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Physical Review A

Minimum-error discrimination among mirror-symmetric mixed quantum states

Chih‐Lung Chou

Published Dec 17, 2004• 19 citations
Publisher

Abstract

We provide an exact solution of finding the optimum measurement strategy for distinguishing three mixed quantum states with mirror symmetry. The signal states reside in two-dimensional Hilbert space. By mirror symmetry, two of the three states have the same prior probabilities and transform into each other under the mirror symmetry. We find that the optimum measurement strategy may consist of three, two, or a single nonzero probability operator measure elements depending upon the prior probabilities of the quantum signals.

DOI
10.1103/physreva.70.062316

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We provide an exact solution of finding the optimum measurement strategy for distinguishing three mixed quantum states with mirror symmetry. The signal states reside in two-dimensional Hilbert space. By mirror symmetry, two of the three states have the same prior probabilities and transform into each other under the mirror symmetry. We find that the optimum measurement strategy may consist of three, two, or a single nonzero probability operator measure elements depending upon the prior probabilities of the quantum signals.
We present the optimal measurement strategy for distinguishing between three quantum states exhibiting a mirror symmetry. The three states live in a two-dimensional Hilbert space, and are thus overcomplete. By mirror symmetry, we understand that the transformation ${|+\stackrel{\ensuremath{\rightarrow}}{〉}|+〉,|\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{〉}\ensuremath{-}|\ensuremath{-}〉}$ leaves the set of states invariant. The obtained measurement strategy minimizes the error probability. An experimental realization for polarized photons, realizable with current technology, is suggested.
A measurement strategy is developed for a different kind of hypothesis testing. It assigns, with minimum probability of error, the state of a quantum system to one or the other of two complementary subsets of a set of N given nonorthogonal quantum states occurring with given a priori probabilities. A general analytical solution is obtained for N states that are restricted to a two-dimensional subspace of the Hilbert space of the system. The result for the special case of three arbitrary but linearly dependent states is applied to a variety of sets of three states that are symmetric and equally probable. It is found that, in this case, the minimum-error probability for distinguishing one of the states from the other two is only about half as large as the minimum-error probability for distinguishing all three states individually.
We consider two different optimized measurement strategies for the discrimination of nonorthogonal quantum states. The first is ambiguous discrimination with a minimum probability of inferring an erroneous result, and the second is unambiguous, i.e., error-free, discrimination with a minimum probability of getting an inconclusive outcome, where the measurement fails to give a definite answer. For distinguishing between two mixed quantum states, we investigate the relation between the minimum-error probability achievable in ambiguous discrimination, and the minimum failure probability that can be reached in unambiguous discrimination of the same two states. The latter turns out to be at least twice as large as the former for any two given states. As an example, we treat the case where the state of the quantum system is known to be, with arbitrary prior probability, either a given pure state, or a uniform statistical mixture of any number of mutually orthogonal states. For this case we derive an analytical result for the minimum probability of error and perform a quantitative comparison with the minimum failure probability.
We formulate an expression for the accessible information in mirror-symmetric ensembles of real qubit states. This expression is used to make a detailed study of optimum quantum measurements for extracting the accessible information in three-state, mirror-symmetric ensembles. Distinct measurement regimes are identified for these ensembles with optimal measurement strategies involving different numbers of measurement operators, similar to results found for minimum error discrimination. Our results generalize known results for the accessible information in two pure states and in the trine ensemble.
We formulate an expression for the accessible information in mirror-symmetric ensembles of real qubit states. This expression is used to make a detailed study of optimum quantum measurements for extracting the accessible information in three-state, mirror-symmetric ensembles. Distinct measurement regimes are identified for these ensembles with optimal measurement strategies involving different numbers of measurement operators, similar to results found for minimum error discrimination. Our results generalize known results for the accessible information in two pure states and in the trine ensemble.