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Physical Review A

Mathematical nature of and a family of lower bounds for the success probability of unambiguous discrimination

Xiaoming Sun, Shengyu Zhang, Yuan Feng, Mingsheng Ying

Published Apr 10, 2002• 30 citations
Publisher

Abstract

Unambiguous discrimination is a strategy to the discrimination problem that identifies the state with certainty, leaving a possibility of undecidability. This paper points out that the optimal success probability of unambiguous discrimination is mathematically the well-known semidefinite programming problem. A family of lower bounds of the optimal success probability is also given.

DOI
10.1103/physreva.65.044306

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Unambiguous discrimination is a strategy to the discrimination problem that identifies the state with certainty, leaving a possibility of undecidability. This paper points out that the optimal success probability of unambiguous discrimination is mathematically the well-known semidefinite programming problem. A family of lower bounds of the optimal success probability is also given.
For the general problem of optimum unambiguous discrimination of quantum states, it is very hard to get the explicit analytical form for optimum detection operators. Therefore, it is necessary to find an efficient numerical method for designing optimum detection operators. According to the theory of semidefinite programming, it is demonstrated that the design of the optimal quantum measurement can be formulated as a standard semidefinite programming problem, and the optimality conditions for semidefinite programming can be used to perspicuously derive a set of optimality conditions for optimum detection operators.By means of the standard semidefinite programming, the maximum value of the success probability of detection and the optimum detection operators can be obtained directly in polynomial time. It is exemplified that this method is simple to be implemented and can be used to design optimum detection operators for quantum-state discrimination efficiently.
We propose a general description for the unambiguous discrimination of mixed states according to the system-environment coupling, and present a procedure to reduce this to a standard semidefinite programming problem. In the two-state case, we introduce the canonical vectors and partly simplify the problem to the case of discrimination between pairs of canonical vectors. By considering the positivity of the $2\ifmmode\times\else\texttimes\fi{}2$ matrices, we obtain a series of new upper bounds for the total success probability, which depends on both the prior probabilities and specific state structures.
Quantum state discrimination is a fundamental task that is meaningful in quantum information theory. In this manuscript, we consider a revised unambiguous discrimination of quantum resources. First, we present an upper bound of the success probability for a revised unambiguous discrimination task in the unasymptotic and asymptotic scenarios. Next, we generalize the task from quantum states to quantum channels. We present an upper bound of the success probability for the task under the adaptive strategy. Furthermore, we show the bound can be computed efficiently. Finally, compared with the classical unambiguous discrimination, we show the advantage of the quantum by considering a quantifier on a set of semidefinite positive operators.
Unambiguous discrimination among non-orthogonal but linearly independent quantum states is possible with a certain probability of success. Here we consider a variant of that problem; rather than discriminating among all of the different states, we shall discriminate between two subsets of them. In particular, for the case of 3 states, we find the optimal strategy to distinguish one from the other two. This procedure has a higher probability of success than distinguishing all 3 states. We show how to construct generalized interferometers that provide an implementation of the optimal strategy.