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 $E_{\vee }^n$ (resp. $E_{\min }^{}$) denote $ \ell_1^n \otimes \ell_1^n\otimes \ell_1^n$ equipped with the injective (resp. minimal) tensor norm. Here ${\mathrm{\ell }}_1^n$ is equipped with its maximal operator space structure. The Bri"et-Vidick method yields that the identity map $I_n$ satisfies (for some $c>0$) $\mathrm{\parallel }{I}_n:E_{\vee }^n\to E_{\min }^{}\mathrm{\parallel }\ge c{n}^{1\mathrm{/}4}(\log n{)}^{-3\mathrm{/}2}\mathrm{.}$ Let $S_2^n$ denote the (Hilbert) space of $n\times n$-matrices equipped with the Hilbert-Schmidt norm. While a lower bound closer to $n^{1\mathrm{/}2}$ 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]}$.