Construction of multipartite unextendible product bases and geometric measure of entanglement of positive-partial-transpose entangled states

TL;DR

This paper constructs new multipartite unextendible product bases in complex quantum systems and analyzes the geometric measure of entanglement for specific PPT entangled states, advancing understanding in quantum entanglement structure.

Contribution

It introduces two families of UPBs in a 6-partite system and constructs a new family of PPT entangled states, providing analytical and upper bound measures of their entanglement.

Findings

Existence of two UPB families in a 6-partite system.

Construction of a new family of 7-qubit PPT entangled states.

Analytical derivation of a geometric measure of entanglement.

Abstract

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.