On Quantum Detection and the Square-Root Measurement

Details On Quantum Detection and the Square-Root Measurement On Quantum Detection and the Square-Root Measurement

2000-05-31

arxiv.org/abs/quant-ph/0005132v2

IEEE Trans. Inform. Theory, vol. 47, pp. 858-872, Mar. 2001

Physics

Quantum Physics

Version of August 29, 2000, with minor revisions. To appear in the IEEE Transactions on Information Theory. RevTex, 48 pages,

Scientific paper

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)].

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