We show that an arbitrary basis of a multipartite quantum state space consisting of $K$ distant parties such that the $k$th party has local dimension $d_k$ always contains at least $N=\sum _{k=1}^K(d_k-1)+1$ members that are unambiguously distinguishable using local operations and classical communication (LOCC). We further show that this lower bound is optimal by analytically constructing a special product basis having only $N$ members unambiguously distinguishable by LOCC. Interestingly, such a special product basis not only gives a stronger form of the weird phenomenon “nonlocality without entanglement,” but also implies the existence of a locally distinguishable entangled basis.

• Received 9 February 2007
• Corrected 11 June 2007

DOI:https://doi.org/10.1103/PhysRevLett.98.230502