Distinguishability and Indistinguishability by Local Operations and Classical Communication

Abstract

It is well known that orthogonal quantum states can be distinguished perfectly. However, if we assume that these orthogonal quantum states are shared by spatially separated parties, the distinguishability of these shared quantum states may be completely different. We show that a set of linearly independent quantum states ${({U}_{m,n}\ensuremath{\bigotimes}I){\ensuremath{\rho}}^{AB}({U}_{m,n}^{\ifmmode\dagger\else\textdagger\fi{}}\ensuremath{\bigotimes}I){}}_{m,n=0}^{d\ensuremath{-}1}$, where ${U}_{m,n}$ are generalized Pauli matrices, cannot be discriminated deterministically or probabilistically by local operations and classical communication. On the other hand, any $l$ maximally entangled states from this set are locally distinguishable if $l(l\ensuremath{-}1)\ensuremath{\le}2d$. The explicit projecting measurements are obtained to locally discriminate these states. As an example, we show that four Werner states are locally indistinguishable.