Tripartite Bell inequality, random matrices and trilinear forms

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 $e_{[i,k,m],[j,l,n]}$.