Quantum nonlocality, tensor norms and operator spaces

A comment on these notes 3 Capítulo 1. Introduction to Quantum nonlocality 5 1. Bell's result: Correlations in EPR 5 2. Tsirelson's theorem and Grothendieck's theorem 7 3. Banach space point of view 15 4. Computer science point of view: XOR games 21 Capítulo 2. Quantum nonlocality: The general case 23 5. Bell functionals and multiplayer games 23 6. Operator spaces 25 7. Description of two-player games with operator spaces 34 8. Bell functionals with signed coefficients 37 Capítulo 3. Large violations of Bell inequalities 41 9. Upper bounds on the ratio ω * /ω in two-player games 41 10. Lower bounds on the largest violations in two-player games 45 11. Some comments about different games 52 Bibliografía 55 1 A comment on these notes These notes are based on a series of lectures that I delivered in July 2016 as a part of the International Graduate Student Summer School: Quantum Correlations and Group C *-Algebra Theory at the University of Zhejiang in Hangzhou (China). The aim of this course was to present an introduction to the use of operator spaces in the field of quantum information theory. Since this course was expected to last no more than 30 hours, in order to give a self-contained and deep enough introduction, I saw myself forced to select only one topic among many different possibilities. As a consequence of this, the course was focused on the connections between quantum nonlocality, a major topic in quantum information theory, and the theory of tensor norms of Banach/operator spaces. In particular, the course was devoted to explaining why the theory of operator spaces is very suitable to study violations of Bell inequalities. The course was thought to be self-contained. To this end, all basic concepts and results that are used (except for well known theorems, for which references are provided) are introduced in detail. At the same time, one of the main goals of the course was to explain some of the recent results in the field so that, at the end, the interested students were able to handle some of the current techniques in the topic. These notes faithfully follow the recent survey, written in a joint work with T. Vidick, However, in the present notes fewer topics than in the previous survey are considered, with the advantage that they are explained in complete detail. In Chapter 1 the basic concepts of quantum nonlocality …