Tripartite Bell inequality, random matrices and trilinear forms

TL;DR

This paper provides a detailed proof of a recent result improving estimates on the growth of deviations in tripartite Bell inequalities, using advanced techniques involving random matrices and trilinear forms.

Contribution

It offers a detailed proof of an improved lower bound on the norms related to tripartite Bell inequalities, connecting tensor norms, Gaussian chaos, and operator space theory.

Findings

Established a lower bound of order n^{1/4} for a specific tensor norm map.

Connected Bell inequality deviations with norms of trilinear forms on Hilbert spaces.

Presented an asymptotically almost sharp estimate for related tensor maps.

Abstract

In this seminar report, we present in detail the proof of a recent result due to J. Bri\"et and T. Vidick, improving an estimate in a 2008 paper by D. P\'erez-Garc\'{\i}a, M. Wolf, C. Palazuelos, I. Villanueva, and M. Junge, estimating the growth of the deviation in the tripartite Bell inequality. The proof requires a delicate estimate of the norms of certain trilinear (or $d$-linear) forms on Hilbert space with coefficients in the second Gaussian Wiener chaos. Let $E^n_{\vee}$ (resp. $E^n_{\min}$) denote $ \ell_1^n \otimes \ell_1^n\otimes \ell_1^n$ equipped with the injective (resp. minimal) tensor norm. Here $ \ell_1^n$ is equipped with its maximal operator space structure. The Bri\"et-Vidick method yields that the identity map $I_n$ satisfies (for some $c>0$) $\|I_n:\ E^n_{\vee}\to E^n_{\min}\|\ge c n^{1/4} (\log n)^{-3/2}.$ Let $S^n_2$ denote the (Hilbert) space of $n\times n$-matrices equipped with the Hilbert-Schmidt norm. While a lower bound closer to $n^{1/2} $ is still open, their method produces an interesting, asymptotically almost sharp, related estimate for the map $J_n:\ S^n_2\stackrel{\vee}{\otimes} S^n_2\stackrel{\vee}{\otimes}S^n_2 \to \ell_2^{n^3} \stackrel{\vee}{\otimes} \ell_2^{n^3} $ taking $e_{i,j}\otimes e_{k,l}\otimes e_{m,n}$ to $e_{[i,k,m],[j,l,n]}$.