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...

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Recently, J. A. Bergou et al. proposed sequential state discrimination as a new quantum state discrimination scheme. In the scheme, by the successful sequential discrimination of a qubit state, receivers Bob and Charlie can share the information of the qubit prepared by a sender Alice. A merit of the scheme is that a quantum channel is established between Bob and Charlie, but a classical communication is not allowed. In this report, we present a method for extending the original sequential state discrimination of two qubit states to a scheme of N linearly independent pure quantum states. Specifically, we obtain the conditions for the sequential state discrimination of N = 3 pure quantum states. We can analytically provide conditions when there is a special symmetry among N = 3 linearly independent pure quantum states. Additionally, we show that the scenario proposed in this study can be applied to quantum key distribution. Furthermore, we show that the sequential state discrimination of three qutrit states performs better than the strategy of probabilistic quantum cloning.

Sequential state discrimination is a strategy for quantum state discrimination of a sender’s quantum states when N receivers are separately located. In this report, we propose optical designs that can perform sequential state discrimination of two coherent states. For this purpose, we consider not only binary phase-shifting-key (BPSK) signals but also general coherent states, with arbitrary prior probabilities. Since our optical designs do not include electric feedback, they can be implemented without difficulty. Furthermore, we analyze our proposal for the case of photon loss. We also demonstrate that sequential state discrimination of two coherent states performs better than the probabilistic quantum cloning strategy. This proposal can facilitate multiparty QKD based on coherent states.

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