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

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  • © 2024

Overview

Authors:
  1. Frank Oertel

    1. Centre for Philosophy of Natural and Social Science (CPNSS), London School of Economics and Political Science (LSE), London, UK

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  • Illuminates in detail a still open question in mathematics
  • Highlights ideas which will lead to new, long-term research projects
  • Reveals surprising links to neighbouring fields, including quantum information theory

Part of the book series: Lecture Notes in Mathematics (LNM, volume 2349)

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About this book

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).

Keywords

Table of contents (9 chapters)

  1. Front Matter

    Pages i-xvi
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  2. Introduction and Motivation: The Outstanding Story of Grothendieck’s Theorem

    • Frank Oertel
    Pages 1-17

  3. Complex Gaussian Random Vectors and the Probability Law \( \mathbb {C} N_{2n}(0, \varSigma _{2n}(\zeta ))\)

    • Frank Oertel
    Pages 19-37

  4. A Quantum Correlation Matrix Version of the Grothendieck Inequality

    • Frank Oertel
    Pages 39-83

  5. Powers of Inner Products of Random Vectors, Uniformly Distributed on the Sphere

    • Frank Oertel
    Pages 85-103

  6. Completely Correlation Preserving Functions

    • Frank Oertel
    Pages 105-115

  7. The Real Case: Towards Extending Krivine’s Approach

    • Frank Oertel
    Pages 117-172

  8. The Complex Case: Towards Extending Haagerup’s Approach

    • Frank Oertel
    Pages 173-193

  9. A Summary Scheme of the Main Result

    • Frank Oertel
    Pages 195-197

  10. Concluding Remarks and Open Problems

    • Frank Oertel
    Pages 199-218

  11. Back Matter

    Pages 219-230
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Authors and Affiliations

  • Centre for Philosophy of Natural and Social Science (CPNSS), London School of Economics and Political Science (LSE), London, UK

    Frank Oertel

About the author

Frank Oertel is an Honorary Research Associate at the LSE Centre for Philosophy of Natural and Social Science (CPNSS) in London. After studying mathematics and physics at the University of Kaiserslautern-Landau (RPTU), he was a researcher at the University of Zurich, ETH Zurich and Heriot-Watt University, Edinburgh and a lecturer at University College Cork (UCC), Ireland and the University of Southampton, UK. His research interests are primarily in functional analysis, including its applications to quantum mechanics, and applications of functional and stochastic analysis to problems in financial mathematics. He has also worked as a mathematical advisor in the financial industry, including banking supervision and audit (Frankfurt, Zurich, Bonn, Munich and London).

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