  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 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 locally
distinguishable entangled basis.
