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

We completely characterize the condition when a tile structure provides an unextendible product basis (UPB) and construct UPBs of different large sizes in Cm- Cn. In particular, we show that there exists a UPB of size (mn-4m-12 ) in Cm - Cn for any n≥m≥3, which solves an open problem [S. Halder et al., Phys. Rev. A 99, 062329 (2019)10.1103/PhysRevA.99.062329]. As an application, we show that this class of UPBs can be perfectly distinguished by local operations and classical communications assisted with a m2 -m2 maximally entangled state.