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

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 $\text{cH}={\text{bbC}}^2\text{ox}{\text{bbC}}^2\text{ox}{\text{bbC}}^4$, three UPBs of size $9$ in $\text{cK}={\text{bbC}}^4\text{ox}{\text{bbC}}^4$, and no UPB of size $7$ in $\text{cH}$ and $\text{cK}$. 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 $\text{cH}$ and $\text{cK}$, respectively. We analytically derive the geometric measure of entanglement of a special \r.