bibkey: divianszkymartonbenevertesi2023certification
authors: Péter Diviánszky; István Márton; Erika Bene; Tamás Vértesi
year: 2023
title: "Certification of qubits in the prepare-and-measure scenario with large input alphabet and connections with the Grothendieck constant"
doi: 10.1038/s41598-023-39529-0
url: https://arxiv.org/abs/2211.17185v2
claim: "The paper certifies qubits in prepare-and-measure scenarios through the matrix functions L_k(M), proves L_k(M) <= k L(M), notes that the bound is tight for k = 2 by the CHSH expression, and conjectures that it is not tight for k >= 3."
strata_touched:
  - D5/S3/Quantum/Optimization/PrepareMeasureLkBound
license: citation-only
triage: anchor

Certification of qubits in the prepare-and-measure scenario with large input alphabet and connections with the Grothendieck constant

P. Diviánszky, I. Márton, E. Bene and T. Vértesi, arXiv:2211.17185v2 (2023-08-29); Sci. Rep. 13, 13200 (2023). Quotations are from the v2 TeX source.

Notation (Methods):

it is convenient to define by $\text{W}nk=\{\text{Id}{k}_j|j=1,2,\dots ,k{\}}^n\subset \text{Mat}nk$ the set of matrices whose rows are all $0$s, but exactly one is $1$. Let $\text{Perm}n\subset \text{Mat}nn$ denote the set of permutation matrices. Let $\text{SM}M={\sum }_{i=1}^n{\Vert M_i\Vert }_1$ denote the Manhattan norm of the matrix $M\in \text{Mat}nm$.

The definition of $L_k$:

&L_k(M) = \max_{W \in \W{n}{k}} \SM{\transp{W}M}.

The definition of $L$:

L(M) = \max_{v \in {-1, +1}^n} \Man{vM}.

Lemma III.4 and the question:

L_k(M) \le kL(M)

It is an open question whether Lemma~\ref{LkkL} is tight or not.

The conjecture:

Note that in the particular case of $k=2$ the matrix $M^{(k)}$ is the CHSH expression~\cite{chsh69}, in which case $L(M^{(2)})=2$ and $L_2(M^{(2)})=4$. Hence, for $k=2$ the upper bound in Lemma~\ref{LkkL} is tight. We conjecture that the bound is not tight for greater values of $k$.

Verified locator

Source: https://arxiv.org/abs/2211.17185v2

• URL: https://arxiv.org/abs/2211.17185v2 (source certification_qubits_v2.tex, md5 728b314751930b8afc32fe1ad74cdc79):
  • notation, l. 461–465;
  • the definition of $L_k$, l. 467–475;
  • the definition of $L$, l. 520;
  • Lemma III.4, l. 561–565;
  • the question, l. 584;
  • the conjecture, l. 608.
• DOI: 10.1038/s41598-023-39529-0 (https://doi.org/10.1038/s41598-023-39529-0).