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Journal of Modern Optics

Generalized measurements for optimally discriminating two mixed states and their linear-optical implementation

Ulrike Herzog, Oliver J. Benson

Published Aug 22, 2009• 7 citations
Publisher

Abstract

Measurement schemes for discriminating two given mixed states of a quantum system have been optimized with respect to various criteria. In many cases the optimum measurements are generalized ones, characterized by detection operators that are not projectors. Here we focus on two generalized measurements. The first refers to the discrimination of two mixed qubit states with maximum confidence at a minimum probability of inconclusive results, and the second to the optimum unambiguous discrimination of two mixed states that are connected by a special similarity transformation. For both measurements we consider linear-optical implementations using single-photon polarization states.

DOI
10.1080/09500340903151286

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Measurement schemes for discriminating two given mixed states of a quantum system have been optimized with respect to various criteria. In many cases the optimum measurements are generalized ones, characterized by detection operators that are not projectors. Here we focus on two generalized measurements. The first refers to the discrimination of two mixed qubit states with maximum confidence at a minimum probability of inconclusive results, and the second to the optimum unambiguous discrimination of two mixed states that are connected by a special similarity transformation. For both measurements we consider linear-optical implementations using single-photon polarization states.
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.
Based on our previous publication [U. Herzog and J. A. Bergou, Phys.Rev. A 71, 050301(R) (2005)] we investigate the optimum measurement for the unambiguous discrimination of two mixed quantum states that occur with given prior probabilities. Unambiguous discrimination of nonorthogonal states is possible in a probabilistic way, at the expense of a nonzero probability of inconclusive results, where the measurement fails. Along with a discussion of the general problem, we give an example illustrating our method of solution. We also provide general inequalities for the minimum achievable failure probability and discuss in more detail the necessary conditions that must be fulfilled when its absolute lower bound, proportional to the fidelity of the states, can be reached.
Based on our previous publication [U. Herzog and J. A. Bergou, Phys. Rev. A 71 , 050301(R)(2005)] we investigate the optimum measurement for the unambiguous discrimination of two mixed quantum states that occur with given prior probabilities. Unambiguous discrimination of nonorthogonal states is possible in a probabilistic way, at the expense of a nonzero probability of inconclusive results, where the measurement fails. Along with a discussion of the general problem, we give an example illustrating our method of solution. We also provide general inequalities for the minimum achievable failure probability and discuss in more detail the necessary conditions that must be fulfilled when its absolute lower bound, proportional to the fidelity of the states, can be reached.