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Zhongguo kexue. Wulixue Lixue Tianwenxue

Universal programmable maximum confidence quantum measurement of two unknown states

Shouya Wang, Xiaodong Zheng, Wenhai Zhang

Published Oct 30, 2024• 0 citations
Publisher

Abstract

In this study, the universal programmable maximum confidence quantum measurement of two unknown states is investigated. The set of measurement operators includes the correct operators and an inconclusive operator. We first determine the two correct operators of the optimal positive-operator valued measure, from which the individual correct probabilities can be obtained. In order to evaluate the efficiency of the correct operators, the average confidence probability is defined. In terms of the positivity of an inconclusive operator, we calculate the value of the average confidence probability and present the explicit expression of the maximum of the average confidence probability. Finally, the application of the research results to quantum key distribution is discussed.

DOI
10.1360/sspma-2024-0381

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In this study, the universal programmable maximum confidence quantum measurement of two unknown states is investigated. The set of measurement operators includes the correct operators and an inconclusive operator. We first determine the two correct operators of the optimal positive-operator valued measure, from which the individual correct probabilities can be obtained. In order to evaluate the efficiency of the correct operators, the average confidence probability is defined. In terms of the positivity of an inconclusive operator, we calculate the value of the average confidence probability and present the explicit expression of the maximum of the average confidence probability. Finally, the application of the research results to quantum key distribution is discussed.
We investigate the universal programmable unambiguous discrimination of two unknown quantum states with two copies. We first present the optimal positive-operator valued measure (POVM) elements that determine the individual success probabilities. In terms of the positivity of an inconclusive operator, we derive the average success probability, which is higher than the average success probability of discriminating two unknown states with one copy given in the reference [Bergou and Hillery, Phys. Rev. Lett. 94, 160501 (2005)].
We study an optimized measurement that discriminates two mixed quantum states with maximum confidence for each conclusive result, thereby keeping the overall probability of inconclusive results as small as possible. When the rank of the detection operators associated with the two different conclusive outcomes does not exceed unity, we obtain a general solution. As an application, we consider the discrimination of two mixed qubit states. Moreover, for the case of higher-rank detection operators we give a solution for particular states. The relation of the optimized measurement to other discrimination schemes is also discussed.
We study an optimum measurement for quantum state discrimination, which maximizes the probability of correct results when the probability of inconclusive results is fixed at a given value. The measurement describes minimum-error discrimination if this value is zero, while under certain conditions it corresponds to optimized maximum-confidence discrimination, or to optimum unambiguous discrimination, respectively, when the fixed value reaches a definite minimum. Using operator conditions that determine the optimum measurement, we derive analytical solutions for the discrimination of two mixed qubit states, including the case of two pure states occurring with arbitrary prior probabilities, and for the discrimination of NNN symmetric states, both pure and mixed. We also consider a case where the given density operators resolve the identity operator, and we specify the optimality conditions for partially symmetric states. Moreover, we show that from the complete solution for arbitrary values of the fixed rate of inconclusive results one can always obtain the optimum measurement in another strategy where the error rate is fixed, and vice versa.
We study an optimized measurement which discriminates NNN mixed quantum states occurring with given prior probabilities. The measurement yields the maximum achievable confidence for each of the NNN conclusive outcomes, thereby keeping the overall probability of inconclusive outcomes as small as possible. It corresponds to optimum unambiguous discrimination when for each outcome the confidence is equal to unity. Necessary and sufficient optimality conditions are derived and general properties of the optimum measurement are obtained. The results are applied to the optimized maximum-confidence discrimination of NNN equiprobable symmetric mixed states. Analytical solutions are presented for a number of examples, including the discrimination of NNN symmetric pure states spanning a ddd-dimensional Hilbert space ($d\ensuremath{\leqslant}N$) and the discrimination of NNN symmetric mixed qubit states.
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.