We show that a set of linearly independent quantum states $\{ (U_{ m,n}\otimes I)\rho ^{ AB}(U_{ m,n}^{ \dagger}\otimes I)\}_{ m,n=0}^{ d-1}$, where $U_{ m,n}$ are generalized Pauli matrices, cannot be discriminated deterministically or probabilistically by local operations and classical communications (LOCC). On the other hand, any $l$ maximally entangled states from this set are locally distinguishable if $l(l-1)\le 2d$. The explicit projecting measurements are obtained to locally discriminate these states. As an example, we show that four Werner states are locally indistinguishable.
Categories: quant-ph