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

TL;DR

This paper determines the minimum size of unextendible product bases in bipartite and multipartite Hilbert spaces, resolving all remaining bipartite cases and many multipartite cases, advancing understanding in quantum information theory.

Contribution

It provides a complete solution to the minimum size problem for unextendible product bases in bipartite spaces and extends results to a broad class of multipartite spaces.

Findings

All remaining bipartite cases are solved.

A large family of multipartite cases are addressed.

The results advance the theoretical understanding of unextendible product bases.

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.