Quantum correlations and causal structures Mohamed Issam Ibnouhsein To cite this version: Mohamed Issam Ibnouhsein. Quantum correlations and causal structures. Quantum Physics [quant- ph]. Université Paris Sud - Paris XI, 2014. English. NNT : 2014PA112426. tel-01146097 HAL Id: tel-01146097 https://tel.archives-ouvertes.fr/tel-01146097 Submitted on 27 Apr 2015 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Universite´ Paris-Sud Ecole Doctorale 564 : Physique en ˆIle-de-France Laboratoire : CEA Saclay/IRFU/LARSIM Discipline : Physique Quantique These` de doctorat Soutenue le 11 d´ecembre 2014 par Mohamed Issam Ibnouhsein Corr´elations quantiques et structures causales Directeur de th`ese : Daniel Est`eve Directeur de recherche au CEA Saclay Encadrant : Alexei Grinbaum Chercheur au CEA Saclay Composition du jury : Pr´esident du jury : Nicolas Gisin Professeur `al’Universit´ede Gen`eve Rapporteurs : Giacomo Mauro D’Ariano Professeur `al’Universit´ede Pavie Jonathan Oppenheim Professeur `aUniversity College London Examinateurs : Alexia Auff`eves Chercheuse `al’Institut N´eel- CNRS Fr´ed´ericGrosshans Chercheur au CNRS R´esum´e Les travaux r´ecents en fondements de la th´eoriequantique (des champs) et en information quantique relativiste tentent de mieux comprendre les effets des contraintes de causalit´eimpos´eesaux op´erationsphysiques sur la structure des corr´elationsquantiques. Le premier chapitre de cette th`eseest consacr´e`al’´etudedes implica- tions conceptuelles de la non-localit´equantique, notion qui englobe celle d’intrication dans un sens pr´ecis. Nous d´etaillonscomment les r´ecentes approches informationnelles tentent de saisir la structure des corr´elations non-locales, ainsi que les questions que ces derni`eressoul`event concernant la capacit´ed’un observateur localis´e`aisoler un syst`emede son environnement. Le second chapitre d´etailleles effets de l’invariance de Poincar´esur la d´etectionet la quantification de l’intrication. Cette invariance impose que tous les syst`emessoient mod´elis´esen derni`ereinstance dans le cadre de la th´eoriedes champs, ce qui implique qu’aucun syst`eme`a´energiefinie ne puisse ˆetrelocalis´e,ainsi que la divergence de toute mesure d’intrication pour des observateurs localis´es. Nous fournissons une solution `aces deux probl`emes en d´emontrant l’´equivalence g´en´eriquequi existe entre une r´esolutionspa- tiale finie des appareils de mesure et l’exclusion des degr´esde libert´ede haute ´energiede la d´efinitiondu syst`emeobserv´e.Cette ´equivalence permet une interpr´etation´epist´emiquedu formalisme quantique standard d´ecrivant les syst`emeslocalis´esnon-relativistes et leurs corr´elations,clarifiant ainsi l’origine des mesures finies d’intrication pour de tels syst`emes. Le dernier chapitre explore un cadre th´eoriquer´ecemment introduit qui pr´editl’existence de corr´elationsquantiques sans ordre causal d´efini.Proc´edant par analogie avec le cas des corr´elations non-locales, nous pr´esentons quelques principes informationnels contraignant la structure de ces corr´elationsdans le but de mieux en comprendre l’origine physique. Mots-cl´es : th´eoriequantique, th´eoriequantique relativiste des champs, informatique quantique, entropie d’intrication, invariance de Poincar´e,sch´ema de localisation, courbes de type temps ferm´ees,ordre causal, information quantique. Abstract Recent works in foundations of quantum (field) theory and relativistic quan- tum information try to better grasp the interplay between the structure of quantum correlations and the constraints imposed by causality on physical operations. Chapter 1 is dedicated to the study of the conceptual implications of quantum nonlocality, a concept that subsumes that of entanglement in a certain way. We detail the recent information-theoretic approaches to un- derstanding the structure of nonlocal correlations, and the issues the latter raise concerning the ability of local observers to isolate a system from its environment. Chapter 2 reviews in what sense imposing Poincar´einvariance affects entanglement detection and quantification procedures. This invariance ul- timately forces a description of all quantum systems within the framework of quantum field theory, which leads to the impossibility of localized finite- energy states and to the divergence of all entanglement measures for local observers. We provide a solution to these two problems by showing that there exists a generic equivalence between a finite spatial resolution of the measurement apparatus and the exclusion of high-energy degrees of freedom from the definition of the observed system. This equivalence allows for an epistemic interpretation of the standard quantum formalism describing non- relativistic localized systems and their correlations, hence a clarification of the origin of the finite measures of entanglement between such systems. Chapter 3 presents a recent theoretical framework that predicts the exis- tence of correlations with indefinite causal order. In analogy to the information- theoretic approaches to nonlocal correlations, we introduce some principles that constrain the structure of such correlations, which is a first step toward a clear understanding of their physical origin. Keywords: quantum theory, relativistic quantum field theory, quan- tum computing, entropy of entanglement, Poincar´einvariance, localization scheme, closed timelike curves, causal order, quantum information. Acknowledgements I address my warmest thanks to my dissertation advisors Daniel Est`eve and Alexei Grinbaum. Their trust and constant support allowed me to enjoy a rare freedom in the choice of topics to study. I am also indebted to Caslavˇ Brukner, Professor at the University of Vienna and Director of the Institute for Quantum Optics and Quantum Information (IQOQI) of Vienna. The in- vitation to visit his group allowed me to better grasp where difficult questions lie in the topics of quantum foundations and quantum information. I have learned a lot from the valuable discussions with Etienne Klein, Vincent Bontems, Ognyan Oreshkov, Magdalena Zych, Igor Pikovski, Ma- teus Ara´ujo,Ma¨elP´egny, Amin¨ Baumeler, Christina Giarmatzi and most importantly Fabio Costa, whose patience and will to answer my (numerous) questions were a valuable help throughout these years. This dissertation is dedicated to my parents for their constant support. Financial support and travel funds over the last five years were provided by the CEA Saclay, the European Commission Q-Essence Project, and the Ministry for Education, Research and Technology of France. Contents Note synth´etique i Introduction . i Chapitre 1 . iii Chapitre 2 . v Chapitre 3 . vii Perspectives . ix Introduction 1 1 Conceptual implications of entanglement 5 1.1 Structure of quantum correlations . 5 1.1.1 General postulates of quantum theory . 6 1.1.2 Entanglement as a resource . 8 1.1.3 Bell’s theorem . 11 1.1.4 Reconstructions of quantum theory . 16 1.2 Entanglement and open systems . 27 1.2.1 Algebraic quantum field theory . 30 1.2.2 Infinite algebras and the Reeh-Schlieder theorem . 35 1.2.3 Type III algebras and split property . 37 2 Entanglement and relativity 41 2.1 Poincar´einvariance . 42 2.1.1 Lorentz group . 42 2.1.2 Poincar´egroup . 45 2.2 Observer-dependent entanglement . 52 2.2.1 Inertial observers . 52 2.2.2 Non-inertial observers . 58 2.2.3 Local detection of entanglement . 64 2.3 Infinite-mode systems . 65 2.3.1 Area law for entanglement entropy . 66 2.3.2 Renormalization at low energy . 68 10 CONTENTS 2.3.3 Area law at high energy and spacetime dynamics . 76 3 Beyond entanglement and definite causal structures 79 3.1 Operational approaches to causal relations . 80 3.2 The process matrix framework . 84 3.2.1 General framework . 84 3.2.2 Theory-independent tests via a causal inequality . 87 3.2.3 Causally non-separable processes . 88 3.3 Boxes compatible with predefined causal order . 89 3.3.1 Local order and causal separability of processes . 89 3.3.2 Generalized probabilistic framework . 91 3.4 Entropic characterizations of causal structures . 92 3.4.1 Constrained signalling and mutual information . 93 3.4.2 Causal games as random access codes . 95 3.4.3 The Hirschfeld-Gebelein-R´enyi maximal correlation . 100 3.5 Beyond causal quantum computation . 101 Conclusions and Outlook 105 Appendix A C∗-algebras and von Neumann algebras 109 A.1 Generalities . 109 A.2 C∗-algebras . 110 A.2.1 General definitions . 110 A.2.2 Commutative C∗-algebras . 110 A.2.3 Gelfand-Naimark-Segal (GNS) construction . 111 A.3 Von Neumann algebras . 113 A.3.1 General definitions . 113 A.3.2 Commutative von Neumann algebras . 113 A.3.3 Representations . 115 A.3.4 Classification of factors . 116 A.4 Spectral theorem . 119 Bibliography 120 Note synth´etique Cette th`esese situe `al’intersection des ´etudessur les fondements de la th´eoriequantique (des champs) et de l’information quantique relativiste. Y sont discut´eesplusieurs probl´ematiques autour de l’imbrication entre les notions de corr´elationsquantiques et de contrainte causale. Le caract`ere math´ematiquedes travaux