Unextendible product bases from tile structures and their local entanglement-assisted distinguishability
Fei Shi, Xiande Zhang, Lin Chen
Abstract
We completely characterize the condition when a tile structure provides an unextendible product basis (UPB) and construct UPBs of different large sizes in ${\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$. In particular, we show that there exists a UPB of size $(mn\ensuremath{-}4\ensuremath{\lfloor}\frac{m\ensuremath{-}1}{2}\ensuremath{\rfloor})$ in ${\mathbb{C}}^{m}\ensuremath{\bigotimes}{\mathbb{C}}^{n}$ for any $n\ensuremath{\ge}m\ensuremath{\ge}3$, which solves an open problem [S. Halder et al., Phys. Rev. A 99, 062329 (2019)]. As an application, we show that this class of UPBs can be perfectly distinguished by local operations and classical communications assisted with a $\ensuremath{\lceil}\frac{m}{2}\ensuremath{\rceil}\ensuremath{\bigotimes}\ensuremath{\lceil}\frac{m}{2}\ensuremath{\rceil}$ maximally entangled state.