2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/130084We 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.4 pages (and a bit more, in RevTex4), no figures. A typographical error in Ref. [16] was corrected. Main results unchanged. Thanks to Dr. S. BandyopadhyayQuantum Physicstext