Metadata
bibkey: cobuccitavakoli2024dimensionality
authors: G. Cobucci, A. Tavakoli
year: 2024
title: Detecting the dimensionality of genuine multi-particle entanglement
doi: 10.1126/sciadv.adq4467
url: https://arxiv.org/abs/2402.06234
claim: "Supplementary Material V conjectures the ten-setting GHZ witness bounds (V3)–(V5) for every odd local dimension."
strata_touched:
- D5/S3/Quantum/Entanglement/GHZ/OddGhzTenSettingWitness
license: citation-only
triage: anchor
Verified locator
DOI: https://doi.org/10.1126/sciadv.adq4467
Source: https://arxiv.org/abs/2402.06234
The arXiv v4 PDF, printed page 22, Supplementary Material V, states:
We have shown this explicitly for odd d ≤ 17 and conjecture it to hold for odd d in general.
Equations (V3) and (V4) assert, respectively,
Equation (V5) is the sufficient operator inequality
The bases and the nine triples are specified in (V1)–(V2) on the same page. The triples are retained as nine labelled settings; their multiplicities also specify the one-dimensional case. The three-party GHZ vector is equation (5). Definition 1, printed page 3, states: “For an arbitrary n-partite state ρ, its GME-dimension is” followed by equation (2), the minimum, over mixtures across bipartitions, of the largest Schmidt number. The Schmidt number is the minimum possible largest pure-state Schmidt rank in a convex decomposition. The matrix rank of the coefficient array is the pure-state Schmidt rank.
The repository proves (V3) and (V4) for every odd dimension, including composite dimensions, using the quadratic-form version of (V5). It does not determine the exact non-GHZ spectrum or prove an even-dimensional analogue.
Printed page 3 also states:
The Schmidt rank is the rank of the reduced density matrix, which is necessarily equal to one for a product state but larger for pure entangled states (it generalizes to the Schmidt number for mixed states [33]).