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arXiv:0810.1919 · Journal of Physics A Mathematical and Theoretical

On the conditions for discrimination between quantum states with minimum error

Stephen M Barnett, Sarah Croke

Published Jan 14, 2009• 58 citations
arXivHTMLPDF

Abstract

We provide a simple proof for the necessity of conditions for discriminating with minimum error between a known set of quantum states.
DOI
10.1088/1751-8113/42/6/062001

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In this paper we consider the general problem of designing the optimal quantum detection for minimum-error quantum state discrimination. Since it is hard to get an explicit solution and we must resort to numerical methods for the general case. We show the problem can be cast into a semidefinite programming(SDP) problem, we use the theory of SDP to solve it. Based on the idea, we propose a method for minimum-error discrimination by formulating the design of optimal quantum detection as a standard SDP problem; furthermore, based on the optimality conditions for SDP, we derive a set of optimality conditions for quantum detection. It is exemplified that this method is simple to implement and can be used to design optimum detection operators for minimum-error discrimination efficiently.
We derive a general lower bound on the minimum-error probability for ambiguous discrimination between arbitrary mmm mixed quantum states with given prior probabilities. When m=2m=2m=2, this bound is precisely the well-known Helstrom limit. Also, we give a general lower bound on the minimum-error probability for discriminating quantum operations. Then we further analyze how this lower bound is attainable for ambiguous discrimination of mixed quantum states by presenting necessary and sufficient conditions related to it. Furthermore, with a restricted condition, we work out an upper bound on the minimum-error probability for ambiguous discrimination of mixed quantum states. Therefore, some sufficient conditions are obtained for the minimum-error probability attaining this bound. Finally, under the condition of the minimum-error probability attaining this bound, we compare the minimum-error probability for ambiguously discriminating arbitrary mmm mixed quantum states with the optimal failure probability for unambiguously discriminating the same states.