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Upper Bounds on Optimal Correct Probability of Quantum State Discrimination with and without Inconclusive Results

Kenji Nakahira, Tsuyoshi Sasaki Usuda, Kentaro Kato

Published Jan 23, 2016• 0 citations
OpenAlex

Abstract

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.

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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 propose upper and lower bounds on the maximum success probability for discriminating given quantum states. The proposed upper bound is obtained from a suboptimal solution to the dual problem of the corresponding optimal state discrimination problem. We also give a necessary and sufficient condition for the upper bound to achieve the maximum success probability; the proposed lower bound can be obtained from this condition. It is derived that a slightly modified version of the proposed upper bound is tighter than that proposed by Qiu et al. [Phys. Rev. A 81, 042329 (2010)]. Moreover, we propose upper and lower bounds on the maximum success probability with a fixed rate of inconclusive results. The performance of the proposed bounds is evaluated through numerical experiments.
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 propose an approach for finding an optimal measurement for quantum state discrimination that maximizes the probability of correct detection with a fixed rate of inconclusive results. In our approach, we obtain the optimal measurement by solving the problem of finding a measurement that maximizes the weighted sum of the probability of correct detection and that of inconclusive results. We show that this problem can be reduced to the widely studied problem of finding a minimum error measurement for a certain state set, which maximizes the probability of correct detection without inconclusive results. As an application of our approach, we show how to solve the problem of finding an optimal measurement for qubit states with a fixed rate of inconclusive results.
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 address a broad class of optimization problems of finding quantum measurements, which includes the problems of finding an optimal measurement in the Bayes criterion and a measurement maximizing the average correct probability with a fixed rate of inconclusive results. Our approach can deal with any problem in which each of the objective and constraint functions is formulated by the sum of the traces of the multiplication of a Hermitian operator and a detection operator. We first derive dual problems and necessary and sufficient conditions for an optimal measurement. We also consider the minimax version of these problems and provide necessary and sufficient conditions for a minimax solution. Finally, for optimization problem having a certain symmetry, there exists an optimal solution with the same symmetry. Examples are shown to illustrate how our results can be used.
We present the solution to the problem of optimally discriminating among quantum states, i.e., identifying the states with maximum probability of success when a certain fixed rate of inconclusive answers is allowed. By varying the inconclusive rate, the scheme optimally interpolates between unambiguous and minimum error discrimination, the two standard approaches to quantum state discrimination. We introduce a very general method that enables us to obtain the solution in a wide range of cases and give a complete characterization of the minimum discrimination error as a function of the rate of inconclusive answers. A critical value of this rate is identified that coincides with the minimum failure probability in the cases where unambiguous discrimination is possible and provides a natural generalization of it when states cannot be unambiguously discriminated. The method is illustrated on two explicit examples: discrimination of two pure states with arbitrary prior probabilities and discrimination of trine states.