Unextendible product bases from tile structures in bipartite systems

Abstract

Abstract Unextendible product basis (UPB) is one of the most fundamental concepts in the field of quantum information theory, and its constructions attract much attention. Recently, Shi et al (2020 Phy. Rev. A 101 062329) constructed UPBs via a tile structure approach. They showed that a tile structure in an m × n grid with certain properties leads to explicit constructions of UPBs in C m ⊗ C n . The size of the UPB is m n − s + 1 if the tile structure contains exactly s tiles. In this paper, we deeply analyze the extent of the tile structure approach, by showing that a desired tile structure in an m × m grid contains at most ⌊ m 2 4 ⌋ + m tiles, and thus the smallest size of a UPB in C m ⊗ C m constructed by this approach is m 2 − ⌊ m 2 4 ⌋ − m + 1 . Such tile structures and UPBs are referred to as extremal tile structures and extremal UPBs. We provide algorithms both for constructing possible candidates of extremal tile structures and checking their validity. For applications, we connect the extremal UPBs to local irreducibility and bound entangled states with large ranks.