# $4\times4$ unextendible product basis and genuinely entangled space

## TL;DR

This paper classifies six inequivalent 4x4 unextendible product bases of size eight, constructs entangled states from them, and explores their relation to genuinely entangled spaces in multipartite quantum systems.

## Contribution

It provides a complete classification of 4x4 UPBs of size eight and links their properties to the existence of genuinely entangled spaces in multipartite systems.

## Key findings

- Six inequivalent 4x4 UPBs of size eight identified
- Constructed PPT entangled states of rank nine from UPBs
- Established conditions for UPBs orthogonal to genuinely entangled spaces

## Abstract

We show that there are six inequivalent $4\times4$ unextendible product bases (UPBs) of size eight, when we consider only 4-qubit product vectors. We apply our results to construct Positive-Partial-Transpose entangled states of rank nine. They are at the same 4-qubit, $2\times2\times4$ and $4\times4$ states, and their ranges have product vectors. One of the six UPBs turns out to be orthogonal to an almost genuinely entangled space, in the sense that the latter does not contain $4\times4$ product vector in any bipartition of 4-qubit systems. We also show that the multipartite UPB orthogonal to a genuinely entangled space exists if and only if the $n\times n\times n$ UPB orthogonal to a genuinely entangled space exists for some $n$. These results help understand an open problem in [Phys. Rev. A 98, 012313, 2018].