Unextendible product bases from tile structures and their local entanglement-assisted distinguishability

Abstract

We completely characterize the condition when a tile structure provides an unextendible product basis (UPB), and construct UPBs of different large sizes in ${\mathbb{C}}^m\otimes {\mathbb{C}}^n$ for any $n\ge m\ge 3$. This solves an open problem in [S. Halder et al., Phys. Rev. A 99, 062329 (2019)]. As an application, we show that our UPBs of size $(mn-4\lfloor \frac{m-1}{2}\rfloor )$ in ${\mathbb{C}}^m\otimes {\mathbb{C}}^n$ can be perfectly distinguished by local operations and classical communications assisted with a $\lceil \frac{m}{2}\rceil \otimes \lceil \frac{m}{2}\rceil$ maximally entangled state.