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
N 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 investigate the optimum strategy for state discrimination of two pure states when the probability of inconclusive answers is fixed. By varying a given rate of inconclusive probabilities, the strategy optimally interpolates between Unambiguous and Minimum-Error discrimination, the two standard approaches to quantum state discrimination. We derive the explicit expressions of all the probabilities using an ancillary system and introducing unitary transformations which act on the input states and produce the output measured on the ancilla. By exploring physically accessible transformation acting on the input states, we present an optical scheme to implement the state discrimination.
We study the discrimination of
N 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
N states in a
d-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
N equiprobable qubit states and for the dicrimination between a pure and a uniformly mixed state in
d 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 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.
In this paper, we investigate the relationship between the minimum-error probability
QE of ambiguous discrimination and the optimal inconclusive probability
QU of unambiguous discrimination. It is known that for discriminating two states, the inequality ${Q}_{U}\ensuremath{\geqslant}2{Q}_{E}$ has been proved in the literature. The main technical results are as follows: (1) We show that, for discriminating more than two states, ${Q}_{U}\ensuremath{\geqslant}2{Q}_{E}$ may not hold again, but the infimum of
QU/QE is
1, and there is no supremum of
QU/QE, which implies that the failure probabilities of the two schemes for discriminating some states may be narrowly or widely gapped. (2) We derive two concrete formulas of the minimum-error probability
QE and the optimal inconclusive probability
QU, respectively, for ambiguous discrimination and unambiguous discrimination among arbitrary
m simultaneously diagonalizable mixed quantum states with given prior probabilities. In addition, we show that
QE and
QU satisfy the relationship that ${Q}_{U}\ensuremath{\geqslant}\frac{m}{m\ensuremath{-}1}{Q}_{E}$.
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.