A measure for genuine tripartite entanglement

We introduce a single real-valued functional $I\left(\vec{n}_1, \vec{n}_2\right)$, built from four three-qubit correlation expectation values, that turns the Greenberger-Horne-Zeilinger (GHZ) algebraic paradox into a quantitative witness of genuine tripartite entanglement. We prove that for every three-qubit state $\rho$ and every pair of measurement directions $\left|I\left(\vec{n}_1, \vec{n}_2 ; \rho\right)\right| \leq 2$, 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 Acín 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 $\max _{\vec{n}_1, \vec{n}_2}\left|I_{\mathrm{W}}\right|=35 / 27 \approx 1.296$, strictly below the GHZ value. Maximising the underlying correlation structure over independent local orthonormal frames on the three parties yields a manifestly local-unitary (LU) invariant quantity $\mathcal{E}_{\mathrm{GHZ}}(\rho) \in[0,1]$ 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 local-frame-independent indicator of GHZ-type genuine tripartite correlation, obtainable from four three-body Pauli correlators alone. We relate $\mathcal{E}_{\mathrm{GHZ}}$ to GHZ fidelity witnesses, analyse its behaviour under local operations and classical communication, and delimit what it does and does not certify about genuine tripartite nonlocality. 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 $d=2$.