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Register a new userOptimal positive-operator-valued measures for unambiguous state discrimination
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Authors
Boris F. Samsonov
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Optimization of the mean efficiency for unambiguous (or error free)discrimination among $N$ given linearly independent nonorthogonal states should be realized in a way to keep the probabilistic quantum mechanical interpretation. This imposes a condition on a certain matrix to be positive semidefinite. We reformulated this condition in such a way that the conditioned optimization problem for the mean efficiency was reduced to finding an unconditioned maximum of a function defined on a unit $N$-sphere for equiprobable states and on an $N$-ellipsoid if the states are given with different probabilities. We established that for equiprobable states a point on the sphere with equal values of Cartesian coordinates, which we call symmetric point, plays a special role. Sufficient conditions for a vector set are formulated for which the mean efficiency for equiprobable states takes its maximal value at the symmetric point. This set, in particular, includes previously studied symmetric states. A subset of symmetric states, for which the optimal measurement corresponds to a POVM requiring a one-dimensional ancilla space is constructed. We presented our constructions of a POVM suitable for the ancilla space dimension varying from 1 till $N$ and the Neumarks extension differing from the existing schemes by the property that it is straightforwardly applicable to the case when it is desirable to present the whole space system + ancilla as the tensor product of a two-dimensional ancilla space and the $N$-dimensional system space.
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We introduce several notions of random positive operator valued measures (POVMs), and we prove that some of them are equivalent. We then study statistical properties of the effect operators for the canonical examples, obtaining limiting eigenvalue distributions with the help of free probability theory. Similarly, we obtain the large system limit for several quantities of interest in quantum information theory, such as the sharpness, the noise content, and the probability range. Finally, we study different compatibility criteria, and we compare them for generic POVMs.
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Evaluating the amount of information obtained from nonorthogonal quantum states is an important topic in the field of quantum information. The commonly used evaluation method is Holevo bound, which only provides an upper bound for quantum measurement. In this paper, we provide a theoretical study of the positive operator-valued measure (POVM) for two nonorthogonal mixed states. We construct a generalized POVM measurement operation, and derive the optimal POVM measurement by Lagrange multiplier method. With simulation, we find that the optimal POVM measurement provides a lower bound for quantum state measurement, which is significantly lower than that predicted by Holevo bound. The derivation of optimal POVM measurement will play an important role in the security research of quantum key distribution.
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