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

Grothendieck inequalities for semidefinite programs with rank constraint