Constructing $2\times2\times4$ and $4\times4$ unextendible product bases and positive-partial-transpose entangled states

TL;DR

This paper classifies unextendible product bases (UPBs) of specific sizes in certain 4-qubit and bipartite systems and constructs a PPT entangled state with a detailed entanglement measure.

Contribution

It provides a complete classification of UPBs of sizes 6 and 9 in certain 4-qubit and bipartite systems and constructs a new PPT entangled state with analytical entanglement quantification.

Findings

Only one UPB of size 6 in ^2d7^2d7^4

Six UPBs of size 9 in ^2d7^2d7^4

No UPB of size 7 in the studied systems

Abstract

The 4-qubit unextendible product basis (UPB) has been recently studied by [Johnston, J. Phys. A: Math. Theor. 47 (2014) 424034]. From this result we show that there is only one UPB of size $6$ and six UPBs of size $9$ in $\cH=\bbC^2\ox\bbC^2\ox\bbC^4$, three UPBs of size $9$ in $\cK=\bbC^4\ox\bbC^4$, and no UPB of size $7$ in $\cH$ and $\cK$. Furthermore we construct a 4-qubit positive-partial-transpose (PPT) entangled state $\r$ of rank seven, and show that it is also a PPT entangled state in $\cH$ and $\cK$, respectively. We analytically derive the geometric measure of entanglement of a special $\r$.