Constructing 2 × 2 × 4 and 4 × 4 unextendible product bases and positive-partial-transpose entangled states

The 4-qubit unextendible product bases (UPBs) has been recently studied by [Johnston, J. Phys. A: Math. Theor. 47 (2014) 424034]. Based on this study we show that there is only one UPB of size 6 and six UPBs of size 9 in (Formula presented.), three UPBs of size 9 in (Formula presented.), and no UPB of size 7 in (Formula presented.) and (Formula presented.). Furthermore we construct a new family of 4-qubit positive-partial-transpose (PPT) entangled states of rank seven, and show that they are also PPT entangled states in (Formula presented.) and (Formula presented.), respectively. We analytically derive a geometric measure of a special PPT entangled states.