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 $\rho$ and every pair of directions, $|I({\vec{n}}_1,{\vec{n}}_2;\rho )|\le 2$, with equality if and only if ${\vec{n}}_1\perp {\vec{n}}_2$ and $\rho$ is locally unitarily equivalent to the GHZ state. On the five-parameter Acín family, $I(\hat{x},\hat{y})$ depends only on ${\lambda }_0{\lambda }_4$ 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.

arXiv comment13 pages, 1 figure