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 $ρ$ and every pair of measurement directions $|I(\vec{n}_1,\vec{n}_2;ρ)|\le 2$, with the bound saturated if and only if the two measurement bases are mutually unbiased and $ρ$ 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. 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. The induced quantity ranges in $[0,1]$, equals one only on the GHZ class, and is therefore a device-independent indicator of GHZ-type genuine tripartite correlation. We also outline a generalisation of $I$ to three-qudit systems built from the Heisenberg--Weyl operators, recovering the standard qubit construction when $d=2$.