Remarks on the Grothendieck norm

TL;DR

This paper discusses key properties of the Grothendieck norm, highlighting its role in relating matrix entries to probabilistic correlations, a duality with the Schur norm, and challenges in finding a noncommutative analog.

Contribution

It provides new insights into the interpretative and computational aspects of the Grothendieck norm and explores duality and noncommutative generalizations.

Findings

Grothendieck norm relates to off-diagonal matrix entry correlations

Identifies a Fourier-type duality between Schur and Grothendieck norms

Highlights the difficulty in computing a noncommutative Grothendieck norm

Abstract

The goal of this short note is to point out three observations around the Grothendieck norm and semidefinite programming. The first is that the Grothendieck norm captures the difficulty of relating the off-diagonal entries of a real, symmetric matrix to a probabilistic correlation, the second is that there is an interesting ``Fourier''-type duality between the Schur and Grothendieck norms of a real matrix, and the third and last centers around the difficulty of finding an efficiently computable noncommutative analog to the Grothendieck norm.