Three-Tangle for Rank-3 Mixed States: mixture of Greenberger-Horne-Zeilinger, W and flipped W states

TL;DR

This paper analytically calculates the three-tangle for rank-3 mixed states composed of GHZ, W, and flipped W states, providing optimal decompositions and an analytical technique for vanishing three-tangle determination.

Contribution

It introduces an analytical method to compute three-tangle for rank-3 states and constructs optimal decompositions across the parameter space.

Findings

Three-tangle is analytically calculated for the mixture of GHZ, W, and flipped W states.

An analytical technique determines when a rank-3 state has zero three-tangle.

The one-tangle exceeds the sum of squared concurrences and three-tangle.

Abstract

Three-tangle for the rank-three mixture composed of Greenberger-Horne-Zeilinger, W and flipped W states is analytically calculated. The optimal decompositions in the full range of parameter space are constructed by making use of the convex-roof extension. We also provide an analytical technique, which determines whether or not an arbitrary rank-3 state has vanishing three-tangle. This technique is developed by making use of the Bloch sphere S^8 of the qutrit system. The Coffman-Kundu-Wootters inequality is discussed by computing one-tangle and concurrences. It is shown that the one-tangle is always larger than the sum of squared concurrences and three-tangle. The physical implication of three-tangle is briefly discussed.