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Bell's Theorem, Non-Separability and Space-Time Individuation in QuantumMechanicsDarrin W. Belousek†Program in History and Philosophy of ScienceDepartment of PhilosophyUniversity of Notre Dame †Support provided by the John J. Reilly Center for Science, Technology and Values and theZahm Research Travel Fund, University of Notre Dame. This research was conducted while aVisiting Scholar in the History and Philosophy of Science Department, Cambridge University, duringthe Lent term 1997. This paper is a shortened version of a chapter from my dissertation, “OntologicalCommitments and Theory Appraisal in the Interpretation of Quantum Mechanics”.2Bell's Theorem, Non-Separability and Space-Time Individuation in QuantumMechanicsAbstract. We first examine Howard's analysis of the Bell factorizability conditionin terms of 'separability' and 'locality' and then consider his claims that the violations ofBell's inequality by the statistical predictions of quantum mechanics should be interpretedin terms of 'non-separability' rather than 'non-locality' and that 'non-separability' implies thefailure of space-time as a principle of individuation for quantum-mechanical systems. AndI find his arguments for both claims to be lacking.1. IntroductionDon Howard has claimed that Bell's theorem and its (meta-)physical implicationscan be fruitfully understood by way of the 'Separation Principle' found in Einstein's ownincompleteness argument (Howard 1985; cf. Einstein 1948). Howard gives an interpretiveanalysis of the 'Separation Principle' in terms of two logically independent, conceptuallydistinct principles he calls 'separability' and 'locality' and shows that the Bell factorizabilitycondition is itself a consequence of these two principles (Howard 1989 and 1992). On thisbasis he argues that the violations of the Bell inequalities by the statistical predictions ofquantum mechanics and the confirmation of those predictions in experiment can (andshould) be interpreted as exhibiting 'non-separability' rather than 'non-locality'. And such'non-separability', he claims, implies the failure of space-time itself as a principle ofindividuation for quantum-mechanical systems generally. In the following I will developand criticize his arguments for each of these claims and find his conclusions less thancompelling.2. Analysis of the Bell Factorizability ConditionWe begin with Howard's notions of 'separability' and 'locality'. Howard states the'separability principle' "asserts that any two spatially separated systems possess their own3separate real states", while the 'locality principle' "asserts that all physical effects arepropagated with finite subluminal velocities, so that no effects can be communicatedbetween systems separated by a space-like interval" (Howard 1985, 173). Regarding the'separability principle' in particular, which will be of primary importance to our discussion,Howard states that itasserts that the contents of any two regions of space-time separated by a nonvanishingspatio-temporal interval constitute separable physical systems, in the sense that [(i)] eachpossesses its own, distinct physical state, and [(ii)] the joint state is wholly determined bythese separate states. In other words, the separability principle asserts that the presenceof a non-vanishing spatio-temporal interval is a sufficient condition for the individuationof physical systems and their associated states.... (Howard 1989, 225–6)For this principle to be applicable, some interpretation of 'state' is required. Howarddefines the 'state' pλ(x |m)λ of a physical system S primarily in terms of a (marginal)conditional probability measure for possible outcomes x of measurements on S givenmeasurement context m, pλ(x|m)(Howard 1989, 226, n. 2; cf. 1992, 310). Accordingto this definition, the quantum state of a physical system would represent a definiteproperty of the system if and only if pλ(x|m)=1, that is, if and only if λ is an eigenstateand x is the corresponding eigenvalue of the observable X, which is precisely theorthodox ‘Eigenstate-Eigenvalue Rule’.Now, beginning with Jarrett's analysis of the Bell factorizability condition into twodistinct independence conditions, outcome independence and parameter independence,Howard aims to show that the factorizability condition is also equivalent to theconjunction of what he calls the 'separability condition', which he claims follows from hisseparability principle, and the 'locality condition', which he claims follows from his localityprinciple (Howard 1989, 1992). Let A(B) be the respective outcome for an EPR-Belltype correlation experiment on S1(S2), a(b) be the respective apparatus setting, and λthe joint state of the composite system. Then the Jarrett (1984) analysis of thefactorizability condition,pλS12(A,B|a,b)=pλS1(A|a)pλS2(B|b),(1)yields the sets of following sets of conditions — outcome and parameter independence,respectively:4pλS1(A|a,b,B)=pλS1(A|a,b)pλS2(B|a,b,A)=pλS2(B|a,b)(2)andpλS1(A|a,b)=pλS1(A|a)pλS2(B|a,b)=pλS2(B|b).(3)Howard's analysis of the factorizability condition (1) begins by making thefollowing identifications:pλ1S1(A|a,b)=pλ12S1(A|a,b)pλ2S2(B|a,b)=pλ12S2(B|a,b).(4)Now, given these identifications (4), Howard's 'locality condition' is just the statement ofparameter independence (3) with λ12, λ1 and λ2 replacing λ where appropriate —pλ1S1(A|a,b)=pλ1S1(A|a)pλ2S2(B|a,b)=pλ2S2(B|b).(5)Howard claims that (5) follows from his locality principle, and this claim appearsunproblematic (but we will return to it below). So, his analysis requires further only thatthe outcome independence condition (2) be shown to be equivalent to his 'separabilitycondition', which he defines as follows:Two physical systems S1 and S2 are 'separable' if there exist 'separate' states λ1 andλ2 for S1 and S2, respectively, such that pλ12S12(A,B|a,b)=pλ1S1(A|a,b)pλ2S2(B|a,b),(6)where λ12 represents the (complete) joint state of the system S12 composed of S1and S2.Howard claims also that this separability condition (6) follows from his separabilityprinciple, and given his definition of 'state' this claim is unobjectionable.5Howard's argument to show that (2) and (6) are equivalent runs as follows. First,to show that (6) implies (2), we apply (6) and (4) successively to the following definitionof conditional probability:pλ12S1(A|a,b,B)≡pλ12S12(A,B|a,b) pλ12S2(B|a,b)=pλ1S1(A|a,b)pλ2S2(B|a,b) pλ12S2(B|a,b)=pλ12S1(A|a,b)