The Real Case: Towards Extending Krivine’s Approach

In this chapter (the heart of the book), we embed Grothendieck’s original approach as well as Krivine’s improvement into a more general framework. We show that the real Grothendieck inequality and Krivine’s approach can be unified in terms of—invertible—CCP-functions and with help of Hermite polynomials. Our general approach is built on a decisive link between “suitably compressed” inverses of CCP functions and quantum correlation matrices. In general, the inverse of an invertible CCP function is not CCP. Moreover, we include a quite recent contribution of Krivine, under the form of an even more general result, again in terms of CCP-functions. We characterise the structure of CCP functions in the real case and give related examples. In doing so, we slightly extend Stein’s Lemma, revisit Noise Stability, apply a few facts from the theory of distributions and test function spaces, introduce the key concept of the hyperbolic CCP transform and show that even Gaussian copulas are lurking.