Tomasz F. Bigaj Non-locality and Possible Worlds EPISTEMISCHE STUDIEN Schriften zur Erkenntnis- und Wissenschaftstheorie Herausgegeben von / Edited by Michael Esfeld • Stephan Hartmann • Albert Newen Band 10 / Volume 10 Tomasz F. Bigaj Non-locality and Possible Worlds A Counterfactual Perspective on Quantum Entanglement ontos verlag Frankfurt I Paris I Ebikon I Lancaster I New Brunswick Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliographie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de North and South America by Transaction Books Rutgers University Piscataway, NJ 08854-8042 [email protected] United Kingdom, Ire, Iceland, Turkey, Malta, Portugal by Gazelle Books Services Limited White Cross Mills Hightown LANCASTER, LA1 4XS [email protected] Livraison pour la France et la Belgique: Librairie Philosophique J.Vrin 6, place de la Sorbonne ; F-75005 PARIS Tel. +33 (0)1 43 54 03 47 ; Fax +33 (0)1 43 54 48 18 www.vrin.fr 2006 ontos verlag P.O. Box 15 41, D-63133 Heusenstamm www.ontosverlag.com ISBN 10: 3-938793-29-5 ISBN 13: 978-3-938793-29-9 2006 No part of this book may be reproduced, stored in retrieval systems or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use of the purchaser of the work Printed on acid-free paper ISO-Norm 970-6 FSC-certified (Forest Stewardship Council) This hardcover binding meets the International Library standard Printed in Germany by buch bücher dd ag Table of Contents Introduction 9 0.1 Outline of the book 13 Chapter 1 Why does the quantum world have to be non-local? 23 1.1 Introduction: realism and locality 23 1.2 The EPR argument: incompleteness or non-locality? 29 1.3 Bell’s theorem and the plight of locality 33 1.4 Generalized Bell’s theorem 39 1.5 Outcome-locality and parameter-locality 47 1.5.1 Non-locality and non-separability 59 1.6 A provisional taxonomy of quantum non-localities 62 Chapter 2 Possible-world semantics for counterfactuals 69 2.1 Stalnaker’s counterfactual logic and the law of conditional excluded middle 72 2.2 Lewis’s counterfactual semantics 77 2.2.1 True-antecedent counterfactuals 81 2.3 Relative similarity between possible worlds 85 2.3.1 Lewis’s similarity ranking 88 2.3.2 Similarity ranking and the EPR correlations 93 2.3.3 Time-asymmetry of counterfactuals and the role of miracles 96 2.4 Conclusions 101 Chapter 3 A counterfactual version of Bell’s theorem and its criticism 105 3.1 Counterfactual strengthening of Bell’s theorem—a first approximation 106 3.2 CHSH inequality across possible worlds 111 3.3 The Matching Condition and the semantics of spatiotemporal counterfactuals 116 3.4 Alternative semantics of quantum counterfactuals 127 3.5 Locality and “might” counterfactuals 132 Chapter 4 The GHZ and Hardy theorems counterfactually strengthened—what went wrong? 141 4.1 The GHZ case 143 4.1.1 Initial assumptions 143 4.1.2 First steps of the proof 147 4.1.3 A semantic model for the GHZ counterfactuals 149 4.1.4 Elimination of Eliminated Conditions 151 4.1.5 Deriving the contradiction 154 4.1.6 Can the proof be rectified? 157 4.2 The Hardy case 160 4.2.1 Physical background 160 4.2.2 The Hardy case and EEC 163 4.2.3 Stapp’s 1997 proof 165 4.2.4 Locality condition LOC2 169 4.2.5 Possible justifications of LOC2 173 4.2.6 Einstein’s criterion of reality 176 4.2.7 Compatibility of LOC1 and quantum precepts 180 4.2.8 Conclusions 182 Chapter 5 Two interpretations of spatiotemporal counterfactuals 185 5.1 “Asymmetry by fiat” solution revived 185 5.2 Generalization of (C2) 190 5.3 Troubles with generalizing (C2) 197 5.4 Generalization of (C2) achieved—a step beyond Lewis’s orthodoxy 204 5.5. The application to the GHZ case 209 5.6 A comparison of the two notions of counterfactuals 219 Chapter 6 Locality explained and the EPR-Bell theorems reconsidered 225 6.1 Two variants of counterfactual locality 226 6.2 Semantic condition of locality and the equivalence theorem 231 6.3 The counterfactual EPR argument at last 236 6.3.1 Basic assumptions 236 6.3.2 The EPR argument and (C1) 239 6.3.3 The failure of the EPR argument under (C2) 246 6.4 Quantum dispositions or elements of reality? 249 6.5 Bell’s theorem with counterfactual hidden variables 255 6.6 Generalized locality condition and counterfactual outcome- dependence 264 Chapter 7 Comparisons and conclusions 271 Bibliography 283 Index 291 INTRODUCTION his book is primarily intended as a philosophical analysis of the phe- T nomenon of non-locality that so famously (or, one may say, infamously) unfolds in quantum theory. However, the scope of the pro- posed analysis is seriously limited by the adopted research methodology. It is not the purpose of the book to present the whole gamut of conceptions of how to interpret and deal with the problem of non-local causation—the causation that seems to propagate instantaneously from place to place— that apparently crops up in the quantum-mechanical realm. Instead, we will adopt a very specific perspective which may be dubbed the “counter- factual” approach to the problem of quantum non-locality and quantum en- tanglement. Thus, the main goal of the book will be to use the logic of counterfactual conditionals, i.e. expressions of the form “If it were (had been) P, then it would be (have been) Q”, in order to shed a new light on the ontological issue of non-locality in quantum mechanics. Actually, a substantial portion of the book will be devoted to the preliminary task of constructing a logic (or logics) of counterfactuals suitable for the interpre- tative job that is called for within the foundations of quantum mechanics. Only after having satisfactory dealt with this challenge, we can venture to seek solutions to the perennial philosophical debate regarding fundamental quantum results such as various versions of Bell-like theorems, where the problem of non-locality plays a pivotal part. Resorting to non-standard logical apparatuses in the foundational analy- sis of quantum mechanics is not an entirely novel idea. There is a well- established and highly regarded method of constructing a formal theory of quantum reality called “quantum logic” which was stimulated by the works of both mathematicians (G. Birkhoff, J. von Neumann) and philosophers (H. Reichenbach).1 The most important feature of these logical ap- 1 Classical texts on the subject of quantum logic are (Birkhoff, von Neumann 1936) and (Reichenbach 1944). A particularly nice and elegant introduction to various versions of quantum logic can be found in (Greechie, Gudder 1973). Other intro- ductions to quantum logic that particularly accentuate its philosophical importance are (Putnam 1969), (van Fraassen 1973), (Haack 1996, chapter 8), (Bub 1997, chapter 1). For the recent developments of this subject see (Dalla Chiara et al. 2004). I took issue with some claims of Birkoff and von Neumann’s version of quantum logic in (Bigaj 2001). Tomasz F. Bigaj • Non-locality and Possible Worlds proaches—for there is actually more than one quantum logic—is that they are all underpinned by the strong conviction that the peculiar properties of quantum reality call for some change in classical logic. The proposed non- classical novelties of quantum logics range from multi-valency to the rejec- tion of the law of distributivity. However, the task of constructing a counterfactual logic that would be helpful for describing quantum- mechanical phenomena is to be differentiated from these ambitious pro- grams. The counterfactual approach to quantum mechanics does not aim to correct classical logic, but rather to extend its expressive powers by intro- ducing some new non-truth-functional operators, similarly to the way modal logics extend, but do not reject classical logic by adding modal op- erators of possibility and necessity. There are, in my opinion, three main reasons why counterfactual condi- tionals may prove themselves useful in the analysis of non-local quantum causality. The first reason is that causality itself admits a clear and intuitive analysis in terms of counterfactuals. It is commonly accepted that there are two predominant conceptions of causality: regularist and singularist. The regularist approach tries to reduce causation to a regular succession of events, thus subsuming singular causal agents under general types of events (“phenomena”, in some terminology) and appealing to the general laws that connect these types of events. An alternative approach, famously developed by David Lewis, views causality as a relation between concrete token-events: this particular event e, as occurring in the actually obtaining circumstances, is a cause of another concrete event e′. Causation under- stood in such a way need not be based on regularities; another instance of event e in different circumstances may not produce an effect of the same type as e′. The key to the relation between the cause and the effect in this case lies in counterfactuality: we say that if the cause e had not occurred in those circumstances, the effect e′ would not have happened. Even without an excursion into the analysis of causation it should be clear that counterfactual conditionals lend themselves naturally to the task of expressing the idea of the lack of influence between events which is so crucial to the notion of locality.