A measure for genuine tripartite entanglement

Abstract

We introduce a real-valued functional $I(\vec{n}_1,\vec{n}_2)$ of 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 $|I(\vec{n}_1,\vec{n}_2;ρ)|\le 2$ for all three-qubit states $ρ$ and all direction pairs, with equality if and only if $\vec{n}_1\perp \vec{n}_2$ and $ρ$ is locally unitarily (LU) equivalent to the GHZ state. We obtain a closed-form $I(\hat{x},\hat{y})$ on the five-parameter Acín canonical family of three-qubit pure states; it depends only on $λ_0λ_4$ and is maximised at $λ_0=λ_4 =1/\sqrt{2}$. For the W state, $I(\hat{x},\hat{y})=0$ and $\max_{\vec{n}_1,\vec{n}_2}|I_{\mathrm{W}}|=35/27\approx 1.296$, strictly below the GHZ value. Maximising the underlying correlation structure over independent local orthonormal frames yields a manifestly LU-invariant quantity $\mathcal{E}_{\mathrm{GHZ}}(ρ)\in[0,1]$ that equals one if and only if $ρ$ is LU equivalent to the GHZ state, equals $35/54\approx0.648$ 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 Pauli correlators. 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, stating 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 Heisenberg--Weyl operators, recovering the standard qubit construction at $d=2$.