A measure for genuine tripartite entanglement

Abstract

We introduce a single real-valued functional $I(\vec{n}_1,\vec{n}_2)$, built from four three-qubit correlation expectation values, that turns the Greenberger--Horne--Zeilinger (GHZ) algebraic paradox into a \emph{quantitative} witness of genuine tripartite entanglement. We prove that for every three-qubit state $\rho$ and every pair of measurement directions $|I(\vec{n}_1,\vec{n}_2;\rho)|\le 2$, with the bound saturated if and only if $\vec{n}_1\perp \vec{n}_2$ and $\rho$ is locally unitarily equivalent to the GHZ state. We obtain a closed-form expression for $I(\hat{x},\hat{y})$ on the five-parameter Ac\'in canonical family of three-qubit pure states; it depends only on the product $\lambda_0\lambda_4$ and is maximised when $\lambda_0=\lambda_4=1/\sqrt{2}$. For the W state we show that $I(\hat{x},\hat{y})=0$ and that $\max_{\vec{n}_1, \vec{n}_2} | I_{W} |=35/27 \approx 1.296$, strictly below the GHZ value. Maximising the underlying correlation structure over \emph{independent} local orthonormal frames on the three parties yields a manifestly local-unitary (LU) invariant quantity $\mathcal{E}_{GHZ}(\rho)\in[0,1]$ that equals one if and only if $\rho$ is LU equivalent to the GHZ state, takes the value $35/54\approx 0.648$ on the W state, and is bounded by $1/2$ on all biseparable and fully separable states; it is therefore a device-independent indicator of GHZ-type genuine tripartite correlation. We carefully delimit which properties are proven and which (notably global convexity and the resulting genuine-multipartite-entanglement witness threshold) are established numerically and remain open analytically. We also outline a generalisation of $I$ to three-qudit systems built from the Heisenberg--Weyl operators, recovering the standard qubit construction when $d=2$.