The Minimum Size of Qubit Unextendible Product Bases

TL;DR

This paper determines the minimal size of unextendible product bases in qubit systems, filling gaps in known cases and employing graph theory and computer-assisted proofs to establish optimality.

Contribution

It constructs the smallest unextendible product bases for all remaining open cases in qubit systems and proves their minimality using graph theory and computational methods.

Findings

Constructed minimal unextendible product bases for all remaining open cases.

Used graph theory techniques to analyze the problem.

Provided computer-assisted proofs of minimality.

Abstract

We investigate the problem of constructing unextendible product bases in the qubit case - that is, when each local dimension equals 2. The cardinality of the smallest unextendible product basis is known in all qubit cases except when the number of parties is a multiple of 4 greater than 4 itself. We construct small unextendible product bases in all of the remaining open cases, and we use graph theory techniques to produce a computer-assisted proof that our constructions are indeed the smallest possible.