A measure for genuine tripartite entanglement

Abstract

We introduce a real-valued functional $I(\vec{n}_1,\vec{n}_2)$ of four three-qubit correlations that turns the Greenberger-Horne-Zeilinger (GHZ) algebraic paradox into a quantitative witness of genuine tripartite entanglement. For every three-qubit state $ρ$ and every pair of directions, $|I(\vec{n}_1,\vec{n}_2;ρ)|\le 2$, with equality if and only if $\vec{n}_1\perp\vec{n}_2$ and $ρ$ is locally unitarily equivalent to the GHZ state. On the five-parameter Acín family, $I(\hat{x},\hat{y})$ depends only on $λ_0λ_4$ and is maximised by the GHZ state; for the W state $I(\hat{x},\hat{y})=0$ and $\max|I_{\mathrm{W}}|=35/27$. Maximising over independent local orthonormal frames yields a local-unitary invariant measure $\mathcal{E}_{\mathrm{GHZ}}\in[0,1]$, computable from four three-body Pauli correlators, that equals one exactly on the GHZ orbit and takes the values $35/54$ on the W state and $1/2$ on product states. We show that $\mathcal{E}_{\mathrm{GHZ}}$ is not convex: an equal mixture of three biseparable states has $\mathcal{E}_{\mathrm{GHZ}}=14/27$, larger than the average $1/2$ of its components. We also relate $\mathcal{E}_{\mathrm{GHZ}}$ to GHZ fidelity witnesses, discuss its behaviour under local operations and classical communication, and outline a generalisation to three qudits via Heisenberg-Weyl operators.