  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)].
