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Research in Optical Sciences

Optimal Measurement for the Discrimination of Quantum States with a Fixed Rate of Inconclusive Outcomes

János A. Bergou, R. Muñoz-Tapia, E. Bagán, G. A. Olivares-Rentería

Published Jan 1, 2012• 0 citations
Publisher

Abstract

We present the optimal measurement scheme for discriminating among quantum states when a certain fixed rate of inconclusive measurement results is allowed and give analytical results for the maximal success probability for all ranges of the parameters.
DOI
10.1364/qim.2012.qw4a.6

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We study the discrimination of NNN mixed quantum states in an optimal measurement that maximizes the probability of correct results while the probability of inconclusive results is fixed at a given value. After considering the discrimination of NNN states in a ddd-dimensional Hilbert space, we focus on the discrimination of qubit states. We develop a method to determine an optimal measurement for discriminating arbitrary qubit states, taking into account that often the optimal measurement is not unique and the maximum probability of correct results can be achieved by several different measurements. Analytical results are derived for a number of examples, mostly for the discrimination between qubit states which possess a partial symmetry, but also for discriminating NNN equiprobable qubit states and for the dicrimination between a pure and a uniformly mixed state in ddd dimensions. In the special case where the fixed rate of inconclusive results is equal to zero, our method provides a treatment for the minimum-error discrimination of arbitrary qubit states which differs from previous approaches.
We propose an upper bound on the maximum correct probability of quantum measurements. The proposed bound is obtained by a suboptimal solution to the dual problem of the optimal state discrimination problems. We derive that a slightly modified version of the proposed bound is tighter than that proposed by Qiu et al. [Phys. Rev. A 81, 042329 (2010)]. We also propose an upper bound on the maximum correct probability with a fixed rate of inconclusive results. The performance of the proposed bounds are evaluated through numerical experiments.
We study an optimal quantum measurement for generalized quantum state discrimination problems, which include the problem of finding an optimal measurement maximizing the average success probability with and without a fixed rate of inconclusive results and the problem of finding an optimal measurement in the Neyman-Pearson strategy. We show that for any generalized quantum state discrimination problem, there is a corresponding minimum error discrimination problem sharing the same optimal measurement. We propose an approach in which the optimal measurement is obtained by solving the corresponding minimum error discrimination problem, and clarify the relationship between optimal solutions to the original and the corresponding problems, with which one can obtain an optimal solution to the original problem in some cases. Moreover, as an example of application of our approach, we present an algorithm for numerically obtaining optimal solutions to generalized quantum state discrimination problems.