Partial K-way negativities and three tangle for three qubit states

TL;DR

This paper analytically derives various entanglement measures for three-qubit states, revealing relationships between negativities and tangles, and explores their behavior in states combining GHZ and W states.

Contribution

It provides analytical expressions for global negativity, partial K-way negativities, and tangles for three-qubit states, establishing new relationships among these measures.

Findings

Product of global negativity and partial three-way negativity equals three tangle.

Partial two-way negativity relates to pairwise qubit tangles.

Identifies conditions for W-like entanglement with zero three tangle.

Abstract

We obtain, analytically, the global negativity, partial $K-$way negativities (K=2, 3), Wooter's tangle and three tangle for the generic three qubit canonical state. It is found that the product of global negativity and partial three way negativity is equal to three tangle, while the partial two way negativity is related to tangle of qubit pairs. We also calculate similar quantities for the state canonical to a single parameter (0<q<1) pure state which is a linear combination of a GHZ state and a W state. In this case for q=0.62685, the state has zero three tangle and zero three-way negativity, having only W-like entanglement. The difference between the product of global and partial three way negativity and three tangle for a given state is a quantitative measure of two qubit coherences transformed by unitary transformations on canonical state into three qubit coherences. The global negativity and partial K-way negativities, obtained by selective partial transpositions on multi-qubit state operator, satisfy inequalities which for three qubits are equivalent to CKW (Coffman-Kundu-Wootter) inequality.