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arXiv:1008.0179 · Physical Review A

Minimum error discrimination between similarity-transformed quantum states

M. A. Jafarizadeh, Rahime Sufiani, Y. Mazhari Khiavi

Published Jul 1, 2011• 11 citations
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Abstract

Using the well-known necessary and sufficient conditions for minimum error discrimination (MED), we extract an equivalent form for the MED conditions. In fact, by replacing the inequalities corresponding to the MED conditions with an equivalent but more suitable and convenient identity, the problem of mixed state discrimination with optimal success probability is solved. Moreover, we show that the mentioned optimality conditions can be viewed as a Helstrom family of ensembles under some circumstances. Using the given identity, MED between $N$ similarity transformed equiprobable quantum states is investigated. In the case that the unitary operators are generating a set of irreducible representation, the optimal set of measurements and corresponding maximum success probability of discrimination can be determined precisely. In particular, it is shown that for equiprobable pure states, the optimal measurement strategy is the square-root measurement (SRM), whereas for the mixed states, SRM is not optimal. In the case that the unitary operators are reducible, there is no closed-form formula in the general case, but the procedure can be applied in each case in accordance to that case. Finally, we give the maximum success probability of optimal discrimination for some important examples of mixed quantum states, such as generalized Bloch sphere $m$-qubit states, spin-$j$ states, particular nonsymmetric qudit states, etc.

DOI
10.1103/physreva.84.012102

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Using the well-known necessary and sufficient conditions for minimum error discrimination (MED), we extract an equivalent form for the MED conditions. In fact, by replacing the inequalities corresponding to the MED conditions with an equivalent but more suitable and convenient identity, the problem of mixed state discrimination with optimal success probability is solved. Moreover, we show that the mentioned optimality conditions can be viewed as a Helstrom family of ensembles under some circumstances. Using the given identity, MED between NNN similarity transformed equiprobable quantum states is investigated. In the case that the unitary operators are generating a set of irreducible representation, the optimal set of measurements and corresponding maximum success probability of discrimination can be determined precisely. In particular, it is shown that for equiprobable pure states, the optimal measurement strategy is the square-root measurement (SRM), whereas for the mixed states, SRM is not optimal. In the case that the unitary operators are reducible, there is no closed-form formula in the general case, but the procedure can be applied in each case in accordance to that case. Finally, we give the maximum success probability of optimal discrimination for some important examples of mixed quantum states, such as generalized Bloch sphere mmm-qubit states, spin-jjj states, particular nonsymmetric qudit states, etc.
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.
Using the convex optimization method and Helstrom family of ensembles introduced in Ref. [1], we have discussed optimal ambiguous discrimination in qubit systems. We have analyzed the problem of the optimal discrimination of N known quantum states and have obtained maximum success probability and optimal measurement for N known quantum states with equiprobable prior probabilities and equidistant from center of the Bloch ball, not all of which are on the one half of the Bloch ball and all of the conjugate states are pure. An exact solution has also been given for arbitrary three known quantum states. The given examples which use our method include: 1. Diagonal N mixed states; 2. N equiprobable states and equidistant from center of the Bloch ball which their corresponding Bloch vectors are inclined at the equal angle from z axis; 3. Three mirror-symmetric states; 4. States that have been prepared with equal prior probabilities on vertices of a Platonic solid. Keywords: minimum-error discrimination, success probability, measurement, POVM elements, Helstrom family of ensembles, convex optimization, conjugate states PACS Nos: 03.67.Hk, 03.65.Ta
Distinguishing different quantum states is a fundamental task having practical applications in information processing. Despite the effort devoted so far, however, strategies for optimal discrimination are known only for specific examples. In this paper we consider the problem of minimum-error quantum state discrimination where one attempts to minimize the average error. We show the general structure of minimum-error state discrimination as well as useful properties to derive analytic solutions. Based on the general structure, we present a geometric formulation of the problem, which can be applied to cases where quantum state geometry is clear. We also introduce equivalent classes of sets of quantum states in terms of minimum-error discrimination: sets of quantum states in an equivalent class that share the same guessing probability. In particular, for qubit states where the state geometry is found with the Bloch sphere, we illustrate that for an arbitrary set of qubit states, the minimum-error state discrimination with equal prior probabilities can be analytically solved, that is, optimal measurement and the guessing probability are explicitly obtained.
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.
Quantum state discrimination is a fundamental task in the field of quantum communication and quantum information theory. Unless the states to be discriminated are mutually orthogonal, there will be some error in any attempt to determine which state was sent. Several strategies to optimally discriminate between quantum states exist, each maximising some figure of merit. In this thesis we mainly investigate the minimum-error strategy, in which the probability of correctly guessing the signal state is maximised. We introduce a method for constructing the optimal Positive-Operator Valued Measure (POVM) for this figure of merit, which is applicable for arbitrary states and arbitrary prior probabilities. We then use this method to solve minimum-error state discrimination for the so-called trine states with arbitrary prior probabilities - the first such general solution for a set of quantum states since the two-state case was solved when the problem of state discrimination was first introduced. We also investigate the difference between local and global measurements for a bipartite ensemble of states, and find that in certain circumstances the local measurement is superior. We conclude by finding a bipartite analogue to the Helstrom conditions, which indicate when a POVM satisfies the minimum-error criteria.
The minimum-error probability of ambiguous discrimination for two quantum states is the well-known {\it Helstrom limit} presented in 1976. Since then, it has been thought of as an intractable problem to obtain the minimum-error probability for ambiguously discriminating arbitrary mmm quantum states. In this paper, we obtain a new lower bound on the minimum-error probability for ambiguous discrimination and compare this bound with six other bounds in the literature. Moreover, we show that the bound between ambiguous and unambiguous discrimination does not extend to ensembles of more than two states. Specifically, the main technical contributions are described as follows: (1) We derive a new lower bound on the minimum-error probability for ambiguous discrimination among arbitrary mmm mixed quantum states with given prior probabilities, and we present a necessary and sufficient condition to show that this lower bound is attainable. (2) We compare this new lower bound with six other bounds in the literature in detail, and, in some cases, this bound is optimal. (3) It is known that if m=2m=2m=2, the optimal inconclusive probability of unambiguous discrimination QUQ_{U}QU​ and the minimum-error probability of ambiguous discrimination QEQ_{E}QE​ between arbitrary given mmm mixed quantum states have the relationship QU≥2QEQ_{U}\geq 2Q_{E}QU​≥2QE​. In this paper, we show that, however, if m>2m>2m>2, the relationship QU≥2QEQ_{U}\geq 2Q_{E}QU​≥2QE​ may not hold again in general, and there may be no supremum of QU/QEQ_{U}/Q_{E}QU​/QE​ for more than two states, which may also reflect an essential difference between discrimination for two-states and multi-states. (4) A number of examples are constructed.