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

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].