不可扩展直积基的新进展

New results on unextendible product bases

不可扩展直积基(unextendible product bases, UPB)是量子信息中的重要概念,在量子信息的诸多领域有着广泛的应用. UPB的构造与组合数学有着密切的联系,著名组合学家Alon和Lovász利用一系列图论工具率先刻画了一组UPB态的数目达到平凡下界时的充分必要条件,进而冯克勤先生将图的1-因子分解等工具引入到此问题的研究之中.本文继续利用图论工具,在部分参数下得到了UPB最小态数目问题的一系列新结果.此外,本文对C^(2)■C^(2)■C^(k)中的所有UPB态的数目的可能取值做了近乎完全的刻画.

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 C^(2)■C^(2)■C^(k).