A measure for genuine tripartite entanglement

TL;DR

This paper introduces a new quantitative measure for genuine tripartite entanglement in three-qubit states, based on correlation expectation values, which distinguishes GHZ states from others and extends to higher-dimensional systems.

Contribution

The authors develop a novel functional that acts as a device-independent witness for GHZ-type entanglement and provide a closed-form expression for pure states, extending to qudits.

Findings

The measure equals 1 only for GHZ states, indicating maximal genuine tripartite entanglement.

For the W state, the measure is strictly below 1, showing it detects different entanglement classes.

The measure ranges from 0 to 1, serving as an indicator of GHZ-type correlations.

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 the two measurement bases are mutually unbiased 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. 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 $d=2$.