bibkey: leskovjanovamista2025gaussiangme
authors: Olga Leskovjanová; Ladislav Mišta Jr.
year: 2025
title: "Minimal criteria for continuous-variable genuine multipartite entanglement"
doi: 10.22331/q-2025-08-27-1837
url: https://arxiv.org/abs/2401.04376v3
claim: "The paper constructs genuine multipartite entanglement criteria for N-mode Gaussian states from tree graphs, labels each tree by a reverse level order labeling rooted at a center vertex or an endpoint of the central edge, and conjectures, after checking all trees up to N = 10, that this labeling always gives an adjacency matrix whose column tails A_{i+1:N,i} each have exactly one nonzero entry, so that the corresponding quadrature combinations have at most two nonzero terms."
strata_touched:
  - D5/S3/Combinatorics/SpectralGraph/ReverseLevelOrderTreeLabeling
license: citation-only
triage: anchor

Minimal criteria for continuous-variable genuine multipartite entanglement

O. Leskovjanová and L. Mišta Jr., arXiv:2401.04376v3; Quantum 9 (2025), 1837. Quotations are from the v3 TeX source (the accepted Quantum version).

The reverse level order labeling (Section 4.1.3, generic trees):

Next, if the center of the tree is a vertex, we choose it as the so-called root of the tree, whereas if it is an edge, we choose one of its endpoints as a root (see red vertices in Tab. 1). Having established a root, we now associate with each vertex a level which is the number of edges from the vertex to the root and arrange the vertices into horizontal layers, where in the topmost layer there is only the root, the second layer is comprised of level-one vertices etc. (see the last column of Tab. 1). This allows us to introduce a labeling of the tree $T$, which can be called fittingly as a reverse level order labeling. Here, we label the vertices by natural numbers in a decreasing order starting by assigning the label $N$ to the root and continuing level by level going from left to right at each level

The conjecture:

We checked by a direct calculation that for all configurations of up to $N=10$ the established reverse level order labeling leads to an adjacency matrix, where each vector $A_{i+1:N,i}$ contains only one non-zero component and therefore the respective quadrature combinations (\ref{uxpi}) contain at most two non-zero terms. This leads us to a conjecture that this statement holds generally for an arbitrary $N$-vertex tree.

Verified locator

Source: https://arxiv.org/abs/2401.04376v3

• URL: https://arxiv.org/abs/2401.04376v3 (the e-print is a tar archive; member Quantum-accepted.tex, md5 b1e8a86c3fefec6858c4ac36ca46f9fd, 3112 lines):
  • the reverse level order labeling, l. 826–836;
  • the conjecture, l. 837–839.
• DOI: 10.22331/q-2025-08-27-1837 (https://doi.org/10.22331/q-2025-08-27-1837; Crossref lists Quantum, volume 9, article 1837, published 27 August 2025).