  In quantum information theory, it is a fundamental problem to construct
multipartite unextendible product bases (UPBs). We show that there exist two
families UPBs in Hilbert space
$\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^4$
by merging two different systems of an existing $7$-qubit UPB of size $11$.
Moreover, a new family of $7$-qubit positive-partial-transpose (PPT) entangled
states of rank $2^7-11$ is constructed. We analytically derive a geometric
measure of entanglement of a special PPT entangled states. Also an upper bound
are given by two methods.

Quantum Physics