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Unextendible product basis is an important object in quantum informationtheory and features a broad spectrum of applications, ranging bound entangledstates, quantum nonlocality without entanglement, and Bell inequalities with noquantum violation. A generalized concept called uncompletable product basisalso attracts much attention. In this paper, we find some unextendible productbases that are uncompletable product bases in every bipartition, which answersa 19 year-old open question proposed by DiVincenzo et al. [Commun. Math. Phys.238, 379 (2003)]. As a consequence, we connect such unextendible product basesto local hiding of information and give a sufficient condition for theexistence of an unextendible product basis, that is still an unextendibleproduct basis in every bipartition. Our results advance the understanding ofthe geometry of unextendible product bases.

Unextendible and uncompletable product bases in every bipartition