Nonlocality of orthogonal product basis quantum states

In this paper, we mainly study the local indistinguishability of mutually orthogonal product basis quantum states in the high-dimensional quantum systems. In the Hilbert space of $3\otimes 3$, Walgate and Hardy [Phys. Rev. Lett. 89, 147901 (2002)] presented a very simple proof for nonlocality of nine orthogonal product basis quantum states which are given by Bennett et al. [Phys. Rev. A 59, 1070 (1999)]. In the quantum system of $d\otimes d$, where $d$ is odd, we construct $d^2$ orthogonal product basis quantum states and prove these states are locally indistinguishable. Then we are able to construct some locally indistinguishable product basis quantum states in the multipartite systems. All these results reveal the phenomenon of “nonlocality without entanglement.”

Figure 1

The structure of 25 orthogonal product basis quantum states in $5\otimes 5$.

Figure 2

The structure of $n^2$ orthogonal product basis quantum states in $n\otimes n$ $(n$ is odd).