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IEEE Transactions on Information Theory

Minimum-Error Discrimination Between Self-Symmetric Mixed Quantum State Signals

Kenji Nakahira, Tsuyoshi Sasaki Usuda

Published Feb 1, 2012• 6 citations
Publisher

Abstract

We derive some properties of minimum-error measurements for mixed quantum state signals where a density operator itself has a certain symmetry. In the first case, we show that if there exists a regular normal operator that commutes with each density operator of mixed state signals, then there exists an optimal measurement that consists of detection operators expressed as the direct sum of positive operators with supports in the eigenspaces of the regular normal operator. In the second case, we show that for abelian geometrically uniform state signals, if a matrix which represents one of the signals in a certain basis has all real elements, then the matrix which represents the corresponding detection operator of an optimal measurement in that basis also has all real elements. In addition, we discuss some examples of applications of these properties and derive analytical solutions of optimal measurements for special cases.

DOI
10.1109/tit.2011.2170954

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We derive some properties of minimum-error measurements for mixed quantum state signals where a density operator itself has a certain symmetry. In the first case, we show that if there exists a regular normal operator that commutes with each density operator of mixed state signals, then there exists an optimal measurement that consists of detection operators expressed as the direct sum of positive operators with supports in the eigenspaces of the regular normal operator. In the second case, we show that for abelian geometrically uniform state signals, if a matrix which represents one of the signals in a certain basis has all real elements, then the matrix which represents the corresponding detection operator of an optimal measurement in that basis also has all real elements. In addition, we discuss some examples of applications of these properties and derive analytical solutions of optimal measurements for special cases.
This article deals with the quantum optimal discrimination among mixed quantum states enjoying geometrical uniform symmetry with respect to a reference density operator ${\ensuremath{\rho}}_{0}$. It is well known that the minimal error probability is given by the positive operator-valued measure obtained as a solution of a convex optimization problem, namely a set of operators satisfying geometrical symmetry, with respect to a reference operator ${\ensuremath{\Pi}}_{0}$ and maximizing $\mathrm{Tr}({\ensuremath{\rho}}_{0}{\ensuremath{\Pi}}_{0})$. In this article, by resolving the dual problem, we show that the same result is obtained by minimizing the trace of a semidefinite positive operator XXX commuting with the symmetry operator and such that $X\ensuremath{\geqslant}{\ensuremath{\rho}}_{0}$. The new formulation gives a deeper insight into the optimization problem and allows to obtain closed-form analytical solutions, as shown by a simple but not trivial explanatory example. In addition to the theoretical interest, the result leads to semidefinite programming solutions of reduced complexity, allowing to extend the numerical performance evaluation to quantum communication systems modeled in Hilbert spaces of large dimension.
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.
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We derive the measurement strategy that optimizes the Bayes criterion, not necessarily the strategy minimizing the error probability, for distinguishing between collections of quantum states. We present a sufficient condition concerning the rank of each detection operator of an optimal measurement for the Bayes criterion. We also derive an analytical expression for an optimal measurement for the Bayes criterion for an Abelian geometrically uniform state set or for a cyclic state set under a certain constraint.