Unextendible and uncompletable product bases in every bipartition

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.

References in corpus (5)

• Quantum secret sharing based on local distinguishability
• Strong quantum nonlocality in $N$-partite systems
• Strong quantum nonlocality for unextendible product bases in heterogeneous systems
• Unextendible product bases from tile structures and their local entanglement-assisted distinguishability
• Negative result about the construction of genuinely entangled subspaces from unextendible product bases

Cited by in corpus (4)

• Strong quantum nonlocality and unextendibility without entanglement in $N$-partite systems with odd $N$
• Genuinely nonlocal sets without entanglement in multipartite systems
• Unextendible and strongly uncompletable product bases
• Progress in the study of the (non)existence of genuinely unextendible product bases