A Quantum Correlation Matrix Version of the Grothendieck Inequality

This chapter describes the Grothendieck inequality in terms of Gram matrices and traces, and makes some connections, through a result of B. S. Tsirel’son, with quantum correlation matrices and thus with Bell’s inequalities, which play a fundamental role in the foundations of quantum physics. In doing so, we show that the sign of any of the \(4^{m}\) entries of the real Walsh-Hadamard transform can be determined in exactly m calculation steps. We point towards a few related open problems, unify some classical results and show how certain norms induced by Gram matrices and trace duality actually fit very well with the operator ideal formulation of the inequality.