Upper bounds for Grothendieck constants, quantum correlation matrices and CCP functions

Abstract

Within the framework of the search for the still unknown exact value of the real and complex Grothendieck constant $K_G^\mathbb{F}$ in the famous Grothendieck inequality (unsolved since 1953), where $\mathbb{F}$ denotes either the real or the complex field, we concentrate our search on their smallest upper bound. To this end, we establish a basic framework, built on functions which map correlation matrices to correlation matrices entrywise by means of the Hadamard product, such as the Krivine function in the real case or the Haagerup function in the complex case. By making use of multivariate real and complex Gaussian analysis, higher transcendental functions, integration over spheres and combinatorics of the inversion of Maclaurin series, we provide an approach by which we also recover all famous upper bounds of Grothendieck himself ($K_G^\mathbb{R} \leq \sinh(\pi/2) \approx 2.301$), Krivine ($K_G^\mathbb{R} \leq \frac{\pi}{2 \ln(1 + \sqrt{2})} \approx 1,782$) and Haagerup ($K_G^\mathbb{C} \leq 1.405$, numerically approximated); each of them as a special case. In doing so, we aim to unify the real and complex case as much as possible and apply our results to several concrete examples, including the Walsh-Hadamard transform (''quantum gate'') and the multivariate Gaussian copula - with foundations of quantum theory and quantum information theory in mind. Moreover, we offer a shortening and a simplification of the proof of the strongest estimation until now; namely that $K_G^\mathbb{R} < \frac{\pi}{2 \ln(1 + \sqrt{2})}$. We summarise our key results in form of an algorithmic scheme and shed light on related open problems and topics for future research.