Grothendieck inequalities for semidefinite programs with rank constraint

TL;DR

This paper extends Grothendieck inequalities to higher ranks, providing new bounds and applications in statistical mechanics and quantum information theory, thereby advancing understanding of semidefinite programs with rank constraints.

Contribution

The paper introduces Grothendieck inequalities for ranks greater than 1 and demonstrates their applications in approximating ground states and XOR games.

Findings

Derived new bounds for higher-rank Grothendieck inequalities

Applied inequalities to approximate ground states in statistical mechanics

Utilized inequalities to analyze XOR games in quantum information

Abstract

Grothendieck inequalities are fundamental inequalities which are frequently used in many areas of mathematics and computer science. They can be interpreted as upper bounds for the integrality gap between two optimization problems: a difficult semidefinite program with rank-1 constraint and its easy semidefinite relaxation where the rank constrained is dropped. For instance, the integrality gap of the Goemans-Williamson approximation algorithm for MAX CUT can be seen as a Grothendieck inequality. In this paper we consider Grothendieck inequalities for ranks greater than 1 and we give two applications: approximating ground states in the n-vector model in statistical mechanics and XOR games in quantum information theory.