On Quantum Detection and the Square-Root Measurement
Abstract
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)].
Cite
@article{arxiv.quant-ph/0005132,
title = {On Quantum Detection and the Square-Root Measurement},
author = {Yonina C. Eldar and G. David Forney},
journal= {arXiv preprint arXiv:quant-ph/0005132},
year = {2007}
}
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 [email protected]