Oertel F. Upper Bounds for Grothendieck Constants, Quantum Correlation Matrices and CCP Functions

Springer Nature Switzerland, 2024. — 238 p.This book concentrates on the famous Grothendieck inequality and the continued search for the still unknown best possible value of the real and complex Grothendieck constant (an open problem since 1953). It describes in detail the state of the art in research on this fundamental inequality, including Krivine's recent contributions, and sheds light on related questions in mathematics, physics and computer science, particularly with respect to the foundations of quantum theory and quantum information theory. Unifying the real and complex cases as much as possible, the monograph introduces the reader to a rich collection of results in functional analysis and probability. In particular, it includes a detailed, self-contained analysis of the multivariate distribution of complex Gaussian random vectors. The notion of Completely Correlation Preserving (CCP) functions plays a particularly important role in the exposition. The prerequisites are a basic knowledge of standard functional analysis, complex analysis, probability, optimisation and some number theory and combinatorics. However, readers missing some background will be able to consult the generous bibliography, which contains numerous references to useful textbooks. The book will be of interest to PhD students and researchers in functional analysis, complex analysis, probability, optimisation, number theory and combinatorics, in physics (particularly in relation to the foundations of quantum mechanics) and in computer science (quantum information and complexity theory).Preface
List of Symbols
Introduction and Motivation: The Outstanding Story of Grothendieck's Theorem
Historical Perspective and Theoretical Framework
Preliminaries, Terminology and Notation
Complex Gaussian Random Vectors and the Probability Law C N2n(0, Σ2n(ζ))
General Complex Gaussian Random Vectors in Cn and Their Probability Distribution
Partitioned Complex Gaussian Random Vectors in C2n and the Probability Law CN2n(0, Σ2n(ζ))
A Quantum Correlation Matrix Version of the Grothendieck Inequality
Gram Matrices, Quantum Correlation, and Beyond
The Grothendieck Inequality, Correlation Matrices and the Matrix Norm ‖·‖∞, 1F
Characterisation of KGF Through Operator Ideals and Violation of Bell Inequalities (A Brief Digression)
KGR(2) and the Walsh-Hadamard Transform: Krivine's Approach Revisited
The Gaussian Inner Product Splitting Property
Powers of Inner Products of Random Vectors, Uniformly Distributed on the Sphere
Gaussian Sign-Correlation
Integration Over Sn-1 and the Gamma Function
Integrating Powers of Inner Products of Random Vectors, Uniformly Distributed on Sn-1
Completely Correlation Preserving Functions
Completely Real Analytic Functions and the Entrywise Matrix Functional Calculus
Completely Correlation Preserving Functions and Schoenberg's Theorem
The Real Case: Towards Extending Krivine's Approach
Some Facts About Real Multivariate Hermite Polynomials
Real CCP Functions and Covariances: A Fourier-Hermite Analysis Approach
Examples of Real CCP Functions, Gaussian Copulas and an Extension of Stein's Lemma
Upper Bounds of KGR and Inversion of Real CCP Functions
The Complex Case: Towards Extending Haagerup's Approach
Multivariate Complex CCP Functions and Their Relation to the Real Case
On Complex Bivariate Hermite Polynomials
Upper Bounds of KGC and Inversion of Complex CCP Functions
A Summary Scheme of the Main Result
Concluding Remarks and Open Problems
Open Problem 1: Grothendieck Constant Versus Taylor Series Inversion
Open Problem 2: Interrelation Between the Grothendieck Inequality and Copulas
Open Problem 3: Non-commutative Dependence Structures in Quantum Mechanics and the Grothendieck Inequality
References
Index