Local indistinguishability of orthogonal product states

In the general bipartite quantum system $m\otimes n$, Wang \emph{et al.} [Y.-L Wang \emph{et al.}, Phys. Rev. A $\mathbf{\text{92}}$, 032313 (2015)] presented $3(m+n)-9$ orthogonal product states which cannot be distinguished by local operations and classical communication (LOCC). In this paper, we aim to construct less locally indistinguishable orthogonal product states in $m\otimes n $\mathrm{.}First,in$3\otimes n (3< n)$quantumsystem,weconstruct$3n-2$ locally indistinguishable orthogonal product states which are not unextendible product bases. Then, for $m\otimes n(4\le m\le n)$, we present $3n+m-4$ orthogonal product states which cannot be perfectly distinguished by LOCC. Finally, in the general bipartite quantum system $m\otimes n(3\le m\le n)$, we show a smaller set with $2n-1$ orthogonal product states and prove that these states are LOCC indistinguishable using a very simple but quite effective method. All of the above results demonstrate the phenomenon of nonlocality without entanglement.