Nonexistence of $n$-qubit unextendible product bases of size $2^n-5$

TL;DR

This paper proves that for n-qubit systems, unextendible product bases of size 2^n-5 do not exist, filling a gap in the understanding of UPB sizes and their limitations.

Contribution

It establishes the nonexistence of n-qubit UPBs of size 2^n-5, extending previous results on UPB sizes and clarifying the structure of these bases.

Findings

UPBs of size 2^n-1, 2^n-2, 2^n-3 do not exist

UPBs of size 2^n-4 exist for all n≥3

UPBs of size 2^n-5 do not exist for any n

Abstract

It is known that the $n$-qubit system has no unextendible product bases (UPBs) of cardinality $2^n-1$, $2^n-2$ and $2^n-3$. On the other hand the $n$-qubit UPBs of cardinality $2^n-4$ exist for all $n\ge3$. We prove that they do not exist for cardinality $2^n-5$.