Yonina C. Eldar, G. David Forney Jr
Published 2000-05-31, updated 2000-08-29Version 2
In this paper we consider the problem of constructing measurements optimized to distinguish between a collection of possibly non-orthogonal quantum states. We consider a collection of pure states and seek a positive operator-valued measure (POVM) consisting of rank-one operators with measurement vectors closest in squared norm to the given states. We compare our results to previous measurements suggested by Peres and Wootters [Phys. Rev. Lett. 66, 1119 (1991)] and Hausladen et al. [Phys. Rev. A 54, 1869 (1996)], where we refer to the latter as the square-root measurement (SRM). We obtain a new characterization of the SRM, and prove that it is optimal in a least-squares sense. In addition, we show that for a geometrically uniform state set the SRM minimizes the probability of a detection error. This generalizes a similar result of Ban et al. [Int. J. Theor. Phys. 36, 1269 (1997)].
Comments: Version of August 29, 2000, with minor revisions. To appear in the IEEE Transactions on Information Theory. RevTex, 48 pages, 3 figures. A briefer version of this paper has also been submitted to Physical Review Letters. A copy is obtainable by writing to the authors at yonina@mit.edu
Journal: IEEE Trans. Inform. Theory, vol. 47, pp. 858-872, Mar. 2001
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