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Research in Optical Sciences

Decomposition of Rank-two Mixed States and Quantum State Discrimination

Luis Roa

Published Jan 1, 2012• 0 citations
Publisher

Abstract

We study how the mathematical property of pure-state decomposition of a mixed state is related to the quantum state discrimination. All the possible linearly independent pure-state decomposition of a rank-two mixed state are characterized by means of the complex overlap between the two involved states, finding that the balanced pure-state decomposition has the smallest probability of being discriminated conclusively and the highest one with minimum-error.

DOI
10.1364/hilas.2012.jt2a.4

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We study how the mathematical property of pure-state decomposition of a mixed state is related to the quantum state discrimination. All the possible linearly independent pure-state decomposition of a rank-two mixed state are characterized by means of the complex overlap between the two involved states, finding that the balanced pure-state decomposition has the smallest probability of being discriminated conclusively and the highest one with minimum-error.
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The problem of discriminating with minimum error between two mixed quantum states is reviewed, with emphasis on the detection operators necessary for performing the measurement. An analytical result is derived for the minimum probability of errors in deciding whether the state of a quantum system is either a given pure state or a uniform statistical mixture of any number of mutually orthogonal states. The result is applied to two-qubit states, and the minimum error probabilities achievable by collective and local measurements on the qubits are compared.
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We study an optimized measurement that discriminates two mixed quantum states with maximum confidence for each conclusive result, thereby keeping the overall probability of inconclusive results as small as possible. When the rank of the detection operators associated with the two different conclusive outcomes does not exceed unity, we obtain a general solution. As an application, we consider the discrimination of two mixed qubit states. Moreover, for the case of higher-rank detection operators we give a solution for particular states. The relation of the optimized measurement to other discrimination schemes is also discussed.