Unextendible and uncompletable product bases in every bipartition

TL;DR

This paper constructs unextendible product bases that are uncompletable in every bipartition, resolving a long-standing open question and deepening understanding of their geometric properties in quantum information theory.

Contribution

It identifies unextendible product bases that are uncompletable in all bipartitions, answering a 19-year-old open problem and linking these bases to local information hiding.

Findings

Constructed unextendible product bases uncompletable in every bipartition.

Connected such bases to local hiding of information.

Provided a sufficient condition for the existence of these bases.

Abstract

Unextendible product basis is an important object in quantum information theory and features a broad spectrum of applications, ranging bound entangled states, quantum nonlocality without entanglement, and Bell inequalities with no quantum violation. A generalized concept called uncompletable product basis also attracts much attention. In this paper, we find some unextendible product bases that are uncompletable product bases in every bipartition, which answers a 19 year-old open question proposed by DiVincenzo et al. [Commun. Math. Phys. 238, 379 (2003)]. As a consequence, we connect such unextendible product bases to local hiding of information and give a sufficient condition for the existence of an unextendible product basis, that is still an unextendible product basis in every bipartition. Our results advance the understanding of the geometry of unextendible product bases.