Unextendible and strongly uncompletable product bases

TL;DR

This paper investigates the existence and properties of unextendible product bases (UPBs) and strongly uncompletable product bases (SUCPBs) in tripartite quantum systems, providing new conditions and constructing smaller UPBs that are SUCPBs in all bipartitions.

Contribution

It introduces a sufficient condition for SUCPBs based on a quasi U-tile structure and constructs a smaller UPB that is an SUCPB in every bipartition.

Findings

Constructed a smaller UPB with size $d^3-3d^2+1$ in $ ext{C}^d ext{C}^d ext{C}^d$

Provided a sufficient condition for the existence of SUCPBs

Analyzed relationships between UPBs and SUCPBs in tripartite systems

Abstract

In 2003, DiVincenzo {\it et al}. put forward the question that whether there exists an unextendible product basis (UPB) which is an uncompletable product basis (UCPB) in every bipartition [\href{https://link.springer.com/article/10.1007/s00220-003-0877-6}{DiVincenzo {\it et al}. Commun. Math. Phys. \textbf{238}, 379-410(2003)}]. Recently, Shi {\it et al}. presented a UPB in tripartite systems that is also a strongly uncompletable product basis (SUCPB) in every bipartition [\href{https://iopscience.iop.org/article/10.1088/1367-2630/ac9e14}{Shi {\it et al}. New J. Phys. \textbf{24}, 113-025 (2022)}]. However, whether there exist UPBs that are SUCPBs in only one or two bipartitions remains unknown. We provide a sufficient condition for the existence of SUCPBs based on a quasi U-tile structure. We analyze all possible cases about the relationship between UPBs and SUCPBs in tripartite systems. In particular, we construct a UPB with smaller size $d^3-3d^2+1$ in $\mathbb{C}^{d}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}$, which is an SUCPB in every bipartition and has a smaller cardinality than the existing one.