Optimal non-signalling violations via tensor norms

  1. Abderramán Amr Rey 1
  2. Carlos Palazuelos 1
  3. Ignacio Villanueva 1
  1. 1 Universidad Complutense de Madrid
    info

    Universidad Complutense de Madrid

    Madrid, España

    ROR 02p0gd045

Revista:
Revista matemática complutense

ISSN: 1139-1138,  1988-2807

Año de publicación: 2020

Volumen: 33

Número: 3

Páginas: 661-694

Tipo: Artículo

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Otras publicaciones en: Revista matemática complutense

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Información de financiación

This research was funded by the Spanish MINECO through Grant No. MTM2017-88385-P, MTM2014-54240-P and by the Comunidad de Madrid through grant QUITEMAD-CM P2018/TCS4342. We also acknowledge funding from SEV-2015-0554-16-3 and “Ramón y Cajal program” RYC-2012-10449 (C. P.). 1 Observe that both quantities depend on N and K , so we should denote L V Q N , K \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$LV^{N,K}_{\mathcal Q}$$\end{document} and L V NS N , K \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$LV^{N,K}_{{\mathcal {NS}}}$$\end{document} , but we will simplify notation when N and K are clear from the context. 2 Formally, the tensor M defines the inequality ⟨ M , P ⟩ ≤ ω L ( M ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\langle M,P\rangle \le \omega _{\mathcal L}(M)$$\end{document} for every P ∈ L \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P\in \mathcal L$$\end{document} . 3 Note that Lemma  5.7 applies on non-negative tensors, so we must use it on - R - = | R - | \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$-R^-=|R^{-}|$$\end{document} . 4 Note that, as we mentioned in Remark 5.3 , in general we cannot replace ‖ α 1 ( R ) ‖ N S G ⊗ π N S G \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \alpha _1(R)\Vert _{NSG\otimes _\pi NSG}$$\end{document} by ‖ α 1 ( R ) ‖ ℓ ∞ N ( ℓ 1 K ) ⊗ π ℓ ∞ N ( ℓ 1 K ) \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Vert \alpha _1(R)\Vert _{\ell _\infty ^N(\ell _1^K)\otimes _\pi \ell _\infty ^N(\ell _1^K)}$$\end{document} . However, for the particular elements of the form Q 1 ⊗ Q 2 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$Q_1\otimes Q_2$$\end{document} , both norms coincide by Lemma 5.2 .

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