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arXiv:1803.09030 · Physical Review A

Quantum-state comparison and discrimination

A. Hayashi, T. Hashimoto, M. Horibe

Published May 21, 2018• 3 citations
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Abstract

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.

DOI
10.1103/physreva.97.052323

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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.
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 quantum state discrimination problem is ubiquitous in quantum information and serves as the foundation for many quantum communication schemes proposed so far. This paper investigates the optimal discrimination of three symmetric quantum states with equal prior probabilities. To obtain completely accessible information about the input pure states, this strategy makes improvements to the two standard strategies of quantum state discrimination, namely the minimum error discrimination and unambiguous discrimination. The new discrimination strategy proposed in this paper can be applied to both linearly independent and dependent quantum states.
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.
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.
A key concept of quantum information theory is that accessing information encoded in a quantum system requires us to discriminate between several possible states the system could be in. A natural generalization of this problem, namely, quantum sequence discrimination, appears in various quantum information processing tasks, the objective being to determine the state of a finite sequence of quantum states. Since such a sequence is a composite quantum system, the fundamental question is whether an optimal measurement is local, i.e., comprising measurements on the individual members, or collective, i.e., requiring joint measurement(s). In some known instances of this problem, the optimal measurement is local, whereas in others, it is collective. But, so far, a definite prescription based solely on the problem description has been lacking. In this paper, we prove that if the members of a given sequence are secretly and independently drawn from an ensemble or even from different ensembles, the optimum success probability is achievable by fixed local measurements on the individual members of the sequence, and no collective measurement is necessary. This holds for both minimum-error and unambiguous state discrimination paradigms.
We determine the optimal measurement that maximizes the average information gain about the state of a qubit system. The qubit is prepared in one of two known states with known prior probabilities. To treat the problem analytically we employ the formalism developed for the maximum confidence quantum state discrimination strategy and obtain the POVM which optimizes the information gain for the entire parameter space of the system. We show that the optimal measurement coincides exactly with the minimum-error quantum measurement only for two pure states, or when the two states have the same Bloch radius or they are on the same diagonal of the Bloch disk.