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

Abstract

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 H=C2⊗C2⊗C4, three UPBs of size 9 in K=C4⊗C4, and no UPB of size 7 in H and K. 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 H and K, respectively. We analytically derive a geometric measure of a special PPT entangled states.