A measure for genuine tripartite entanglement

• URL: http://arxiv.org/abs/2605.02876v1
• Date: Mon, 04 May 2026 17:49:01 GMT
• Title: A measure for genuine tripartite entanglement
• Abstract summary: We prove that for every three-qubit state $$ is a device-independent indicator of GHZ-type genuine tripartite entanglement.<n>We also outline a generalisation of $I$ to three-qudit systems built from the Heisenberg--Weyl operators.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• 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 $d=2$.

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