The Minimum Size of Unextendible Product Bases in the Bipartite Case (and Some Multipartite Cases)

Abstract

A long-standing open question asks for the minimum number of vectors needed to form an unextendible product basis in a given bipartite or multipartite Hilbert space. A partial solution was found by Alon and Lovasz in 2001, but since then only a few other cases have been solved. We solve all remaining bipartite cases, as well as a large family of multipartite cases.

17 pages, 4 figures

References in corpus (1)

• The Minimum Size of Qubit Unextendible Product Bases

Cited by in corpus (36)

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• Several nonlocal sets of multipartite pure orthogonal product states
• Local indistinguishability of orthogonal product states
• Genuinely Nonlocal Product Bases: Classification and Entanglement Assisted Discrimination
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• Constructing locally indistinguishable orthogonal product bases in an $m\otimes n$ system
• Genuinely entangled subspace with all-encompassing distillable entanglement across every bipartition
• From unextendible product bases to genuinely entangled subspaces
• Strongly nonlocal unextendible product bases do exist
• Strong quantum nonlocality for unextendible product bases in heterogeneous systems
• Distinguishability classes, resource sharing, and bound entanglement distribution
• Unextendible product bases from tile structures and their local entanglement-assisted distinguishability
• Local distinguishability based genuinely quantum nonlocality without entanglement
• Constructing UMEB from maximally entangled basis
• Local Distinguishability of Generic Unentangled Orthonormal Bases
• Unextendible Product Basis for Fermionic Systems
• Nonlocality of tripartite orthogonal product states
• Multiqubit UPB: The method of formally orthogonal matrices
• Unextendible product bases, bound entangled states, and the range criterion
• Nonlocality without entanglement: Party asymmetric case
• Nonexistence of $n$-qubit unextendible product bases of size $2^n-5$
• Alternative method for deriving nonlocal multipartite product states
• Small set of orthogonal product states with nonlocality
• Additivity of entanglement of formation via entanglement-breaking space
• Strong quantum nonlocality: Unextendible biseparability beyond unextendible product basis
• A complete picture of the four-party linear inequalities in terms of the 0-entropy
• Negative result about the construction of genuinely entangled subspaces from unextendible product bases
• Locally unidentifiable subset of quantum states and its resourcefulness in secret password distribution
• More assistance of entanglement, less rounds of classical communication
• Exploring strong locality : Quantum state discrimination regime and beyond
• Classifying locally distinguishable sets: No activation across bipartitions
• Unextendible and strongly uncompletable product bases
• Maximum Dimension of Subspaces with No Product Basis
• Genuinely entangled subspaces beyond strongly nonlocal unextendible biseparable bases
• Constructing unextendible product bases from multiqubit ones
• Progress in the study of the (non)existence of genuinely unextendible product bases