Tripartite Bell inequality, random matrices and trilinear forms

Details Tripartite Bell inequality, random matrices and trilinear forms Tripartite Bell inequality, random matrices and trilinear forms

2012-03-12

arxiv.org/abs/1203.2509v1

Mathematics

Operator Algebras

Scientific paper

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

Affiliated with

Also associated with

No associations

Rating

Tripartite Bell inequality, random matrices and trilinear forms does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Tripartite Bell inequality, random matrices and trilinear forms, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Tripartite Bell inequality, random matrices and trilinear forms will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-488480