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arXiv:0906.4884 · Physical Review A

Discrimination with error margin between two states: Case of general occurrence probabilities

H. Sugimoto, T. Hashimoto, M. Horibe, A. Hayashi

Published Nov 17, 2009• 44 citations
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Abstract

We investigate a state discrimination problem which interpolates minimum error and unambiguous discrimination by introducing a margin for the probability of error. We closely analyze discrimination of two pure states with general occurrence probabilities. The optimal measurements are classified into three types. One of the three types of measurement is optimal depending on parameters (occurrence probabilities and error margin). We determine the three domains in the parameter space and the optimal discrimination success probability in each domain in a fully analytic form. It is also shown that when the states to be discriminated are multipartite, the optimal success probability can be attained by local operations and classical communication. For discrimination of two mixed states, an upper bound of the optimal success probability is obtained.

DOI
10.1103/physreva.80.052322

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We investigate a state discrimination problem which interpolates minimum error and unambiguous discrimination by introducing a margin for the probability of error. We closely analyze discrimination of two pure states with general occurrence probabilities. The optimal measurements are classified into three types. One of the three types of measurement is optimal depending on parameters (occurrence probabilities and error margin). We determine the three domains in the parameter space and the optimal discrimination success probability in each domain in a fully analytic form. It is also shown that when the states to be discriminated are multipartite, the optimal success probability can be attained by local operations and classical communication. For discrimination of two mixed states, an upper bound of the optimal success probability is obtained.
There are two common settings in a quantum-state discrimination problem. One is minimum-error discrimination where a wrong guess (error) is allowed and the discrimination success probability is maximized. The other is unambiguous discrimination where errors are not allowed but the inconclusive result ``I do not know'' is possible. We investigate a discrimination problem with a finite margin imposed on the error probability. The two common settings correspond to the error margins 1 and 0. For arbitrary error margin, we determine the optimal discrimination probability for two pure states with equal occurrence probabilities. We also consider the case where the states to be discriminated are multipartite and show that the optimal discrimination probability can be achieved by local operations and classical communication.
We consider a state discrimination problem which deals with settings of minimum-error and unambiguous discrimination systematically by introducing a margin for the probability of an incorrect guess. We analyze discrimination of three symmetric pure states of a qubit. The measurements are classified into three types, and one of the three types is optimal depending on the value of the error margin. The problem is formulated as one of semidefinite programming. Starting with the dual problem derived from the primal one, we analytically obtain the optimal success probability and the optimal measurement that attains it in each domain of the error margin. Moreover, we analyze the case of three symmetric mixed states of a qubit.
Abstract Minimum error discrimination (MED) and Unambiguous discrimination (UD) are two common strategies for quantum state discrimination that can be modified by imposing a finite error margin on the error probability. Error margins 0 and 1 correspond to two common strategies. In this paper, for an arbitrary error margin m , the discrimination problem of equiprobable quantum symmetric states is analytically solved for four distinct cases. A generating set of irreducible and reducible representations of a subgroup of a unitary group are considered, separately, as unitary operators that produce one set of the symmetric states. In the irreducible case, for N ⩾ d mixed and pure qudit states, one critical m which divides the parameter space into two domains is obtained. The number of critical values m in the reducible case is two, for both N mixed and pure qubit states. The reason for this difference between numbers of critical values m is explained. The optimal set of measurements and corresponding maximum success probability in fully analytical form are determined for all values of the error margin. The relationship between the amount of error that is imposed on error probability and geometrical situation of states with changes in rank of element corresponding to inconclusive result is determined. The behaviors of elements of measurement are explained geometrically in order to decrease the error probability in each domain. Furthermore, the problem of the discrimination with error margin among elements of two different sets of symmetric quantum states is studied. The number of critical values m is equivalent to one set in both reducible and irreducible cases. In addition, optimal measurements in each domain are obtained.
We investigate the performance of discrimination strategy in the comparison task of known quantum states. In the discrimination strategy, one infers whether or not two quantum systems are in the same state on the basis of the outcomes of separate discrimination measurements on each system. In some cases with more than two possible states, the optimal strategy in minimum-error comparison is that one should infer the two systems are in different states without any measurement, implying that the discrimination strategy performs worse than the trivial ``no-measurement'' strategy. We present a sufficient condition for this phenomenon to happen. For two pure states with equal prior probabilities, we determine the optimal comparison success probability with an error margin, which interpolates the minimum-error and unambiguous comparison. We find that the discrimination strategy is not optimal except for the minimum-error case.
We investigate a discrimination scheme between unitary processes. By introducing a margin for the probability of an erroneous guess, this scheme interpolates the two standard discrimination schemes: minimum-error and unambiguous discrimination. We present solutions for two cases. One is the case of two unitary processes with general prior probabilities. The other is the case with a group symmetry: The processes comprise a projective representation of a finite group. In the latter case, we found that unambiguous discrimination is a kind of ``all or nothing'': The maximum success probability is either 0 or 1. We also thoroughly analyze how entanglement with an auxiliary system improves discrimination performance.
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.