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On Graph Approaches to Contextuality and Their Role in Quantum Theory SPRINGER BRIEFS IN MATHEMATICS Barbara Amaral · Marcelo Terra Cunha On Graph Approaches to Contextuality and their Role in Quantum Theory 123 SpringerBriefs in Mathematics Series editors Nicola Bellomo Michele Benzi Palle Jorgensen Tatsien Li Roderick Melnik Otmar Scherzer Benjamin Steinberg Lothar Reichel Yuri Tschinkel George Yin Ping Zhang SpringerBriefs in Mathematics showcases expositions in all areas of mathematics and applied mathematics. Manuscripts presenting new results or a single new result in a classical field, new field, or an emerging topic, applications, or bridges between new results and already published works, are encouraged. The series is intended for mathematicians and applied mathematicians. More information about this series at http://www.springer.com/series/10030 SBMAC SpringerBriefs Editorial Board Carlile Lavor University of Campinas (UNICAMP) Institute of Mathematics, Statistics and Scientific Computing Department of Applied Mathematics Campinas, Brazil Luiz Mariano Carvalho Rio de Janeiro State University (UERJ) Department of Applied Mathematics Graduate Program in Mechanical Engineering Rio de Janeiro, Brazil The SBMAC SpringerBriefs series publishes relevant contributions in the fields of applied and computational mathematics, mathematics, scientific computing, and related areas. Featuring compact volumes of 50 to 125 pages, the series covers a range of content from professional to academic. The Sociedade Brasileira de Matemática Aplicada e Computacional (Brazilian Society of Computational and Applied Mathematics, SBMAC) is a professional association focused on computational and industrial applied mathematics. The society is active in furthering the development of mathematics and its applications in scientific, technological, and industrial fields. The SBMAC has helped to develop the applications of mathematics in science, technology, and industry, to encourage the development and implementation of effective methods and mathematical tech- niques for the benefit of science and technology, and to promote the exchange of ideas and information between the diverse areas of application. http://www.sbmac.org.br/ Barbara Amaral • Marcelo Terra Cunha On Graph Approaches to Contextuality and their Role in Quantum Theory 123 Barbara Amaral Marcelo Terra Cunha Department of Statistics, Physics IMECC – Department of and Mathematics Applied Mathematics Federal University of São João del-Rei University of Campinas Ouro Branco, Minas Gerais, Brazil Campinas, São Paulo, Brazil International Institute of Physics, Federal University of Rio Grande do Norte Natal, Rio Grande do Norte, Brazil ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs in Mathematics ISBN 978-3-319-93826-4 ISBN 978-3-319-93827-1 (eBook) https://doi.org/10.1007/978-3-319-93827-1 Library of Congress Control Number: 2018946783 © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2018 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface With all the [...] things going on in the world, just the fact that we can wake up in the morning is kind of a miracle. John Zorn This book focuses on two of the most striking features of quantum theory— contextuality and nonlocality—fascinating, intrinsically quantum phenomena with both fundamental and practical implications. We are specially interested in graph approaches to contextuality, which were developed from the perception that the knowledge in graph theory could be applied directly to solve several different open problems. This led to a breakthrough in the field, providing new tools and insights, some of them borrowed from other applications of graph theory. One of the main contributions of the graph approach to contextuality comes from the fact that many sets of probability distributions in contextuality scenarios can be described using well-known convex sets from graph theory, leading to a beautiful geometric characterisation of such sets. This brings into play the tools of convex optimisation, linear programming and semidefinite programming, that can now be used to solve tasks that we did not know how to tackle before the graph approach was developed. The connection of contextuality and graphs helps us with many practical problems, but its contributions can be even deeper. Many results regarding graph invariants, such as the Lovász theta function, lead to important developments in foundations of quantum physics. This was the main problem discussed in the PhD thesis of one of us, supervised by the other, including important contributions by Adán Cabello and an internship with Andreas Winter (two of the founding fathers of graph approaches to contextuality), the starting point for the book that you are reading now. In the following pages, you will find our attempt to present the main contributions of graph theory to the study of contextuality and how they help us to understand the subtleties of quantum theory. v vi Preface We are indebted to many friends for their interest and help with this project. Let us start by Carlile Lavor, who invited us for writing the book, and Adán Cabello, who started our interest for the field. Many colleagues and students were essential, for many different reasons, and we take this opportunity to thank them: Ernesto Galvão, Matthias Kleinmann, Raphael Drumond, Roberto Baldijão, Rui Soares Barbosa, Saulo Moreira and Steve Walborn made valuable comments on previous versions of the text. Shane Mansfield has not only suggested important changes to the text but also sent us the beautiful bundle diagrams, which appear on Sect. 2.6.1. Cristhiano Duarte, for discussions regarding the book, graphs, football, beer, the good and the bad of life in academia...AnaBelénSainz,for endless discussions that helped us understanding much better our own work (and ourselves). Rafael Chaves and all the International Institute of Physics for support and hospitality in Natal. Agencies, groups and networks always play important roles in science. We are indebted to Capes, CNPq and INCT-IQ for support. EnLight, MathFoundQ and QiQm were essential in bringing the appropriate environment for scientific work. We do thank our colleagues that are part of them! We also thank SBMAC and Springer for this opportunity. Another part of life was also essential during this process of writting, revising, writting...Forkeeping us reasonably sane, we thank our friends from wherever and specially our families (dogs included!). Natal, Brazil Barbara Amaral Barão Geraldo, Brazil Marcelo Terra Cunha February 2018 Contents 1 Introduction .................................................................. 1 1.1 States and Measurements .............................................. 4 1.1.1 Repeatability and Compatibility .............................. 5 1.1.2 Classical Probability Theory .................................. 6 1.1.3 Quantum Probability Theory.................................. 7 1.2 Completing a Probabilistic Model ..................................... 8 1.2.1 The Assumption of Noncontextuality ........................ 9 1.3 Outline of the Book .................................................... 10 2 Contextuality: The Compatibility-Hypergraph Approach .............. 13 2.1 Compatibility Scenarios................................................ 14 2.1.1 Bell Scenarios.................................................. 18 2.2 Probability Distributions and Physical Theories ..................... 19 2.2.1 Classical Realisations and Noncontextuality ................. 19 2.2.2 Quantum Realisations ......................................... 20 2.3 Noncontextuality Inequalities.......................................... 21 2.3.1 The CHSH Inequality ......................................... 22 2.3.2 The KCBS Inequality ......................................... 24 2.3.3 The n-Cycle Inequalities ...................................... 26 2.4 The Exclusivity Graph ................................................. 28 2.4.1 Vertex-Weighted Exclusivity Graph .......................... 33 2.5 The Geometry of the Case H = G ................................... 34 2.5.1 Description of the Nondisturbing, Quantum and Noncontextual Behaviours ............................... 34 2.5.2 The Cut Polytope
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