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Optimum Unambiguous Discrimination of Quantum States Based on Semidefinite Programming

Guohua Dong

Published Jan 1, 2008• 0 citations
OpenAlex

Abstract

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.

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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.
In this paper we consider the general problem of designing the optimal quantum detection for minimum-error quantum state discrimination. Since it is hard to get an explicit solution and we must resort to numerical methods for the general case. We show the problem can be cast into a semidefinite programming(SDP) problem, we use the theory of SDP to solve it. Based on the idea, we propose a method for minimum-error discrimination by formulating the design of optimal quantum detection as a standard SDP problem; furthermore, based on the optimality conditions for SDP, we derive a set of optimality conditions for quantum detection. It is exemplified that this method is simple to implement and can be used to design optimum detection operators for minimum-error discrimination efficiently.
Abstract—In this paper, we consider the problem of unambiguous discrimination between a set of linearly independent pure quantum states. We show that the design of the optimal measurement that minimizes the probability of an inconclusive result can be formulated as a semidefinite programming problem. Based on this formulation, we develop a set of necessary and sufficient conditions for an optimal quantum measurement. We show that the optimal measurement can be computed very efficiently in polynomial time by exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum. Using the general conditions for optimality, we derive necessary and sufficient conditions so that the measurement that results in an equal probability of an inconclusive result for each one of the quantum states is optimal. We refer to this measurement as the equal-probability measurement (EPM). We then show that for any state set, the prior probabilities of the states can be chosen such that the EPM is optimal. Finally, we consider state sets with strong symmetry properties and equal prior probabilities for which the EPM is optimal. We first consider geometrically uniform (GU) state sets that are defined over a group of unitary matrices and are generated by a single generating vector. We then consider compound GU state sets which are generated by a group of unitary matrices using multiple generating vectors, where the generating vectors satisfy a certain (weighted) norm constraint. Index Terms—Compound geometrically uniform (CGU) quantum states, equal-probability measurement (EPM), geometrically uniform (GU) quantum states, quantum detection, semidefinite programming, unambiguous discrimination. I.
We consider the problem of unambiguous discrimination between a set of linearly independent pure quantum states. We show that the design of the optimal measurement that minimizes the probability of an inconclusive result can be formulated as a semidefinite programming problem. Based on this formulation, we develop a set of necessary and sufficient conditions for an optimal quantum measurement. We show that the optimal measurement can be computed very efficiently in polynomial time by exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum. Using the general conditions for optimality, we derive necessary and sufficient conditions so that the measurement that results in an equal probability of an inconclusive result for each one of the quantum states is optimal. We refer to this measurement as the equal-probability measurement (EPM). We then show that for any state set, the prior probabilities of the states can be chosen such that the EPM is optimal. Finally, we consider state sets with strong symmetry properties and equal prior probabilities for which the EPM is optimal. We first consider geometrically uniform (GU) state sets that are defined over a group of unitary matrices and are generated by a single generating vector. We then consider compound GU state sets which are generated by a group of unitary matrices using multiple generating vectors, where the generating vectors satisfy a certain (weighted) norm constraint.
We consider the problem of designing an optimal quantum detector to minimize the probability of a detection error when distinguishing among a collection of quantum states, represented by a set of density operators. We show that the design of the optimal detector can be formulated as a semidefinite programming problem. Based on this formulation, we derive a set of necessary and sufficient conditions for an optimal quantum measurement. We then show that the optimal measurement can be found by solving a standard (convex) semidefinite program. By exploiting the many well-known algorithms for solving semidefinite programs, which are guaranteed to converge to the global optimum, the optimal measurement can be computed very efficiently in polynomial time within any desired accuracy. Using the semidefinite programming formulation, we also show that the rank of each optimal measurement operator is no larger than the rank of the corresponding density operator. In particular, if the quantum state ensemble is a pure-state ensemble consisting of (not necessarily independent) rank-one density operators, then we show that the optimal measurement is a pure-state measurement consisting of rank-one measurement operators.
We consider the problem of determining the state of an unknown quantum sequence without error. The elements of the given sequence are drawn with equal probability from a known set of linearly independent pure quantum states with the property that their mutual inner products are all real and equal. This problem can be posed as an instance of unambiguous state discrimination where the states correspond to that of all possible sequences having the same length as the given one. We calculate the optimum probability by solving the optimality conditions of a semidefinite program. The optimum value is achievable by measuring individual members of the sequence, and no collective measurement is necessary.
We consider the problem of designing an optimal quantum detector that distinguishes unambiguously between a collection of mixed quantum states. Using arguments of duality in vector space optimization, we derive necessary and sufficient conditions for an optimal measurement that minimizes the probability of an inconclusive result. We show that the previous optimal measurements that were derived for certain special cases satisfy these optimality conditions. We then consider state sets with strong symmetry properties, and show that the optimal measurement operators for distinguishing between these states share the same symmetries, and can be computed very efficiently by solving a reduced size semidefinite program.