For any quantum ensemble and any positive score function, the optimal expected gain is no larger than the square root of the gain of the generalized pretty good measurement, implying a two-fold mean-square-error bound in Bayesian estimation.
On Quantum Detection and the Square-Root Measurement
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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)].
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Near-optimal performance of square-root measurement for general score functions and quantum ensembles
For any quantum ensemble and any positive score function, the optimal expected gain is no larger than the square root of the gain of the generalized pretty good measurement, implying a two-fold mean-square-error bound in Bayesian estimation.