Minimum-error discrimination between three mirror-symmetric states

Abstract

We present the optimal measurement strategy for distinguishing between three quantum states exhibiting a mirror symmetry. The three states live in a two-dimensional Hilbert space, and are thus overcomplete. By mirror symmetry, we understand that the transformation ${|+\stackrel{\ensuremath{\rightarrow}}{〉}|+〉,|\ensuremath{-}\stackrel{\ensuremath{\rightarrow}}{〉}\ensuremath{-}|\ensuremath{-}〉}$ leaves the set of states invariant. The obtained measurement strategy minimizes the error probability. An experimental realization for polarized photons, realizable with current technology, is suggested.