bibkey: szalaytoth2025entanglementdepth
authors: Szilárd Szalay; Géza Tóth
year: 2025
title: "Alternatives of entanglement depth and metrological entanglement criteria"
doi: 10.22331/q-2025-04-18-1718
url: https://arxiv.org/abs/2408.15350v3
claim: "The paper labels multipartite entanglement properties by integer partitions, introduces generator functions on partitions such as the power sum s_q and the entanglement dimension Dim_d, notes integer collisions of both, and expects s_q to be injective for noninteger q and Dim_d to be injective for noninteger d."
strata_touched:
  - D5/S3/Combinatorics/Partitions/EntanglementDepthGeneratorInjectivityRefutation
license: citation-only
triage: anchor

Alternatives of entanglement depth and metrological entanglement criteria

Sz. Szalay and G. Tóth, arXiv:2408.15350v3; Quantum 9 (2025), 1718. Quotations are from the v3 TeX source, with the macros \field, \pinv and \Dim expanded to , $\hat{}$ and $\mathrm{Dim}$, \mset written as braces, and citation commands omitted.

Integer partitions:

Such properties are labeled by \emph{integer partitions} of $n$, $\hat{\xi }=\{x_1,x_2,\dots ,x_{|\hat{\xi }|}\}$, which are multisets, containing the $x\in \hat{\xi }$ parts $x\in \mathbb{N}$, such that ${\sum }_{x\in \hat{\xi }}x=n$.

The power sum:

For $q\in \mathbb{R}$, let us have the \emph{power sum} $s_q(\hat{\xi }):={\sum }_{x\in \hat{\xi }}{x}^q,$

The entanglement dimension:

For $d=\dim ({\mathcal{H}}_i)\ge 2$, let us have the \emph{entanglement dimension} (or \emph{effective dimension}) ${\mathrm{Dim}}_d(\hat{\xi }):={\sum }_{x\in \hat{\xi }}{d}^x,$

The two expectations (Section 4.7):

Although several of these equations have nontrivial solutions (for example, $2^2+2^2+2^2=3^2+1^2+1^2+1^2=12$ for $n=6$, $q=2$, or $2^3+2^3+2^3+2^3=3^3+1^3+1^3+1^3+1^3+1^3=32$ for $n=8$, $q=3$), these seem to be rather sporadic. On the other hand, for $q\notin \mathbb{N}$ the equation becomes transcendental, and we expect the injectivity of $s_q$.

For the case $d\notin \mathbb{N}$, we expect the injectivity of ${\mathrm{Dim}}_d$, although the meaning of ${\mathrm{Dim}}_d$ is not established for noninteger $d$, which is just a parameter then.

Verified locator

Source: https://arxiv.org/abs/2408.15350v3

• URL: https://arxiv.org/abs/2408.15350v3 (the e-print is a tar archive; member depths.tex, md5 bbacd570d915c7fdda31d547fe476fec, 6004 lines):
  • integer partitions, l. 462–466;
  • the power sum, Eq. eq:sq, l. 1903–1907;
  • the entanglement dimension, Eq. eq:Dimd, l. 2306–2312;
  • the integer collisions and the two expectations, l. 2428–2447.
• DOI: 10.22331/q-2025-04-18-1718 (https://doi.org/10.22331/q-2025-04-18-1718; Crossref lists Quantum, volume 9, article 1718, published 18 April 2025).