New results on unextendible product bases
Yiwei Zhang, Fei Shi, Xiande Zhang, Yang Yiting, Gennian Ge · Scientia Sinica Mathematica · 2021
Unextendible product bases (UPB) is an important concept and has extensive applications in various fields of quantum information theory. The construction of UPBs is closely related to combinatorics. Alon and Lovász (2001) first used a series of graph theoretic tools to characterize the necessary and sufficient conditions when the size of a UPB can attain the trivial lower bound. Later Feng (2006) brought the 1-factorization of graphs into the study of UPBs. In this paper, we further apply some graph theoretic tools to analyze the minimum size of a UPB under certain parameters and obtain a series of new results. Moreover, we have an almost complete characterization of all the potential sizes of UPBs in $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^k$.