessing probability, the necessary and sufficient condition for the exact solution, and the optimal measurement are analytically obtained. We apply this result to the problem of discrimination to arbitrary three-qubit mixed states with given a priori probabilities and obtain the complete analytic solution to the guessing probability and optimal measurement.The goal of quantum-state discrimination is to distinguish between states of a given set as well as possible. In other words, it can be regarded as a problem to find the optimal measurement for discriminating among the given quantum states. In fact, every state in classical physics can be orthogonal to each other and therefore distinguished perfectly [1]. However, in quantum physics, a state cannot be perfectly discriminated because of the existence of nonorthogonal states [2][3][4]. Quantum-state discrimination[5] is classified into minimum error discrimination, originally introduced by Helstrom [2], unambiguous discrimination [6][7][8], and maximum confidence discrimination [9]. The purpose of minimum error strategy is to find the optimal measurement and the minimum error probability (or guessing probability) for arbitrary N -qudit mixed quantum states with arbitrary a priori probabilities. In the N = 2 case, regardless of the dimension, the Helstrom bound [2] gives an analytic solution to the problem. In the N = 3 case the analytic solution for pure qubit states is provided by [10,11]. In [12] the analytic solution for mixed qubit states is considered without the necessary and sufficient conditions for the solution. In other words, the full understanding for discrimination of three-qubit mixed quantum states is not provided yet.The optimal measurement for linearly independent quantum states is the von Neumann measurement [13]. But if the given quantum states are linearly dependent, the von Neumann measurement may not be optimal. Therefore, the positive-operator-valued-measure (POVM) should be used for arbitrary quantum states. From the point where POVM can be used as a measurement and the probability to guess the quantum states correctly becomes convex, the minimum error discrimination problem may be solved by convex optimization [14]. There have also been some efforts to solve it using a dual problem [15] or complementarity problem [16]. By applying qubit-state geometry to the optimality conditions for measurement operators and complementary states, Bae [17] obtained a geometric method to find the * Electronic address: yyhkwon@hanyang.ac.kr guessing probabilit...

No abstract

No 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 ρ0 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 (1) 0 , 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 q0 = q 0 . In fact, by maximum confidences of two qubit states and non-diagonal element of ρ0, the our problem is completely understood. We provide an analytic solution of our problem when q0 = q (1) 0 , 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.

No abstract