A measure for genuine tripartite entanglement
Abstract
We introduce a real-valued functional I(n⃗1,n⃗2) of four three-qubit correlations that turns the Greenberger-Horne-Zeilinger (GHZ) algebraic paradox into a quantitative witness of genuine tripartite entanglement. For every three-qubit state ρ and every pair of directions, |I(n⃗1,n⃗2;ρ)|≤ 2, with equality if and only if n⃗1⊥n⃗2 and ρ is locally unitarily equivalent to the GHZ state. On the five-parameter Acín family, I(x̂,ŷ) depends only on λ0λ4 and is maximised by the GHZ state; for the W state I(x̂,ŷ)=0 and max|IW|=35/27. Maximising over independent local orthonormal frames yields a local-unitary invariant measure ℰGHZ∈[0,1], computable from four three-body Pauli correlators, that equals one exactly on the GHZ orbit and takes the values 35/54 on the W state and 1/2 on product states. We show that ℰGHZ is not convex: an equal mixture of three biseparable states has ℰGHZ=14/27, larger than the average 1/2 of its components. We also relate ℰGHZ to GHZ fidelity witnesses, discuss its behaviour under local operations and classical communication, and outline a generalisation to three qudits via Heisenberg-Weyl operators.
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BibTeX
@misc{wu2026measure,
title = {{A measure for genuine tripartite entanglement}},
author = {Shengjun Wu and Kaichen Zhong and Jeffery Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2605.02876},
eprint = {2605.02876},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2605.02876},
}
RIS
TY - UNPB
TI - A measure for genuine tripartite entanglement
AU - Shengjun Wu
AU - Kaichen Zhong
AU - Jeffery Wu
PY - 2026
DA - 2026/10/18
PB - arXiv
DO - 10.48550/arxiv.2605.02876
UR - https://arxiv.org/abs/2605.02876
AB - We introduce a real-valued functional I(n⃗_1,n⃗_2) of four three-qubit correlations that turns the Greenberger-Horne-Zeilinger (GHZ) algebraic paradox into a quantitative witness of genuine tripartite entanglement. For every three-qubit state ρ and every pair of directions, |I(n⃗_1,n⃗_2;ρ)|≤ 2, with equality if and only if n⃗_1⊥n⃗_2 and ρ is locally unitarily equivalent to the GHZ state. On the five-parameter Acín family, I(x̂,ŷ) depends only on λ_0λ_4 and is maximised by the GHZ state; for the W state I(x̂,ŷ)=0 and max|I_W|=35/27. Maximising over independent local orthonormal frames yields a local-unitary invariant measure ℰ_GHZ∈[0,1], computable from four three-body Pauli correlators, that equals one exactly on the GHZ orbit and takes the values 35/54 on the W state and 1/2 on product states. We show that ℰ_GHZ is not convex: an equal mixture of three biseparable states has ℰ_GHZ=14/27, larger than the average 1/2 of its components. We also relate ℰ_GHZ to GHZ fidelity witnesses, discuss its behaviour under local operations and classical communication, and outline a generalisation to three qudits via Heisenberg-Weyl operators.
ER -