View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by CERN Document Server NONLOCALITY AS AN AXIOM FOR QUANTUM THEORY* Daniel Rohrlich and Sandu Pop escu School of Physics and Astronomy, Tel-Aviv University Ramat-Aviv, Tel-Aviv 69978 Israel ABSTRACT Quantum mechanics and relativistic causality together imply nonlo cality: nonlo- cal correlations (that violate the CHSH inequality) and nonlo cal equations of motion (the Aharonov-Bohm e ect). Can weinvert the logical order? We consider a conjecture that nonlo cality and relativistic causality together imply quantum mechanics. We show that correlations preserving relativistic causality can violate the CHSH inequality more strongly than quantum correlations. Also, we describ e nonlo cal equations of motion, preserving rel- ativistic causality, that do not arise in quantum mechanics. In these nonlo cal equations of motion, an exp erimenter \jams" nonlo cal correlations b etween quantum systems. 1. INTRODUCTION Two asp ects of quantum nonlo cality are nonlo cal correlations and nonlo cal equations of motion. Nonlo cal correlations arise in settings such as the one discussed by Einstein, 1 2 Po dolsky and Rosen . As Bell showed (and Asp ect has reviewed in his lecture here) no 3 theory of lo cal variables can repro duce these correlations. The Aharonov-Bohm e ect is also nonlo cal in that an electromagnetic eld in uences an electron in a region where the eld vanishes. The eld induces a relative phase b etween two sets of paths available to an electron, displacing the interference pattern b etween the two sets of paths. Thus, the 4 Aharonov-Bohm e ect implies nonlo cal equations of motion. Both asp ects of quantum non- lo cality arise within nonrelativistic quantum theory.However, the very de nition of a lo cal variable is relativistic: a lo cal variable can b e in uenced only byevents in its backward light cone, and can in uence events only in its forward light cone. In this sense, quantum mechanics and relativity together imply nonlo cality. They co exist b ecause quantum correla- tions preserve relativistic causality (i.e. they do not allow us to transmit signals faster than *Talk presented at 60 Years of E.P.R. (Workshop on the Foundations of Quantum Mechanics, in honor of Nathan Rosen), Technion, Israel, March 1995 1 light). But quantum mechanics do es not allow us to consider isolated systems as separate, 1 as Einstein, Po dolsky and Rosen assumed. This violation of not the letter but the spirit of sp ecial relativity has left manyphysicists (including Bell) deeply unsettled. Today, quantum nonlo cality seems as fundamental|and as unsettling|as ever. If nonlo cality is fundamen- tal, why not make nonlo cality an axiom of quantum theory rather than a consequence? Can we then invert the logical order, showing that nonlo cality and relativistic causality together imply quantum theory? 2. NONLOCALITY I: NONLOCAL CORRELATIONS Quantum mechanics and relativistic causality together give rise to nonlo cal corre- lations, which manyphysicists regard as a negative asp ect of quantum theory. Here, we regard quantum nonlo cality as a p ositive asp ect of quantum theory. What new p ossibilities do es quantum nonlo cality o er us? In particular, if we make nonlo cality an axiom, what 57 b ecomes of the logical structure of quantum theory? The sp ecial theory of relativity can b e deduced in its entirety from two axioms: the equivalence of inertial reference frames, and 7 the constancy of the sp eed of light. Aharonov has prop osed such a logical structure for quantum theory. Let us take, as axioms of quantum theory, relativistic causality and nonlo- cality. As an initial, immediate result, we deduce that quantum theory is not deterministic, 7 otherwise these two axioms would b e incompatible. Two \negative" asp ects of quantum mechanics|indeterminacy and limits on measurements|then app ear as a consequence of a fundamental \p ositive" asp ect: the p ossibility of nonlo cal action. Moreover, by taking nonlo cality as an axiom, we free ourselves of the need to explain it. Wehave not yet de ned the axiom of nonlo cality. Relativistic causalityiswell de ned, but quantum nonlo cality arises b oth in nonlo cal correlations and in the Aharonov-Bohm e ect. In this section we consider nonlo cal correlations. We ask which theories yield nonlo cal correlations while preserving causality. Our result is indep endent of quantum mechanics or 8 any particular mo del. We nd that quantum mechanics is only one of a class of theories consistent with our two axioms, and, in a certain sense, not even the most nonlo cal theory. 9 The Clauser, Horne, Shimony, and Holt (CHSH) form of Bell's inequality, holds in any classical theory (that is, any theory of lo cal hidden variables). It states that a certain 2 combination of correlations lies b etween -2 and 2: 0 0 0 0 2 E (A; B )+E(A; B )+E(A ;B) E(A ;B ) 2 : (1) p 2 and 4, are imp ortant b ounds on the CHSH sum of Besides 2, two other numb ers, 2 correlations. If the four correlations in Eq. (1) were indep endent, the absolute value of the sum could b e as muchas4. For quantum correlations, however, the CHSH sum of p 10 2. Where do es this b ound come from? correlations is b ounded in absolute value by2 Rather than asking why quantum correlations violate the CHSH inequality,we might ask why they do not violate it more. Let us say that of the two axioms prop osed ab ove, the axiom of nonlo cality implies that quantum correlations violate the CHSH inequality at least sometimes. Wemay then guess that the other axiom, relativistic causality, might imply that quantum correlations do not violate it maximally. Could it b e that relativistic causality restricts the violation p 2 instead of 4? If so, then the two axioms determine the quantum violation of the to 2 CHSH inequality. To answer this question, we ask what restrictions relativistic causality imp oses on joint probabilities. Relativistic causality forbids sending messages faster than light. Thus, if one observer measures the observable A, the probabilities for the outcomes A = 1 and A = 1must b e indep endent of whether the other observer cho oses to measure 0 8;11 B or B .However, it can b e shown that this constraint do es not limit the CHSH sum of p 2. For example, imagine a \sup erquantum" correlation function quantum correlations to 2 E for spin measurements along given axes. Assume E dep ends only on the relative angle between axes. For any pair of axes, the outcomes j ""i and j ##i are equally likely, and similarly for j "#i and j #"i. These four probabilities sum to 1, so the probabilities for j "#i and j ##i sum to 1=2. In any direction, the probabilityof j"i or j#i is 1=2 irresp ectiveof a measurement on the other particle. Measurements on one particle yield no information ab out measurements on the other, so relativistic causality holds. The correlation function then satis es E ( )=E(). Now let E ( )have the form (i) E ( ) = 1 for 0 =4; (ii) E ( ) decreases monotonically and smo othly from 1 to -1 as increases from =4 to 3=4; 3 (iii) E ( )=1 for 3=4 . 0 0 ^ ^ Consider four measurements along axes de ned by unit vectorsa ^ , b,^a, and b sepa- rated by successive angles of =4 and lying in a plane. If wenow apply the CHSH inequality Eq. (1) to these directions, we nd that the sum of correlations 0 0 0 0 ^ ^ ^ ^ E (^a; b)+E(^a ; b)+E(^a; b ) E (^a ; b )=3E(=4) E (3=4) = 4 (2) violates the CHSH inequality with the maximal value 4. Thus, a correlation function could satisfy relativistic causality and still violate the CHSH inequality with the maximal value 4. 3. NONLOCALITY I I: NONLOCAL EQUATIONS OF MOTION In one version of the Aharonov-Bohm e ect, an isolated magnetic ux, inserted b e- tween two slits, shifts the interference pattern of electrons passing through the slits. It thereby a ects the electron's momentum, since the electron arrives at a di erent p oint than it would without the electromagnetic eld. Thus, the Aharonov-Bohm e ect implies non- 4 7 lo cal equations of motion. Aharonov has shown that a physical quantity, the modular 12 momentum of the ux, is uncertain exactly as required to keep us from seeing a violation of causality. In general, mo dular momentum is measurable and ob eys a nonlo cal equation of motion. But when the ux is lo cated b etween the slits, its mo dular momentum is completely uncertain. Is quantum mechanics the only relativistically causal theory with nonlo cal equations of motion? As in the last section, wemay approach this question by lo oking for a theory 13 not equivalent to quantum mechanics that ob eys relativisitic causality and nonlo cality. Wehave considered a mo del in which action by an exp erimenter a ects (\jams") nonlo cal correlations b etween systems measured at spacelike separations from the action. For exam- 5 ple, Shimony considers the e ect of a laser b eam crossing the path of one of the photons in a singlet pair, after the photon has already passed. We nd that while nonlo cal \jamming" is not p ossible in quantum mechanics, it could b e consistent with relativisitic causality.If jamming is realized in nature, then p erhaps, as suggested by Grunhaus, it is p ossible to jam nonlo cal quantum correlations.