bibkey: cuimehtamousavinezhadi2020chsh
authors: D. Cui; A. Mehta; H. Mousavi; S. S. Nezhadi
year: 2020
title: "A generalization of CHSH and the algebraic structure of optimal strategies"
doi: 10.22331/q-2020-10-21-346
url: https://arxiv.org/abs/1911.01593
claim: "Characterize the groups G_n in terms of direct and semidirect products of small well-known groups (Section 1.4, item 2)."
strata_touched:
  - D5/S3/Quantum/Algebra/GeneralizedCHSHGroupDecomposition
license: citation-only
triage: anchor

A generalization of CHSH and the algebraic structure of optimal strategies

D. Cui, A. Mehta, H. Mousavi and S. S. Nezhadi, Quantum 4, 346 (2020). The locators below identify the journal article and the arXiv source; the quotations refer to arXiv:1911.01593v5.

Verified locator

DOI: https://doi.org/10.22331/q-2020-10-21-346

Source: https://arxiv.org/abs/1911.01593

Version used for quotations: https://arxiv.org/abs/1911.01593v5

Question and presentation

Section 1.4 (Further work), item 2, printed p. 7:

In Section 5, we give efficient explicit presentations for $G_n$ and its multiplication table. Can we go further and characterize these groups in terms of direct and semidirect products of small well-known groups? The first few cases are as follows

The displayed cases are $G_3\cong {\mathbb{Z}}_3\times A_4$, $G_4\cong ({\mathbb{Z}}_2^3⋊{\mathbb{Z}}_4)⋊{\mathbb{Z}}_4$, $G_5\cong ({\mathbb{Z}}_2^4⋊{\mathbb{Z}}_5)\times {\mathbb{Z}}_5$, and $G_6\cong {\mathbb{Z}}_3\times ((({\mathbb{Z}}_4\times {\mathbb{Z}}_2^3)⋊{\mathbb{Z}}_2)⋊{\mathbb{Z}}_2)⋊{\mathbb{Z}}_3$.

Section 5, printed p. 24, defines

$G_n=⟨P_0,P_1,J\mid P_0^n,P_1^n,J^n,[J,P_0],[J,P_1],J^i(P_0^i{P}_1^{-i}{)}^2\text{ for }i=1,2,\dots ,\lfloor n/2\rfloor ⟩$.

Here the listed words are relators equal to the identity. Lemma 5.2, printed p. 25, states:

For all $i,j$, the elements $P_0^i{P}_1^{-i}$ and $P_0^j{P}_1^{-j}$ commute.

Theorem 5.7, printed p. 26, states:

$H_n\cong G_n$ for all $n\in \mathbb{N}$.

The group $H_n$ is generated by Alice’s observables $A_0=X$ and $A_1=z_n^2{D}_{0}X$. The theorem identifies this matrix group with the presentation; it does not give the uniform product decomposition. The repository’s decomposition concerns every $n\ge 2$, with cyclic factors represented by Multiplicative (ZMod m) and the elementary abelian factor by functions on Fin (n-2).