The three kinds of three-qubit entanglement

TL;DR

This paper introduces a polynomial measure for W-entanglement in three-qubit states, complementing existing measures for GHZ and bipartite entanglement, and establishes an order among these types of entanglement.

Contribution

It constructs a new polynomial measure for W-entanglement and demonstrates the ordered relationship among bipartite, W, and GHZ entanglement measures.

Findings

Bipartite measure is larger than W measure.

W measure is larger than GHZ measure.

The three types of entanglement are ordered in strength.

Abstract

We construct an important missing piece in the entanglement theory of pure three-qubit states, which is a polynomial measure of W-entanglement, working in parallel to the three-tangle, which is a polynomial measure of GHZ-entanglement, and to the bipartite concurrence, which is a polynomial measure of bipartite entanglement. We also show that these entanglement measures are ordered, the bipartite measure is larger than the W measure, which is larger than the GHZ measure. It is meaningful then to consider these three types of three-qubit entanglement, which are also ordered, bipartite is weaker than W, which is weaker than GHZ, in parallel to the order of the three equivalence classes of entangled three-qubit states.