bibkey: pozsgaywanless2024tensor
authors: Balázs Pozsgay; Ian M. Wanless
year: 2024
title: "Tensor network decompositions for absolutely maximally entangled states"
doi: 10.22331/q-2024-05-08-1339
url: "https://arxiv.org/abs/2308.07042v3"
claim: "The paper studies decompositions of six-leg perfect tensors (absolutely maximally entangled states) into networks of four-leg tensors, treats states from classical maps through orthogonal arrays, solves the linear case over finite fields, where the factors of a decomposition of a perfect tensor are themselves perfect, and leaves the nonlinear case of this statement open."
strata_touched:
  - D5/S3/Combinatorics/Latin/ClassicalPerfectTensorFactorization
license: citation-only
triage: anchor

Pozsgay and Wanless, Tensor network decompositions for absolutely maximally entangled states

B. Pozsgay and I. M. Wanless, Quantum 8 (2024) 1339, DOI 10.22331/q-2024-05-08-1339; arXiv:2308.07042v3. Quotations are from the arXiv v3 TeX source.

An orthogonal array of strength $s$ on $D$ symbols has $D^s$ rows and every $s$ columns orthogonal: every $s$-tuple appears exactly once among their rows. A map $u:X^{k/2}\to X^{k/2}$ gives the array with rows $(a,u(a))$, and the corresponding state is absolutely maximally entangled when this array has strength $k/2$. For $k=6$ the paper considers the decomposition $u^{(123)}={\mathcal{C}}^{(23)}{\mathcal{B}}^{(13)}{\mathcal{A}}^{(12)}$, “where now $\mathcal{A},\mathcal{B},\mathcal{C}$ are classical functions acting from $X^2$ to $X^2$”. For linear maps over a finite field it solves the decomposition explicitly and concludes:

In other words, if $G$ represents a linear perfect tensor with 6 legs, and it has a $▹$-decomposition into linear tensors with 4 legs, then those factors will themselves be perfect. It is an open question whether this statement is true in the nonlinear case for $k=6$.

Verified locator

Source: https://arxiv.org/abs/2308.07042v3

• DOI: https://doi.org/10.22331/q-2024-05-08-1339 (journal version; title, authors, journal, volume and article number from Crossref).
• URL: https://arxiv.org/abs/2308.07042v3. The e-print member factoringtriunit.tex has 2465 lines and md5 09f50a3ea0a8bd02b8a8fc0caa5c1b5e. Locations:
  • orthogonal arrays, l. 891–902;
  • arrays from classical maps, l. 929–937;
  • the decomposition into classical functions on $X^2$, l. 945–964;
  • the linear conclusion and the open question, l. 1196–1200.