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. Briët and T. Vidick, improving an estimate in a 2008 paper by D. Pérez-Garćıa, 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 ∨ (resp. E min) denote l1 ⊗ l1 ⊗ l1 equipped with the injective (resp. minimal) tensor norm. Here l1 is equipped with its maximal operator space structure. The Briët-Vidick method yields that the identity map In satisfies (for some c > 0) ‖In : E ∨ → E min‖ ≥ cn(logn). Let S 2 denote the (Hilbert) space of n× n-matrices equipped with the Hilbert-Schmidt norm. While a lower bound closer to n is still open, their method produces an interesting, asymptotically almost sharp, related estimate for the map Jn : S n 2 ∨ ⊗ S 2 ∨ ⊗ S 2 → l 3 2 ∨ ⊗ ln3 2 taking ei,j ⊗ ek,l ⊗ em,n to e[i,k,m],[j,l,n]. 1. Tripartite Bell inequality We will prove the following theorem due to J. Briët and T. Vidick, improving an estimate in Junge etal. The proof in [1] was kindly explained to me in detail by T. Vidick. The improvements below (improving the power of the logarithmic term) are routine refinements of the ideas in [1]. Theorem 1.1. Let Y (N) be N ×N a Gaussian random matrix with Gaussian entries all i.i.d. of mean zero and L2-norm equal to N −1/2. Let Y (N) j (j = 1, 2, · · · ) be a sequence of independent copies of Y (N). There is a constant C such that with large probability we have for all scalars aij with 1 ≤ i, j ≤ N ‖ N