In the quantum realm, there are curious correlations between theproperties of distant systems. An example of such correlations isprovided by the famous Einstein-Podolsky-Rosen/Bohm experiment. Thecorrelations in the EPR/B experiment strongly suggest that there arenon-local influences between distant systems, i.e., systems betweenwhich no light signal can travel, and indeed orthodox quantummechanics and its various interpretations postulate the existence ofsuch non-locality. Yet, the question of whether the EPR/B correlationsimply non-locality and the exact nature of this non-locality is amatter of ongoing controversy. Focusing on EPR/B-type experiments, inthis entry we consider the nature of the various kinds of non-localitypostulated by different interpretations of quantum mechanics. Based onthis consideration, we briefly discuss the compatibility of theseinterpretations with the special theory of relativity.

• 1. Introduction
• 2. Bell's theorem and non-locality
• 3. The analysis of factorizability
• 4. Action at a distance, holism and non-separability
  • 4.1 Action at a distance
  • 4.2 Holism
  • 4.3 Non-separability
• 5. Holism, non-separability and action at a distance in quantum mechanics
  • 5.1 Collapse theories
  • 5.2 Can action-at-a-distance co-exist with non-separability and holism?
  • 5.3 No-collapse theories
• 6. Superluminal causation
• 7. Superluminal signaling
  • 7.1 Necessary and sufficient conditions for superluminal signaling
  • 7.2 No-collapse theories
  • 7.3 Collapse theories
  • 7.4 The prospects ofcontrollable probabilistic dependence
  • 7.5 Superluminal signaling and action-at-a-distance
• 8. The analysis of factorizability: implications for quantum non-locality
  • 8.1 Non-separability, holism and action at a distance
  • 8.2 Superluminal signaling
  • 8.3 Relativity
  • 8.4 Superluminal causation
  • 8.5 On the origin and nature of parameter dependence
• 9. Can there be ‘local’ quantum theories?
• 10. Can quantum non-locality be reconciled with relativity?
  • 10.1 Collapse theories
  • 10.2 No-collapse theories
  • 10.3 Quantum causal loops and relativity
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Introduction

The quantum realm involves curious correlations between distantevents. A well-known example is David Bohm's (1951) version ofthe famous thought experiment that Einstein, Podolsky and Rosenproposed in 1935 (henceforth, the EPR/B experiment). Pairs of particlesare emitted from a source in the so-called spin singlet state and rushin opposite directions (see Fig. 1 below). When the particles arewidely separated from each other, they each encounter a measuringapparatus that can be set to measure their spin components alongvarious directions. Although the measurement events are distant fromeach other, so that no slower-than-light or light signal can travelbetween them, the measurement outcomes are curiously correlated.^[1] That is, while the outcome of each of the distant spin measurementsseems to be a matter of pure chance, they are correlated with eachother: The joint probability of the distant outcomes is different fromthe product of their single probabilities. For example, theprobability that each of the particles will spin clockwise about thez-axis in az-spin measurement (i.e., a measurementof the spin component along thez direction) appears to be½. Yet, the outcomes of such measurements are perfectlyanti-correlated: If the left-hand-side (L-) particle happens to spinclockwise (anti-clockwise) about thez-axis, theright-hand-side (R-) particle will spin anti-clockwise (clockwise)about that axis. And this is true even if the measurements are madesimultaneously.

Figure 1: A schematic illustration of the EPR/Bexperiment. Particle pairs in the spin singlet state are emitted inopposite directions and when they are distant from each other(i.e., space-like separated), they encounter measurement apparatusesthat can be set to measure spin components along variousdirections.

The curious EPR/B correlations strongly suggest the existence ofnon-local influences between the two measurement events, and indeedorthodox ‘collapse’ quantum mechanics supports thissuggestion. According to this theory, before the measurements theparticles do not have any definite spin. The particles come to possessa definite spin only with the first spin measurement, and the outcomeof this measurement is a matter of chance. If, for example, the firstmeasurement is az-spin measurement on the L-particle, theL-particle will spin either clockwise or anti-clockwise about thez-axis with equal chance. And the outcome of theL-measurement causes an instantaneous change in the spin properties ofthe distant R-particle. If the L-particle spins clockwise(anti-clockwise) about thez-axis, the R-particle willinstantly spin anti-clockwise (clockwise) about the same axis. (It iscommon to call spins in opposite directions ‘spin up’ and‘spin down,’ where by convention a clockwise spinning maybe called ‘spin up’ and anti-clockwise spinning may becalled ‘spin down.’)

It may be argued that orthodox quantum mechanics is false, and thatthe non-locality postulated by it does not reflect any non-locality inthe quantum realm. Alternatively, it may be argued that orthodoxquantum mechanics is a good instrument for predictions rather than afundamental theory of the physical nature of the universe. On thisinstrumental interpretation, the predictions of quantum mechanics arenot an adequate basis for any conclusion about non-locality: Thistheory is just an incredible oracle (or a crystal ball), which providesa very successful algorithm for predicting measurement outcomes andtheir probabilities, but it offers little information about ontologicalmatters, such as the nature of objects, properties and causation in thequantum realm.

Einstein, Podolsky and Rosen (1935) thought that quantum mechanics isincomplete and that the curious correlations between distant systemsdo not amount to action at a distance between them. The apparentinstantaneous change in the R-particle's properties during theL-measurement is not really a change of properties, but rather achange of knowledge. (For more about the EPR argument, see the entryon the EPR argument, Redhead 1987, chapter 3, and Albert 1992, chapter3. For discussions of the EPR argument in the relativistic context,see Ghirardi and Grassi 1994 and Redhead and La Riviere 1997.) On thisview, quantum states of systems do not always reflect their completestate. Quantum states of systems generally provide information aboutsome of the properties that systems possess and informationabout the probabilities of outcomes of measurements on them, and thisinformation does not generally reflect the complete state of thesystems. In particular, the information encoded in the spin singletstate is about the probabilities of measurement outcomes of spinproperties in various directions, about the conditional probabilitiesthat the L- (R-) particle has a certain spin property given that theR- (L-) particle has another spin property, and about theanti-correlation between the spins that the particles may have in anygiven direction (for more details, see section 5.1). Thus, the outcomeof az-spin measurement on the L-particle and the spinsinglet state (interpreted as a state of knowledge) jointly provideinformation about thez-spin property of the R-particle. Forexample, if the outcome of the L-measurement isz-spin‘up,’ we know that the R-particle hasz-spin‘down’; and if we assume, as EPR did, that there is nocurious action at a distance between the distant wings (and that thechange of the quantum-mechanical state of the particle pair in theL-measurement is only a change in state of knowledge), we could alsoconclude that the R-particle hadz-spin ‘down’even before the L-measurement occurs.

How could the L-outcome change ourknowledge/ignorance about the R-outcome if it has no influence on it?The simplest and most straightforward reply is that the L- and the R-outcome have a common cause that causes them to be correlated, so thatknowledge of one outcome provides knowledge about the other.^[2] Yet, the question is whether the predictions of orthodox quantummechanics, which have been highly confirmed by various experiments,are compatible with the quantum realm being local in the sense ofinvolving no influences between systems between which light andslower-than-light signals cannot travel (i.e., space-like separatedsystems). More particularly, the question is whether it is possible toconstruct a local, common-cause model of the EPR/B experiment, i.e., amodel that postulates no influence between systems/events in thedistant wings of the experiment, and that the correlation between themare due to the state of the particle pair at the source. In 1935,Einstein, Podolsky and Rosen believed that this is possible. But, asJohn Bell demonstrated in 1964, this belief is difficult touphold.

2. Bell's theorem and non-locality

In a famous theorem, John Bell (1964) demonstrated that granted someplausible assumptions, any local model of the EPR/B experiment iscommitted to certain inequalities about the probabilities ofmeasurement outcomes, ‘the Bell inequalities,’ which areincompatible with the quantum-mechanical predictions. When Bell provedhis theorem, the EPR/B experiment was only a thought experiment. Butdue to technological advances, various versions of this experiment havebeen conducted since the 1970s, and their results have overwhelminglysupported the quantum-mechanical predictions (for brief reviews ofthese experiments and further references, see the entry on Bell'stheorem and Redhead 1987, chapter 4, section 4.3 and ‘Notes andReferences’). Thus, a wide consensus has it that the quantumrealm involves some type of non-locality.

The basic idea of Bell's theorem is as follows. A model of the EPR/Bexperiment postulates that the state of the particle pair togetherwith the apparatus settings to measure (or not to measure) certainspin properties determine the probabilities for single and jointspin-measurement outcomes. A local Bell model of this experiment alsopostulates that probabilities of joint outcomes factorize into thesingle probabilities of the L- and the R- outcomes: The probability ofjoint outcomes is equal to the product of the probabilities of thesingle outcomes. More formally, let λ denote the pair's statebefore any measurement occurs. Letl denote the setting ofthe L-measurement apparatus to measure spin along thel-axis(i.e., thel-spin of the L-particle), and letrdenote the setting of the R-measurement apparatus to measure spinalong ther-axis (i.e., ther-spin of theR-particle). Letx_l be the outcome of al-spin measurement in the L-wing, and lety_r be the outcome of ar-spin measurementin the R-wing; wherex_l is either the L-outcomel-spin ‘up’ or the L-outcomel-spin‘down,’ andy_r is either the R-outcomer-spin ‘up’ or the R-outcomer-spin‘down.’ LetP_λlr(x_l &y_r) bethe joint probability of the L- and the R-outcome, andP_λl(x_l) andP_λr(y_r) be the singleprobabilities of the L- and the R-outcome, respectively; where thesubscripts λ,l andr denote the factors thatare relevant for the probabilities of the outcomesx_l andy_r. Then, for anyλ,l,r,x_l andy_r:^[3]

Factorizability
P_λ l r(x_l &y_r) =P_λ l(x_l) ·P_λ r(y_r).

(Here and henceforth, for simplicity's sake we shall denote events andstates, such as the measurement outcomes, and the propositions thatthey occur by the same symbols.)

The state λ is typically thought of as the pair's state at theemission time, and it is assumed that this state does not change inany relevant sense between the emission and the first measurement. Itis (generally) a different state from the quantum-mechanical pair'sstate ψ. ψ is assumed to be an incomplete state of the pair,whereas λ is supposed to be a (more) complete state of thepair. Accordingly, pairs with the same state ψ may have differentstates λ which give rise to different probabilities of outcomesfor the same type of measurements. Also, the states λ may beunknown, hidden, inaccessible or uncontrollable.

Factorizability is commonly motivated as a locality condition. Innon-local models of the EPR/B experiment, the correlations between thedistant outcomes are accounted for by non-local influences between thedistant measurement events. For example, in orthodox quantum mechanicsthe first spin measurement on, say, the L-particle causes an immediatechange in the spin properties of the R-particle and in theprobabilities of future outcomes of spin measurements on this particle.By contrast, in local models of this experiment the correlations aresupposed to be accounted for by a common cause—the pair'sstate λ (see Fig. 2 below): The pair's state and theL-setting determine the probability of the L-outcome; the pair'sstate and the R-setting determine the probability of theR-outcome; and the pair's state and the L- and theR-setting determine the probability of joint outcomes, which (asmentioned above) is simply the product of these single probabilities.The idea is that the probability of each of the outcomes is determinedby ‘local events,’ i.e., events that are confined to itsbackward light-cone, and which can only exert subluminal or luminalinfluences on it (see Figure 3 below); and the distant outcomes arefundamentally independent of each other, and thus their jointprobability factorizes. (For more about this reasoning, see sections 6and 8-9.)

Figure 2: A schematic common-cause model of the EPR/Bexperiment. Arrows denote causal connections.

Figure 3: A space-time diagram of a local model ofthe EPR/B experiment. The circles represent the measurement events,and the cones represent their backward light cones, i.e., theboundaries of all the subluminal and luminal influences on them. Thedotted lines denote the propagation of the influences of the pair'sstate at the emission and of the settings of the measurementapparatuses on the measurement outcomes.

A Bell model of the EPR/B experiment also postulates that for eachquantum-mechanical state ψ there is a distribution ρ over allthe possible pair states λ, which isindependent of the settings of the apparatuses. That is, thedistribution of the (‘complete’) states λ depends onthe (‘incomplete’) state ψ, and this distributionis independent of the particular choice of measurements in the L- andR-wing (including the choice not to measure any quantity). Orformally, for any quantum-mechanical state ψ, L-settingsl andl′, and R-settingsr andr′:

λ-independence
ρ_ψ l r(λ) = ρ_{ψ l′ r}(λ) = ρ_{ψ l r′}(λ) = ρ_{ψ l′ r′}(λ) = ρ_ψ(λ)

where the subscripts denote the factors that are potentiallyrelevant for the distribution of the states λ.

Although the model probabilities (i.e., the probabilities of outcomesprescribed by the states λ) are different from thecorresponding quantum-mechanical probabilities of outcomes (i.e., theprobabilities prescribed by the quantum-mechanical states ψ),the quantum mechanical probabilities (which have been systematicallyconfirmed) are recovered by averaging over the model probabilities.That is, it is supposed that the quantum-mechanical probabilitiesP_ψlr(x_l &y_r),P_ψl(x_l)andP_ψr(y_r)are obtained by averaging over the model probabilitiesP_λlr(x_l &y_r),P_λl(x_l) andP_λr(y_r),respectively: For any ψ,l,r,x_l andy_r,

Empirical Adequacy
P_ψ l r(x_l &y_r) =_λ P_λ l r(x_l &y_r) · ρ_ψ l r(λ)
P_ψ l(x_l) =_λ P_λ l(x_l ) · ρ_ψ l(λ)
P_ψ r(y_r) =_λ P_λ r(y_r) · ρ_ψ r(λ).^[4]

The assumption of λ-independence is very plausible. Itpostulates that (complete) pair states at the source are uncorrelatedwith the settings of the measurement apparatuses. And independently ofone's philosophical view about free will, this assumption is stronglysuggested by our experience, according to which it seems possible toprepare the state of particle pairs at the source independently of theset up of the measurement apparatuses.

There are two ways to try to explain a failure ofλ-independence. One possible explanation is thatpairs' states and apparatus settings share a common cause, whichalways correlates certain types of pairs' states λ withcertain types of L- and R-setting. Such a causal hypothesis will bedifficult to reconcile with the common belief that apparatus settingsare controllable at experimenters' will, and thus could be setindependently of the pair's state at the source. Furthermore,thinking of all the different ways one can measure spin properties andthe variety of ways in which apparatus settings can be chosen, thepostulation of such common cause explanation for settings andpairs' states would seem highlyad hoc and its existenceconspiratorial.

Another possible explanation for the failure of λ-independenceis that the apparatus settings influence the pair's state at thesource, and accordingly the distribution of the possible pairs' statesλ is dependent upon the settings. Since the settings can bemade after the emission of the particle pair from the source, thiskind of violation of λ-independence would require backwardcausation. (For advocates of this way out of non-locality, see Costade Beauregard 1977, 1979, 1985, Sutherland 1983, 1998, 2006 and Price1984, 1994, 1996, chapters 3, 8 and 9.) On some readings of JohnCramer's (1980, 1986) transactional interpretation of quantummechanics (see Maudlin 1994, pp. 197-199), such violation ofλ-independence is postulated. According to this interpretation,the source sends ‘offer’ waves forward to the measurementapparatuses, and the apparatuses send ‘confirmation’ waves(from the space-time regions of the measurement events) backward tothe source, thus affecting the states of emitted pairs according tothe settings of the apparatuses. The question of whether such atheory can reproduce the predictions of quantum mechanics is acontroversial matter (see Maudlin 1994, pp. 197-199, Berkovitz 2002,section 5, and Kastner 2006). It is noteworthy, however, that whilethe violation of λ-independence is sufficient for circumventingBell's theorem, the failure of this conditionper se does notsubstantiate locality. The challenge of providing a local model of theEPR/B experiment also applies to models that violateλ-independence. (For more about these issues, see sections 9and 10.3.)

In any case, as Bell's theorem demonstrates, factorizability,λ-independence and empirical adequacy jointly imply the Bellinequalities, which are violated by the predictions of orthodox quantummechanics (Bell 1964, 1966, 1971, 1975a,b). Granted the systematicconfirmation of the predictions of orthodox quantum mechanics and theplausibility of λ-independence, Bell inferred thatfactorizability fails in the EPR/B experiment. Thus, interpretingfactorizability as a locality condition, he concluded that the quantumrealm is non-local. (For further discussions of Bell's theorem,the Bell inequalities and non-locality, see Bell 1966, 1971, 1975a,b,1981, Clauseret al 1969, Clauser and Horne 1974, Shimony1993, chapter 8, Fine 1982a,b, Redhead 1987, chapter 4, Butterfield1989, 1992a, Pitowsky 1989, Greenberger, Horne and Zeilinger 1989,Greenberger, Horne, Shimony and Zeilinger 1990, Mermin 1990, and theentry on Bell's theorem.)

3. The analysis of factorizability

Following Bell's work, a broad consensus has it that thequantum realm involves some type of non-locality (for examples, seeClauser and Horne 1974, Jarrett 1984,1989, Shimony 1984, Redhead 1987,Butterfield 1989, 1992a,b, 1994, Howard 1989, Healey 1991, 1992, 1994,Teller 1989, Clifton, Butterfield and Redhead 1990, Clifton 1991,Maudlin 1994, Berkovitz 1995a,b, 1998a,b, and references therein).^[5] But there is an ongoing controversy as to its exact nature and itscompatibility with relativity theory. One aspect of this controversyis over whether the analysis of factorizability and the different waysit could be violated may shed light on these issues. Factorizabilityis equivalent to the conjunction of two conditions (Jarrett 1984,1989, Shimony 1984):^[6]

Parameter independence. The probability of a distantmeasurement outcome in the EPR/B experiment is independent of thesetting of the nearby measurement apparatus. Or formally, for anypair's state λ, L-settingl, R-settingr,L-outcomex_l and R-outcomey_r:

PI

Pλlr(xl) =Pλl(xl)

and

Pλlr(yr) =Pλr(yr).

Outcome independence. The probability of a distantmeasurement outcome in the EPR/B experiment is independent of thenearby measurement outcome. Or formally, for any pair's stateλ, L-settingl, R-settingr, L-outcomex_l and R-outcomey_r:

Pλlr(xl /yr) =Pλlr(xl)	and	Pλlr(yr /xl) =Pλlr(yr)
Pλlr(yr) > 0		Pλlr(xl) > 0,

or more generally,

OI
P_λlr(x_l &y_r) =P_λlr(x_l) ·P_λlr(y_r).

Assuming λ-independence (see section 2), any empiricallyadequate theory will have to violate OI or PI. A common view has itthat violations of PI involve a different type of non-locality thanviolations of OI: Violations of PI involve some type ofaction-at-a-distance that is impossible to reconcile with relativity(Shimony 1984, Redhead 1987, p. 108), whereas violations of OI involvesome type of holism, non-separability and/or passion-at-a-distance thatmay be possible to reconcile with relativity (Shimony 1984, Readhead1987, pp. 107, 168-169, Howard 1989, Teller 1989).

On the other hand, there is the view that the analysis above (aswell as other similar analyses of factorizability^[7]) is immaterial for studying quantum non-locality (Butterfield 1992a,pp. 63-64, Jones and Clifton 1993, Maudlin 1994, pp. 96 and 149) andeven misleading (Maudlin 1994, pp. 94-95 and 97-98). On thisalternative view, the way to examine the nature of quantumnon-locality is to study the ontology postulated by the variousinterpretations of quantum mechanics and alternative quantum theories.^[8] In sections 4-7, we shall follow this methodology and discuss thenature of non-locality postulated by several quantum theories. Thediscussion in these sections will furnish the ground for evaluatingthe above controversy in section 8.

4. Action at a distance, holism and non-separability

4.1 Action at a distance

In orthodox quantum mechanics as well as in any other current quantumtheory that postulates non-locality (i.e., influences between distant,space-like separated systems), the influences between the distantmeasurement events in the EPR/B experiment do not propagatecontinuously in space-time. They seem to involve action at adistance. Yet, a common view has it that these influences are due tosome type of holism and/or non-separability of states of compositesystems, which are characteristic of systems in entangled states (likethe spin singlet state), and which exclude the very possibility ofaction at a distance. The paradigm case of action at a distance is theNewtonian gravitational force. This force acts between distinctobjects that are separated by some (non-vanishing) spatial distance,its influence is symmetric (in that any two massive objects influenceeach other), instantaneous and does not propagate continuously inspace. And it is frequently claimed or presupposed that such action ata distance could only exist between systems with separate states innon-holistic universes (i.e., universes in which the states ofcomposite systems are determined by, or supervene upon the states oftheir subsystems and the spacetime relations between them), which arecommonly taken to characterize the classical realm.^[9]

In sections 4.2 and 4.3, we shall briefly review the relevantnotions of holism and non-separability (for a more comprehensivereview, see the entry on holism and nonseparability in physics andHealey 1991). In section 5, we shall discuss the nature of holism andnon-separability in the quantum realm as depicted by various quantumtheories. Based on this discussion, we shall consider whether thenon-local influences in the EPR/B experiment constitute action at adistance.

4.2 Holism

In the literature, there are various characterizations of holism.Discussions of quantum non-locality frequently focus on propertyholism, where certain physical properties of objects are not determinedby the physical properties of their parts. The intuitive idea is thatsome intrinsic properties of wholes (e.g. physical systems) are notdetermined by the intrinsic properties of their parts and thespatiotemporal relations that obtain between these parts. This idea canbe expressed in terms of supervenience relations.

Property Holism. Some objects have intrinsicqualitative properties and/or relations that do not supervene upon theintrinsic qualitative properties and relations of their parts and thespatiotemporal relations between these parts.

It is difficult to give a general precise specification of the terms‘intrinsic qualitative property’ and‘supervenience.’ Intuitively, a property of an object isintrinsic just in case that object has this property in and for itselfand independently of the existence or the state of any other object. Aproperty is qualitative (as opposed to individual) if it does notdepend on the existence of any particular object. And the intrinsicqualitative properties of an objectO supervene upon theintrinsic qualitative properties and relations of its parts and thespatiotemporal relations between them just in case there is no changein the properties and relations ofO without a change in theproperties and relations of its parts and/or the spatiotemporalrelations between them. (For attempts to analyze the term‘intrinsic property,’ see for example Langton and Lewis1998 and the entry on intrinsic vs. extrinsic properties. For a reviewof different types of supervenience, see for example Kim 1978,McLaughlin 1994 and the entry on supervenience.)

Paul Teller (1989, p. 213) proposes a related notion of holism,‘relational holism,’ which is characterized as theviolation of the following condition:

Particularism. The world is composed ofindividuals. All individuals have non-relational properties and allrelations supervene on the non-relational properties of therelata.

Here, by a non-relational property Teller means an intrinsic property(1986a, p. 72); and by ‘the supervenience of a relationalproperty on the non-relational properties of the relata,’ hemeans that ‘if two objects, 1 and 2, bear a relationRto each other, then,necessarily, if two further objects,1′ and 2′ have the same non-relational properties, then1′ and 2′ will also bear the same relationR toeach other’ (1989, p. 213). Teller (1986b, pp. 425-7) believesthat spatiotemporal relations between objects supervene upon theobjects’ intrinsic physical properties. Thus, he does notinclude the spatiotemporal relations in the supervenience basis. Thisview is controversial, however, as many believe that spatiotemporalrelations between objects are neither intrinsic nor supervene upon theintrinsic qualitative properties of these objects. But, if suchsupervenience does not obtain, particularism will also be violated inclassical physics, and accordingly relational holism will fail to markthe essential distinction between the classical and the quantumrealms. Yet, one may slightly revise Teller's definition ofparticularism as follows:

Particularism*. The world is composed ofindividuals. All individuals have non-relational properties and allrelations supervene upon the non-relational properties of the relataand the spatiotemporal relations between them.

In what follows in this entry, by relational holism we shall mean aviolation of particularism*.

4.3 Non-separability

Like holism, there are various notions of non-separability on offer.The most common notion in the literature is state non-separability,i.e., the violation of the following condition:

State separability. Each system possesses aseparate state that determines its qualitative intrinsic properties,and the state of any composite system is wholly determined by theseparate states of its subsystems.

The term ‘wholly determined’ is vague. But, as before, onemay spell it out in terms of supervenience relations: Stateseparability obtains just in case each system possesses a separatestate that determines its qualitative intrinsic properties andrelations, and the state of any composite system is supervenient uponthe separate states of its subsystems.

Another notion of non-separability is spatiotemporal non-separability.Inspired by Einstein (1948), Howard (1989, pp. 225-6) characterizesspatiotemporal non-separability as the violation of the followingseparability condition:

Spatiotemporal separability. The contents of anytwo regions of space-time separated by a non-vanishing spatiotemporalinterval constitute two separate physical systems. Each separatedspace-time region possesses its own, distinct state and the joint stateof any two separated space-time regions is wholly determined by theseparated states of these regions.

A different notion of spatiotemporal non-separability, proposed byHealey (see the entry on holism and nonseparability in physics), isprocess non-separability. It is the violation of the followingcondition:

Process separability. Any physical processoccupying a spacetime regionR supervenes upon an assignmentof qualitative intrinsic physical properties at spacetime points inR.

5. Holism, non-separability and action at a distance in quantum mechanics

The quantum realm as depicted by all the quantum theories thatpostulate non-locality, i.e., influences between distant (space-likeseparated) systems, involves some type of non-separability orholism. In what follows in this section, we shall consider the natureof the non-separability and holism manifested by variousinterpretations of quantum mechanics. On the basis of thisconsideration, we shall address the question of whether theseinterpretations predicate the existence of action at a distance. Westart with the so-called ‘collapse theories.’

5.1 Collapse theories

5.1.1 Orthodox quantum mechanics

In orthodox quantum mechanics, normalized vectors in Hilbert spacesrepresent states of physical systems. When the Hilbert space is ofinfinite dimension, state vectors can be represented by functions, theso-called ‘wave functions.’ In any given basis, there is aunique wave function that corresponds to the state vector in thatbasis. (For an entry level review of the highlights of themathematical formalism and the basic principles of quantum mechanics,see the entry on quantum mechanics, Albert 1992, Hughes 1989, Part I,and references therein; for more advanced reviews, see Bohm 1951 andRedhead 1987, chapters 1-2 and the mathematical appendix.)

For example, the state of the L-particle havingz-spin‘up’ (i.e., spinning ‘up’ about thez-axis) can be represented by the vector |z-up>in the Hilbert space associated with the L-particle, and the state ofthe L-particle havingz-spin ‘down’ (i.e.,spinning ‘down’ about thez-axis) can berepresented by the orthogonal vector, |z-down>. Particlepairs may be in a state in which the L-particle and the R-particlehave opposite spins, for instance either a state|ψ₁> in which the L-particle hasz-spin‘up’ and the R-particle hasz-spin‘down,’ or a state |ψ₂> in which theL-particle hasz-spin ‘down’ and the R-particlehasz-spin ‘up.’ Each of these states isrepresented by a tensor product of vectors in the Hilbert space of theparticle pair: |ψ₁> = |z-up>_L|z-down>_R and |ψ₂> =|z-down>_L|z-up>_R; where the subscripts L and Rrefer to the Hilbert spaces associated with the L- and the R-particle,respectively. But particle pairs may also be in a superposition ofthese states, i.e., a state that is a linear sum of the states|ψ₁> and |ψ₂>, e.g. the staterepresented by

|ψ3>	=	1/√2 (|ψ1>− |ψ2>)
	=	1/√2(|z-up>L|z-down>R− |z-down>L|z-up>R).

In fact, this is exactly the case in the spin singlet state. In thisstate, the particles are entangled in a non-separable state (i.e., astate that cannot be decomposed into a product of separate states ofthe L- and the R-particle), in which (according to theproperty-assignment rules of orthodox quantum mechanics) the particlesdo not possess any definitez-spin (or definite spin in anyother direction). Thus, the condition of state separability fails: Thestate of the particle pair (which determines its intrinsic qualitativeproperties) is not wholly determined by the separate states of theparticles (which determine their intrinsic qualitative properties). Ormore precisely, the pair's state is not supervenient upon theseparable states of the particles. In particular, the superpositionstate of the particle pair assigns a ‘correlational’property that dictates that the outcomes of (ideal)z-spinmeasurements on both the L- and the R-particle will beanti-correlated, and this correlational property is not supervenientupon properties assigned by any separable states of the particles (formore details, see Healey 1992, 1994). For similar reasons, the spinsinglet state also involves property and relational holism; for theabove correlational property of the particle pair also fails tosupervene upon the intrinsic qualitative properties of the particlesand the spatiotemporal relations between them. Furthermore, theprocess that leads to each of the measurement outcomes is alsonon-separable, i.e., process separability fails (see Healey 1994 andthe entry on holism and nonseparability in physics).

This correlational property is also ‘responsible’ for theaction at a distance that the orthodox theory seems to postulatebetween the distant wings in the EPR/B experiment. Recall (section 1)that Einstein, Podolsky and Rosen thought that this curious action ata distance reflects the incompleteness of this theory rather than astate of nature. The EPR argument for the incompleteness of theorthodox theory is controversial. But the orthodox theory seems to beincomplete for a different reason. This theory postulates that innon-measurement interactions, the evolution of states obeys a linearand unitary equation of motion, the so-called Schrödingerequation (see the entry on quantum mechanics), according to which theparticle pair in the EPR/B experiment remains in an entangledstate. This equation of motion also dictates that in a spinmeasurement, the pointers of the measurement apparatuses get entangledwith the particle pair in a non-separable state in which (according tothe theory's property assignment, see below) the indefiniteness ofparticles’ spins is ‘transmitted’ to the pointer'sposition: In this entangled state of the particle pair and thepointer, the pointer lacks any definite position, in contradiction toour experience of perceiving it pointing to either ‘up’ or‘down.’

The above problem, commonly called ‘the measurementproblem,’ arises in orthodox no-collapse quantum mechanics fromtwo features that account very successfully for the behavior ofmicroscopic systems: The linear dynamics of quantum states asdescribed by the Schrödinger equation and the property assignmentrule called ‘eigenstate-eigenvalue link.’ According to theeigenstate-eigenvalue link, a physical observable, i.e., a physicalquantity, of a system has definite value (one of its eigenvalues) justin case the system is in the corresponding eigenstate of thatobservable (see the entry on quantum mechanics, section4). Microscopic systems may be in a superposition state of spincomponents, energies, positions, momenta as well as other physicalobservables. Accordingly, microscopic systems may be in a state ofindefinitez-spin, energy, position, momentum and variousother quantities. The problem is that given the linear and unitarySchrödinger dynamics, these indefinite quantities are alsoendemic in the macroscopic realm. For example, in az-spinmeasurement on a particle in a superposition state ofz-spin‘up’ andz-spin ‘down,’ the positionof the apparatus’s pointer gets entangled with the indefinitez-spin of the particle, thus transforming the pointer into astate of indefinite position, i.e., a superposition of pointing‘up’ and pointing ‘down’ (see Albert 1992,chapter 4, and the entry on collapse theories, section 3). Inparticular, in the EPR/B experiment the L-measurement causes theL-apparatus pointer to get entangled with the particle pair,transforming it into a state of indefinite position:

|ψ₄> = 1/√2(|z-up>_L |z-down>_R|up>_LA − |z-down>_L |z-up>_R |down>_LA)

where |up>_LA and|down>_LA are the states of the L-apparatuspointer displaying the outcomesz-spin ‘up’ andz-spin ‘down,’ respectively. Since the above typeof indefiniteness is generic in orthodox no-collapse quantummechanics, in this theory measurements typically have no definiteoutcomes, in contradiction to our experience.

In order to solve this problem, the orthodox theory postulates that inmeasurement interactions, entangled states of measured systems and thecorresponding measurement apparatuses do not evolve according to theSchrödinger equation. Rather, they undergo a‘collapse’ into product (non-entangled) states, where thesystems involved have the relevant definite properties. For example,the entangled state of the particle pair and the L-apparatus in theEPR/B experiment may collapse into a product state in which theL-particle comes to possessz-spin ‘up,’ theR-particle comes to possessz-spin ‘down’ and theL-apparatus pointer displaying the outcomez-spin‘up’:

|ψ₅> = |z-up>_L |z-down>_R|up>_LA.

The problem is that in the orthodox theory, the notions ofmeasurement and the time, duration and nature of state collapses remaincompletely unspecified. As John Bell (1987b, p. 205) remarks, thecollapse postulate in this theory, i.e., the postulate that dictatesthat in measurement interactions the entangled states of the relevantsystems do not follow the Schrödinger equation but rather undergoa collapse, is no more than ‘supplementary, imprecise, verbal,prescriptions.’

This problem of accounting for our experience of perceiving definitemeasurement outcomes in orthodox quantum mechanics, is an aspect of themore general problem of accounting for the classical-like behavior ofmacroscopic systems in this theory.

5.1.2 Dynamical models for state vector reduction

The dynamical models for state-vector reduction were developed toaccount for state collapses as real physical processes (for a reviewof the collapse models and a detailed reference list, see the entry oncollapse theories). The origin of the collapse models may be dated toBohm and Bub's (1966) hidden variable theory and Pearle's (1976)spontaneous localization approach, but the program has received itscrucial impetus with the more sophisticated models developed byGhirardi, Rimini and Weber in 1986 (see also Bell 1987a and Albert1992) and their consequent development by Pearle (1989) (see alsoGhirardi, Pearle and Rimini 1990, and Butterfield et al. 1993).Similarly to orthodox collapse quantum mechanics, in the GRW modelsthe quantum-mechanical state of systems (whether it is expressed by avector or a wave function) provides a complete specification of theirintrinsic properties and relations. The state of systems follows theSchrödinger equation, except that it has a probability forspontaneous collapse, independently of whether or not the systems aremeasured. The chance of collapse depends on the ‘size’ ofthe entangled systems—in the earlier models the ‘size’of systems is predicated on the number of the elementary particles,whereas in later models it is measured in terms of mass densities. Inany case, in microscopic systems, such as the particle pairs in theEPR/B experiment, the chance of collapse is very small andnegligible—the chance of spontaneous state collapse in suchsystems is cooked up so that it will occur, on average, every hundredmillion years or so. This means that the chance that the entangledstate of the particle pair in the EPR/B experiment will collapse to aproduct state between the emission from the source and the firstmeasurement is virtually zero. In an earlier L-measurement, the stateof the particle pair gets entangled with the state of theL-measurement apparatus. Thus, the state of the pointer of theL-apparatus evolves from being ‘ready’ to measure acertain spin property to an indefinite outcome. For instance, in az-spin measurement the L-apparatus gets entangled with theparticle pair in a superposition state of pointing to ‘up’and pointing to ‘down’ (corresponding to the states of theL-particle havingz-spin ‘up’ and havingz-spin ‘down’), and the R-apparatus remainsun-entangled with these systems in the state of being ready to measurez-spin. Or formally:

|ψ₆> = 1/√2(|z-up>_L |up>_AL |z-down>_R − |z-down>_L |down>_AL |z-up>_R) |ready>_AR

where, as before, |up>_AL and|down>_AL denote the states of the L-apparatusdisplaying the outcomesz-spin ‘up’ and‘down’ respectively, and |ready>_ARdenotes the state of the R-apparatus being ready to measurez-spin. In this state, a gigantic number of particles of theL-apparatus pointer are entangled together in the superposition stateof being in the position (corresponding to pointing to)‘up’ and the position (corresponding to pointing to)‘down.’ For assuming, for simplicity of presentation, thatthe position of all particles of the L-apparatus pointer in the stateof pointing to ‘up’ (‘down’) is the same, thestate |ψ₆> can be rewritten as:

|ψ₇> = 1/√2(|z-up>_L |up>_p1 |up>_p2 |up>_p3 … |z-down>_R −
                 |z-down>_L |down>_p1 |down>_p2 |down>_p3 … |z-up>_R) |ready>_AR

wherep_i denotes thei-particle of theL-apparatus pointer, and |up>_pi(|down>_pi) is the state of thei-particle being in the position corresponding to the outcomez-spin ‘up’ (‘down’).^[10] The chance that at least one of the vast number of the pointer'sparticles will endure a spontaneous localization towardbeing in the position corresponding to either the outcomez-spin ‘up’ or the outcomez-spin‘down’ within a very short time (a split of a microsecond) is very high. And since all the particles of the pointer andthe particle pair are entangled with each other, such a collapse willcarry with it a collapse of the entangled state of the pointer of theL-apparatus and the particle pair toward either

|z-up>_L|up>_p1|up>_p2|up>_p3 … |z-down>_R

or

|z-down>_L|down>_p1|down>_p2|down>_p3 … |z-up>_R.

Thus, the pointer will very quickly move in the direction of pointingto either the outcomez-spin ‘up’ or the outcomez-spin ‘down.’

If (as portrayed above) the spontaneous localization of particles wereto a precise position, i.e., to the position corresponding to theoutcome ‘up’ or the outcome ‘down,’ the GRWcollapse models would successfully resolve the measurementproblem. Technically speaking, a precise localization is achieved bymultiplying |ψ₇> by a delta function centered on theposition corresponding to either the outcome ‘up’ or theoutcome ‘down’ (see the entry on collapse theories,section 5 and Albert 1992, chapter 5); where the probability of eachof these mutually exhaustive possibilities is ½. The problem isthat it follows from the uncertainty principle (see the entry on theuncertainty principle) that in such localizations the momenta and theenergies of the localized particles would be totally uncertain, sothat gases may spontaneously heat up and electrons may be knocked outof their orbits, in contradiction to our experience. To avoid thiskind of problems, GRW postulated that spontaneous localizations arecharacterized by multiplications by Gaussians that are centered aroundcertain positions, e.g. the position corresponding to either theoutcome ‘up’ or the outcome ‘down’ in thestate |ψ₇>. This may be problematic, because ineither case the state of the L-apparatus pointer at (what wecharacteristically conceive as) the end of the L-measurement would bea superposition of the positions ‘up’ and‘down.’ For although this superposition‘concentrates’ on either the outcome ‘up’ orthe outcome ‘down’ (i.e., the peak of the wave functionthat corresponds to this state concentrates on one of thesepositions), it also has ‘tails’ that go everywhere: Thestate of the L-apparatus is a superposition of an infinite number ofdifferent positions. Thus, it follows from the eigenstate-eigenvaluelink that the position observable of the L-apparatus has no definitevalue at the end of the measurement. But if the position observablehaving a definite value is indeed required in order for theL-apparatus to have a definite location, then the pointer will pointto neither ‘up’ nor ‘down,’ and the GRWcollapse models will fail to reproduce the classical-like behavior ofsuch systems.^[11]

In later models, GRW proposed to interpret the quantum state as adensity of mass and they postulated that if almost all the density ofmass of a system is concentrated in a certain region, then the systemis located in that region. Accordingly, pointers ofmeasurement apparatuses do have definite positions at the end ofmeasurement interactions. Yet, this solution has also given rise to adebate (see Albert and Loewer 1995, Lewis 1997, 2003a, 2004, Ghirardiand Bassi 1999, Bassi and Ghirardi 1999, 2001, Clifton and Monton 1999,2000, Frigg 2003, and Parker 2003).

The exact details of the collapse mechanism and its characteristics inthe GRW/Pearle models have no significant implications for the type ofnon-separability and holism they postulate—all these modelsbasically postulate the same kinds of non-separability and holism asorthodox quantum mechanics (see section 5.1.1). And action at adistance between the L- and the R-wing will occur if the L-measurementinteraction, a supposedly local event in the L-wing, causes some localevents in the R-wing, such as the event of the pointer of themeasurement apparatus coming to possess a definite measurement outcomeduring the R-measurement. That is, action at a distance will occur ifthe L-measurement causes the R-particle to come to possess a definitez-spin and this in turn causes the pointer of the R-apparatusto come to possess the corresponding measurement outcome in theR-measurement. Furthermore, if the L-measurement causes the R-particleto come to possess (momentarily) a definite position in the R-wing,then the action at a distance between the L- and the R-wing will occurindependently of whether the R-particle undergoes a spinmeasurement.

The above discussion is based on an intuitive notion of action at adistance and it presupposes that action at a distance is compatiblewith non-separability and holism. In the next section we shall providemore precise characterizations of action at a distance and in light ofthese characterizations reconsider the question of the nature ofaction at a distance in the GRW/Pearle collapse models.

5.2 Can action-at-a-distance co-exist with non-separability and holism?

The action at a distance in the GRW/Pearle models is different fromthe Newtonian action at a distance in various respects. First, incontrast to Newtonian action at a distance, this action is independentof the distance between the measurement events. Second, whileNewtonian action is symmetric, the action in the GRW/Pearle models is(generally) asymmetric: Either the L-measurement influences theproperties of the R-particle or the R-measurement influences theproperties of the L-particle, depending on which measurement occursfirst (the action will be symmetric when both measurements occursimultaneously). Third (and more important to our consideration), incontrast to Newtonian action at a distance, before the end of theL-measurement the state of the L-apparatus and the R-particle is notseparable and accordingly it is not clear that the influence isbetween separate existences, as the case is supposed to be inNewtonian gravity.

This non-separability of the states of the particle pair and theL-measurement apparatus, and more generally the fact that thenon-locality in collapse theories is due to state non-separability, hasled a number of philosophers and physicists to think that wavecollapses do not involve action at a distance. Yet, the question ofwhether there is an action at a distance in the GRW/Pearle models (andvarious other quantum theories) depends on how we interpret the term‘action at a distance.’ And, as I will suggest below, on anatural reading of Isaac Newton's and Samuel Clarke'scomments concerning action at a distance, there may be a peacefulcoexistence between action at a distance and non-separability andholism.

Newton famously struggled to find out the cause of gravity.^[12] In a letter to Bentley, dated January 17 1692/3, he said:

You sometimes speak of Gravity as essential and inherent to Matter.Pray do not ascribe that Notion to me, for the Cause of Gravity is whatI do not pretend to know, and therefore would take more Time toconsider it. (Cohen 1978, p. 298)

In a subsequent letter to Bentley, dated February 25, 1692/3, headded:

It is inconceivable that inanimate Matter should, without theMediation of something else, which is not material, operate upon, andaffect other matter without mutual Contact…That Gravity shouldbe innate, inherent and essential to Matter, so that one body may actupon another at a distance thro’ a Vacuum, without the Mediationof any thing else, by and through which their Action and Force may beconveyed from one to another, is to me so great an Absurdity that Ibelieve no Man who has in philosophical Matters a competent Faculty ofthinking can ever fall into it. Gravity must be caused by an Agentacting constantly according to certain laws; but whether this Agent bematerial or immaterial, I have left to the Consideration of myreaders. (Cohen 1978, pp. 302-3)

Samuel Clarke, Newton's follower, similarly struggled with thequestion of the cause of gravitational phenomenon. In his famouscontroversy with Leibniz, he said:^[13]

That one body attracts another without any intermediate means, isindeed not a miracle but a contradiction; for 'tis supposingsomething to act where it is not. But the means by which two bodiesattract each other, may be invisible and intangible and of a differentnature from mechanism …

And he added:

That this phenomenon is not producedsans moyen, that iswithout a cause capable of producing such an effect, is undoubtedlytrue. Philosophers therefore can search after and discover that cause,if they can; be it mechanical or not. But if they cannot discover thecause, is therefore the effect itself, the phenomenon, or the matter offact discovered by experience … ever the less true?

Newton's and Clarke's comments suggest that for themgravity was a law-governed phenomenon, i.e., a phenomenon in whichobjects influence each other at a distance according to the Newtonianlaw of gravity, and that this influence is due to some means which maybe invisible and intangible and of a different nature from mechanism.On this conception of action at a distance, there seems to be no reasonto exclude the possibility of action at a distance in the quantum realmeven if that realm is holistic or the state of the relevant systems isnon-separable. That is, action at a distance may be characterized asfollows:

Action at a distance is a phenomenon in which achange in intrinsic properties of one system induces a change in theintrinsic properties of a distant system,independently of theinfluence of any other systems on the distant system, and withoutthere being a process that carries this influence contiguously in spaceand time.

We may alternatively characterize action at a distance in a moreliberal way:

Action* at a distance is a phenomenon in which achange in intrinsic properties of one system induces a change in theintrinsic properties of a distant system without there being a processthat carries this influence contiguously in space and time.

And while Newton and Clarke did not have an explanation for the actionat a distance involved in Newtonian gravity, on the abovecharacterizations action at a distance in the quantum realm would beexplained by the holistic nature of the quantum realm and/ornon-separability of the states of the systems involved. In particular,if in the EPR/B experiment the L-apparatus pointer has a definiteposition before the L-measurement and the R-particle temporarily comesto possess definite position during the L-measurement, then theGRW/Pearle models involve action at a distance and thus also action*at a distance. On the other hand, if the R-particle never comes topossess a definite position during the L-measurement, then theGRW/Pearle models only involve action* at a distance.

5.3 No-collapse theories

5.3.1 Bohm's theory

In 1952, David Bohm proposed a deterministic, ‘hiddenvariables’ quantum theory that reproduces all the observablepredictions of orthodox quantum mechanics (see Bohm 1952, Bohm,Schiller and Tiomno 1955, Bell 1982, Dewdney, Holland and Kyprianidis1987, Dürr, Goldstein and Zanghì 1992a, 1997, Albert 1992,Valentini 1992, Bohm and Hiley 1993, Holland 1993, Cushing 1994, andCushing, Fine and Goldstein 1996; for an entry level review, see theentry on Bohmian mechanics and Albert 1992, chapter 5).

In contrast to orthodox quantum mechanics and the GRW/Pearlecollapse models, in Bohm's theory wave functions always evolveaccording to the Schrödinger equation, and thus they nevercollapse. Wave functions do not represent the states of systems.Rather, they are states of a ‘quantum field (on configurationspace)’ that influences the states of systems.^[14] Also, particles always have definite positions, and the positions ofthe particles and their wave function at a certain time jointlydetermine the trajectories of the particles at all future times. Thus,particles’ positions and their wave function determine theoutcomes of any measurements (so long as these outcomes are recordedin the positions of some physical systems, as in any practicalmeasurements).

There are various versions of Bohm's theory. In the‘minimal’ Bohm theory, formulated by Bell (1982),^[15] the wave function is interpreted as a ‘guiding’ field(which has no source or any dependence on the particles) thatdeterministically governs the trajectories of the particles accordingto the so-called ‘guiding equation’ (which expresses thevelocities of the particles in terms of the wave function).^[16] The states of systems are separable (the state of any compositesystem is completely determined by the state of its subsystems), andthey are completely specified by the particles’positions. Spins, and any other properties which are not directlyderived from positions, are not intrinsic properties ofsystems. Rather, they are relational properties that are determined bythe systems’ positions and the guiding field. In particular,each of the particles in the EPR/B experiment has dispositions to‘spin’ in various directions, and these dispositions arerelational properties of the particles— they are(generally) determined by the guiding field and the positions of theparticles relative to the measurement apparatuses and to eachother.

Figure 4. The EPR/B experiment with Stern-Gerlachmeasurement devices. Stern-Gerlach 1 is on, set up to measure thez-spin of the L-particle, and Stern-Gerlach 2 is off. Thehorizontal lines in the left-hand-side denote the trajectories of sixL-particles in the spin singlet state after an (impulsive)z-spin measurement on the L-particle, and the horizontallines in the right-hand-side denote the trajectories of thecorresponding R-particles. The center plane is aligned orthogonally tothez-axis, so that particles that emerge above this planecorrespond toz-spin ‘up’ outcome and particlesthat emerge below this plane correspond toz-spin‘down’ outcome. The little arrows denote thez-spin components of the particles in the‘non-minimal’ Bohm theory (where spins are intrinsicproperties of particles), and are irrelevant for the‘minimal’ Bohm theory (where spins are not intrinsicproperties of particles).

To see the nature of non-locality postulated by the minimal Bohmtheory, consider again the EPR/B experiment and suppose that themeasurement apparatuses are Stern-Gerlach (S-G) magnets which areprepared to measurez-spin. In any run of the experiment, themeasurement outcomes will depend on the initial positions of theparticles and the order of the measurements. Here is why. In theminimal Bohm theory, the spin singlet state denotes the relevant stateof the guiding field rather than the intrinsic properties of theparticle pair. If the L-measurement occurs before the R-measurement,the guiding field and the position of the L-particle at the emissiontime jointly determine the disposition of the L-particle to emergefrom the S-G device either above or below a plane aligned in thez-direction; where emerging above (below) the plane meansthat the L-particlez-spins ‘up’(‘down’) about thez-axis and the L-apparatus‘pointer’ points to ‘up’ (‘down’)(see Fig. 4 above). All the L-particles that are emitted above thecenter plane aligned orthogonally to thez-direction, likethe L-particles 1-3, will be disposed to spin ‘up’; andall the particles that are emitted below this plane, like theL-particles 4-6, will be disposed to spin ‘down.’Similarly, if the R-measurement occurs before the L-measurement, theguiding field and the position of the R-particle at the emission timejointly determine the disposition of the R-particle to emerge eitherabove thez-axis (i.e., toz-spin ‘up’)or below thez-axis (i.e., toz-spin‘down’) according to whether it is above or below thecenter plane, independently of the position of the L-particle alongthez-axis.

But thez-spin disposition of the R-particle changesimmediately after an (earlier)z-spin measurement on theL-particle: The R-particles 1-3 (see Fig. 4), which were previouslydisposed toz-spin ‘up,’ will now be disposed toz-spin ‘down,’ i.e., to emerge below the centerplane aligned orthogonally to thez-axis; and the R-particles4-6, which were previously disposed toz-spin‘down,’ will now be disposed toz-spin‘up,’ i.e., to emerge above this center plane. Yet, theL-measurementper se does not have any immediate influence onthe state of the R-particle: The L-measurement does not influence theposition of the R-particle or any other property that is directlyderived from this position. It only changes the guiding field, andthus grounds new spin dispositions for the R-particle. But thesedispositions are not intrinsic properties of the R-particle. Rather,they are relational properties of the R-particle, which are groundedin the positions of both particles and the state of the guiding field.^[17] (Note that in the particular case in which the L-particle is emittedabove the center plane aligned orthogonally to thez-axis andthe R-particle is emitted below that plane, an earlierz-spinon the L-particle will have no influence on the outcome of az-spin on the R-particle.)

While there is no contiguous process to carry the influence of theL-measurement outcome on events in the R-wing, the question of whetherthis influence amounts to action at a distance depends on the exactcharacterization of this term. In contrast to the GRW/Pearle collapsemodels, the influence of the L-measurement outcome on the intrinsicproperties of the R-particle is dependent on the R-measurement: Beforethis measurement occurs, there are no changes in the R-particle'sintrinsic properties. Yet, the influence of the L-measurement on theR-particle is at a distance. Thus, the EPR/B experiment as depicted bythe minimal Bohm theory involves action* at a distance but not actionat a distance.

Bohm's theory portrays the quantum realm as deterministic. Thus, thesingle-case objective probabilities, i.e., the chances, it assigns toindividual spin-measurement outcomes in the EPR/B experiment aredifferent from the corresponding quantum-mechanical probabilities. Inparticular, while in quantum mechanics the chances of the outcomes‘up’ and ‘down’ in an earlier L- (R-) spinmeasurement are both ½, in Bohm's theory these chances areeither one or zero. Yet, Bohm's theory postulates a certaindistribution, the so-called ‘quantum-equilibriumdistribution,’ over all the possible positions of pairs with thesame guiding field. This distribution is computed from thequantum-mechanical wave function, and it is typically interpreted asignorance over the actual position of the pair; an ignorance that maybe motivated by dynamical considerations and statistical patternsexhibited by ensembles of pairs with the same wave function (for moredetails, see the entry on bohmian mechanics, section 9). And thesum-average (or more generally the integration) over this distributionreproduces all the quantum-mechanical observable predictions.

What is the status of this probability postulate? Is it a law ofnature or a contingent fact (if it is a fact at all)? The answers tothese questions vary (see Section 7.2.1, Bohm 1953, Valentini 1991a,b,1992, 1996, 2002, Valentini and Westman 2004, Dürr, Goldstein andZanghì 1992a,b, 1996, fn. 15, and Callender 2006).

Turning to the question of non-separability, the minimal Bohm theorydoes not involve state non-separability. For recall that in thistheory the state of a system does not consist in its wave function,but rather in the system's position, and the position of a compositesystem always factorizes into the positions of its subsystems. Here,the non-separability of the wave function reflects the state of theguiding field. This state propagates not in ordinary three-space butin configuration space, where each point specifies the configurationof both particles. The guiding field of the particle pair cannot befactorized into the guiding field that governs the trajectory of theL-particle and the guiding field that governs the trajectory of theR-particle. The evolution of the particles’ trajectories,properties and dispositions is non-separable, and accordingly theparticles’ trajectories, properties and dispositions arecorrelated even when the particles are far away from each other and donot interact with each other. Thus, process separability fails.

In the non-minimal Bohm theory^[18], the behavior of anN-particle system is determined by itswave function and the intrinsic properties of the particles. But, incontrast to the minimal theory, in the non-minimal theory spins areintrinsic properties of particles. The wave function always evolvesaccording to the Schrödinger equation, and it is interpreted as a‘quantum field’ (which has no sources or any dependence onthe particles). The quantum field guides the particles via the‘quantum potential,’ an entity which is determined fromthe quantum field, and the evolution of properties is fully deterministic.^[19]

Like in the minimal Bohm theory, the non-separability of the wavefunction in the EPR/B experiment dictates that the evolution of theparticles’ trajectories, properties and dispositions isnon-separable, but the behavior of the particles is somewhat different.In the earlierz-spin measurement on the L-particle, thequantum potential continuously changes, and this change induces animmediate change in thez-spin of the R-particle. If theL-particle starts to spin ‘up’ (‘down’) in thez-direction, the R-particle will start to spin‘down’ (‘up’) in the same direction (see thelittle arrows in Fig. 4).^[20] Accordingly, the L-measurement induces instantaneous action at adistance between the L- and the R-wing. Yet, similarly to the minimalBohm theory, while the disposition of the R-particle to emerge aboveor below the center plane aligned orthogonally to thez-direction in az-spin measurement may changeinstantaneously, the actual trajectory of the R-particle along thez-direction does not change before the measurement of theR-particle'sz-spin occurs. Only during the R-measurement,the spin and the position of the R-particle get correlated and theR-particle's trajectory along thez-direction is dictated bythe value of its (intrinsic)z-spin.

Various objections have been raised against Bohm's theory (for adetailed list and replies, see the entry on Bohmian mechanics, section15). One main objection is that in Bohmian mechanics, the guidingfield influences the particles, but the particles do not influence theguiding field. Another common objection is that the theory is involvedwith a radical type of non-locality, and that this type ofnon-locality is incompatible with relativity. While it may be verydifficult, or even impossible, to reconcile Bohm's theory withrelativity, as is not difficult to see from the above discussion, thetype of non-locality that the minimal Bohm theory postulates in theEPR/B experiment does not seem more radical than the non-localitypostulated by the orthodox interpretation and the GRW/Pearle collapsemodels.

5.3.2 Modal interpretations

Modal interpretations of quantum mechanics were designed to solve themeasurement problem and to reconcile quantum mechanics withrelativity. They are no-collapse, (typically) indeterministichidden-variables theories. Quantum-mechanical states of systems(which may be construed as denoting their states or information aboutthese states) always evolve according to unitary and linear dynamicalequations (the Schrödinger equation in the non-relativisticcase). And the orthodox quantum-mechanical state description ofsystems is supplemented by a set of properties, which depends on thequantum-mechanical state and which is supposed to be rich enough toaccount for the occurrence of definite macroscopic events and theirclassical-like behavior, but sufficiently restricted to escape all theknown no-hidden-variables theorems. (For modal interpretations, seevan Fraassen 1973, 1981, 1991, chapter 9, Kochen 1985, Krips 1987,Dieks 1988, 1989, Healey 1989, Bub 1992, 1994, 1997, Vermaas and Dieks1995, Clifton 1995, Bacciagaluppi 1996, Bacciagaluppi and Hemmo 1996,Bub and Clifton 1996, Hemmo 1996b, Bacciagaluppi and Dickson 1999,Clifton 2000, Spekkens and Sipe 2001a,b, Bene and Dieks 2002, andBerkovitz and Hemmo 2006a,b. For an entry-level review, see the entryon modal interpretations of quantum theory. For comprehensive reviewsand analyses of modal interpretations, see Bacciagaluppi 1996, Hemmo1996a, chapters 1-3, Dieks and Vermaas 1998, Vermaas 1999, and theentry on modal interpretations of quantum theory. For theno-hidden-variables theorems, see Kochen and Specker 1967,Greenberger, Horne and Zeilinger 1989, Mermin 1990 and the entry onthe Kochen-Specker theorem.)^[21]

Modal interpretations vary in their property assignment. Forsimplicity, we shall focus on modal interpretations in which theproperty assignment is based on the so-called Schmidtbiorthogonal-decomposition theorem (see Kochen 1985, Dieks 1989, andHealey 1989). LetS₁ andS₂ besystems associated with the Hilbert spacesH^S1 andH^S2, respectively. There exist bases{|α_i>} and{|β_i>} forH^S1 andH^S2 respectively such that the state ofS₁+S₂ can be expressed as alinear combination of the following form of vectors from thesebases:

|ψ₈>_S1+S2 =∑_i c_i|α_i>_S1 |β_i>_S2.

When the absolute values of the coefficientsc_iare all unequal, the bases{|α_i>} and{|β_i>} and the abovedecomposition of |ψ₈>_S1+S2 areunique. In that case, it is postulated thatS₁ hasa determinate value for each observable associated withH^S1 with the basis{|α_i>} andS₂ has a determinate value for each observableassociated withH^S2 with the basis{|β_i>}, and |c_i|²provide the (ignorance)probabilities of the possible values that these observables may have.^[22] For example, suppose that the state of the L- and the R-particle inthe EPR/B experiment before the measurements is:

|ψ₉> = (1/√2+ε) |z-up>_L|z-down>_R − (1/√2-ε′) |z-down>_L|z-up>_R

where 1/√2 >> ε,ε′,(1/√2+ε)²+(1/√2-ε′)²= 1, and (as before) |z-up>_L(|z-up>_R) and |z-down>_L (|z-down>_R) denote the states of the L- (R-)particle havingz-spin ‘up’ andz-spin ‘down’, respectively.^[23] Then, either the L-particle spins ‘up’ and the R-particlespins ‘down’ in thez-direction, or theL-particle spins ‘down’ and the R-particle spins‘up’ in thez-direction. Thus, in contrast to theorthodox interpretation and the GRW/Pearle collapse models, in modalinterpretations the particles in the EPR/B experiment may havedefinite spin properties even before any measurement occurs.

To see how the modal interpretation accounts for the curiouscorrelations in EPR/B-type experiments, let us suppose that the stateof the particle pair and the measurement apparatuses at the emissiontime is:

|ψ₁₀> = ((1/√2+ε)|z-up>_L|z-down>_R −(1/√2−ε′)|z-down>_L|z-up>_R)|ready>_AL|ready>_AR

where |ready>_AL(|ready>_AR) denotes the state of theL-apparatus (R-apparatus) being ready to measurez-spin. Inthis state, the L- and the R-apparatus are in the definite state ofbeing ready to measurez-spin, and (similarly to the state|ψ₉>) the L- and the R-particle have definitez-spin properties: Either the L-particle hasz-spin‘up’ and the R-particle hasz-spin‘down,’ or the L-particle hasz-spin‘down’ and the R-particle hasz-spin ‘up,’^[24] where the probability of the realization of each of thesepossibilities is approximately 1/2. In the (earlier)z-spinmeasurement on the L-particle, the state of the particle pair and theapparatuses evolves to the state:

|ψ₁₁> = ((1/√2+ε) |z-up>_L|up>_AL|z-down>_R − (1/√2-ε′) |z-down>_L|down>_AL|z-up>_R) |ready>_AR

where (as before) |up>_AL and|down>_AL denote the states of the L-apparatuspointing to the outcomesz-spin ‘up’ andz-spin ‘down’, respectively. In this state,either the L-particle has az-spin ‘up’ and theL-apparatus points to ‘up,’ or the L-particle hasz-spin ‘down’ and the L-apparatus points to‘down.’ And, again, the probability of each of thesepossibilities is approximately 1/2. The evolution of the propertiesfrom the state |ψ₁₀> to the state|ψ₁₁> depends on the dynamical laws. In almost allmodal interpretations, if the particles have definitez-spinproperties before the measurements, the outcomes ofz-spinmeasurements will reflect these properties. That is, the evolution ofthe properties of the particles and the measurement apparatuses willbe deterministic, so that the spin properties of the particles do notchange in the L-measurement and the pointer of the L-apparatus comesto display the outcome that corresponds to thez-spinproperty that the L-particle had before the measurement. If, forexample, before the measurements the L- and the R-particle haverespectively the propertiesz-spin ‘up’ andz-spin ‘down’, the (earlier)z-spinmeasurement on the L-particle will yield the outcome ‘up’and the spin properties of the particles will remainunchanged. Accordingly, az-spin measurement on theR-particle will yield the outcome ‘down’. Thus, in thiscase the modal interpretation involves neither action at a distancenor action* at a distance.

However, if the measurement apparatuses are set up to measurex-spin rather thanz-spin, the evolution of theproperties of the L-particle and the L-apparatus will beindeterministic. As before, the L-measurement will not cause any changein the actual spin properties of the R-particle. But the L-measurementoutcome will cause an instant change in the spin dispositions of theR-particle and the R-measurement apparatus. If, for example, theL-measurement outcome isx-spin ‘up’ and theL-particle comes to possesx-spin ‘up,’ then theR-particle and the R-apparatus will have respectivelythe dispositions to possessx-spin ‘down’ and todisplay the outcomex-spin ‘down’ on ax-spin measurement. Thus, like the minimal Bohm theory, themodal interpretation may involve action* at a distance in the EPR/Bexperiment. But, unlike the minimal Bohm theory, here spins areintrinsic properties of particles.

In the above modal interpretation, property composition fails: Theproperties of composite systems are not decomposable into theproperties of their subsystems. Consider, again, the state|ψ₁₀>. As ‘separated’ systems (i.e., inthe decompositions of the composite system of the particlepair+apparatuses into the L-particle and the R-particle+apparatusesand into the R-particle and the L-particle+apparatuses) the L- and theR-particle have definitez-spin properties. But, assubsystems of the composite system of the particle pair (e.g. in thedecomposition of the composite system of the particle pair+apparatusesinto the particle pair and the apparatuses), they have no definitez-spin properties.

A failure of property composition occurs also in the state |ψ₁₁>, where the L- and the R-particle havedefinitez-spin properties both as ‘separated’systems and as subsystems of the particle pair (though in contrast with |ψ₁₀>, in |ψ₁₁> the rangeof the possible properties of the particles as separated systems andas subsystems of the pair is the same). For nothing in the above propertyassignment implies that in |ψ₁₁> the spinproperties that the L-particle has as a ‘separated’ systemand the spin properties that it has as a subsystem of the particle pairbe the same: The L-particle may havez-spin ‘up’as a separated system andz-spin ‘down’ as asubsystem of the particle pair.

Furthermore, the dynamics of the properties that the L-particle(R-particle) has as a separated system and the dynamics of itsproperties as a subsystem of the particle pair are generally different.^[25] Consider, again, the state |ψ₁₀>. In the (earlier)z-spin measurement on the L-particle, the spin propertiesthat the L-particle has as a separated system follow a deterministicevolution — the L-particle has eitherz-spin‘up’ orz-spin ‘down’ before andafter the L-measurement; whereas as a subsystem of the particle pair,the spin properties of the L-particle follow an indeterministicevolution — the L-particle has no definite spin propertiesbefore the L-measurement and eitherz-spin ‘up’(with approximately chance ½) orz-spin‘down’ (with approximately chance ½) after theL-measurement.

The failure of property composition implies that the quantum realm asdepicted by the above version of the modal interpretation involvesstate non-separability and property and relational holism. Stateseparability fails because the state of the particle pair is notgenerally determined by the separate states of the particles. Indeed,as is easily shown, the actual properties that the L- and theR-particle each has in the state |ψ₉> are alsocompatible with product states in which the L- and the R-particle arenot entangled. Property and relational holism fail because in thestate |ψ₉> the properties of the pair do notsupervene upon the properties of its subsystems and the spatiotemporalrelations between them. Furthermore, process separability fails forsimilar reasons.

The failure of property composition in the modal interpretationcalls for explanation. It may be tempting to postulate that theproperties that a system (e.g. the L-particle) has, as a separatedsystem, are the same as the properties that it has as a subsystem ofcomposite systems. But, as Bacciagaluppi (1995) and Clifton (1996a)have shown, such property assignment will be inconsistent: It will besubject to a Kochen and Specker-type contradiction. Furthermore, asVermaas (1997) demonstrates, the properties of composite systems andthe properties of their subsystems cannot be correlated (in wayscompatible with the Born rule).

For what follows in the rest of this subsection, the views ofdifferent authors differ widely. Several variants of modalinterpretations were developed in order to fix the problem of thefailure of property composition. The most natural explanation of thefailure of property composition is that quantum states assignrelational rather than intrinsic properties to systems (see Kochen1985, Bene and Dieks 2002, and Berkovitz and Hemmo 2006a,b). Forexample, in the relational modal interpretation proposed by Berkovitzand Hemmo (2006a,b), the main idea is that quantum states assignproperties to systems only relative to other systems, and propertiesof a system that are related to different systems are generallydifferent. In particular, in the state |ψ₁₀> theL-particle has a definitez-spin property relative to theR-particle, the measurement apparatuses and the rest of the universe,but (as a subsystem of the particle pair) it has no definitez-spin relative to the measurement apparatuses and the restof universe.^[26] On this interpretation, the properties of systems are highlynon-local by their very nature. Properties like pointing to‘up’ and pointing to ‘down’ are not intrinsicto the measurement apparatuses. Rather, they are relations betweenthe apparatuses and other systems. For example, the property of theL-apparatus pointing to ‘up’ relative to the particlepair, the R-apparatus and the rest of the universe is not intrinsic tothe L-apparatus; it is a relation between the L-apparatus and theparticle pair, the R-apparatus and the rest of the universe. As such,this property is highly non-local: It is located in neither the L-wingnor any other subregion of the universe. Yet, due to the dynamicallaws, properties like the position of pointers of measurementapparatuses, which appear to us to be local, behave like localproperties in any experimental circumstances, and accordingly thisradical type of non-locality is unobservable (for more details, seeBerkovitz and Hemmo 2006b, sections 8.1 and 9).

Another way to try to explain the failure of property composition isto interpret the properties of composite systems as holistic,non-decomposable properties. On this interpretation, thez-spin ‘up’ property that the L-particle has as asubsystem of the particle pair in the state |ψ₉> iscompletely different from thez-spin ‘up’property that the L-particle has as a separated system, and the use ofthe term ‘z-spin up’ in both cases is misleading(for more details, see Berkovitz and Hemmo 2006a).^[27]

The relational and holistic interpretations of properties mark aradical shift from the standard interpretation of properties inorthodox quantum mechanics. Other advocates of the modal interpretationhave chosen not to follow this interpretation, and opted for a modalinterpretation that does not violate property composition. While theproperty assignment above does not assume any preferred partition ofthe universe (the partition of the universe into a particle pair andthe rest of the universe is as good as the partition of the universeinto the L-particle and the rest of the universe), proponents ofproperty composition postulated that there is a preferred partition ofthe universe into ‘atomic’ systems and accordingly apreferred factorization of the Hilbert space of the universe. Thispreferred factorization is supposed to be the basis for the‘core’ property assignment: Properties are prescribed toatomic systems according to a property assignment that is ageneralization of the bi-orthogonal decomposition property assignment.^[28] And the properties of complex systems are postulated to becompositions of the properties of their atomic systems (see the entryon modal interpretations of quantum theory, section 2, andBacciagaluppi and Dickson 1999). The challenge for this atomic modalinterpretation is to justify the assumption that there is a preferredpartition of the universe, and to provide some idea about how suchfactorization should look like.

Finally, while the modal interpretation was designed to solve themeasurement problem and reconcile quantum mechanics with specialrelativity, it faces challenges on both accounts. First, in certainimprefect measurements (where there are imprefections in the couplingbetween the measured system and the pointer of the measurementapparatus and/or the pointer and the environment), modalinterpretations that are based on the Schmidtbiorthogonal-decomposition theorem (and more generally the spectraldecomposition theorem) fail to account for definite measurementoutcomes, in contradiction to our experience (see Bacciagaluppi andHemmo 1996 and Bacciagaluppi 2000). For versions of the modalinterpretations that seem to escape this problem, see Van Fraassen(1973, 1991), Bub (1992, 1997), Bene and Dieks (2002) and Berkovitzand Hemmo (2006a,b). Second, as we shall see in section 10.2, a numberof no-go theorems challenge the view that modal interpretations couldbe genuinely relativistic.

5.3.3 Everett-like interpretations

In 1957, Everett proposed a new no-collapse interpretation of orthodoxquantum mechanics (see Everett 1957a,b, 1973, Barrett 1999, the entryon Everett's relative-state formulation of quantum mechanics, theentry on the many-worlds interpretation of quantum mechanics, andreferences therein). The Everett interpretation is a no-collapseinterpretation of quantum mechanics, where the evolution of quantumstates is always according to unitary and linear dynamical equations(the Schrödinger equation in the non-relativistic case). In thisinterpretation, quantum states are fundamentally relative. Systems haverelative states, which are derivable from the various branches of theentangled states. For example, consider again |ψ₁₁>.

|ψ₁₁> = (1/√2+ε) |z-up>_L|up>_AL|z-down>_R |ready>_AR− (1/√2-ε′) |z-down>_L|down>_AL|z-up>_R |ready>_AR.

In this quantum-mechanical state, the L-apparatus is in the state ofpointing to the outcomez-spin ‘up’relative to the L-particle being in the statez-spin‘up,’ the R-particle being in the statez-spin‘down’ and the R-apparatus being ready to measurez-spin; and in the state of pointing to the outcomez-spin ‘down’relative to the L-particlebeing in the statez-spin ‘down,’ the R-particlebeing in the statez-spin ‘up’ and theR-apparatus being ready to measurez-spin. Likewise, theL-particle is in the statez-spin ‘up’ relativeto the L-apparatus being in the state of pointing to the outcomez-spin ‘up,’ the R-particle being in the statez-spin ‘down’ and the R-apparatus being ready tomeasurez-spin; and in the statez-spin‘down’ relative to the L-apparatus being in the state ofpointing to the outcomez-spin ‘down,’ theR-particle being in the statez-spin ‘up’ and theR-apparatus being ready to measurez-spin. And similarly,mutatis mutandis, for the relative state of the R-particleand the R-apparatus.

Everett's original formulation left the exact meaning of theserelative states and their relations to observers’ experience andbeliefs open, and there have been different Everett-likeinterpretations of these states. Probably the most popular reading ofEverett is the splitting-worlds interpretation (see DeWitt 1971,Everett's relative-state formulation of quantum mechanics, Barrett1999, and references therein). In the splitting-worlds interpretation,each of the branches of the state |ψ₁₁> refers to adifferent class of worlds (all of which are real) where the states ofthe L-apparatus, R-apparatus and the particles are all separable:Class-1 worlds in which the L-particle is in the statez-spin‘up,’ the R-particle is in the statez-spin‘down,’ the L-apparatus is in the state of pointing to theoutcomez-spin ‘up’ and the R-apparatus in thestate of being ready to measurez-spin; and class-2 worlds inwhich the L-particle is in the statez-spin‘down,’ the R-particle is in the statez-spin‘up,’ the L-apparatus is in the state of pointing to theoutcomez-spin ‘down’ and the R-apparatus is inthe state of being ready to measurez-spin. More generally,each term in state of the universe, as represented in a certainpreferred basis, reflects the states of its systems in some class ofworlds; where the range of the different classes of worlds increaseswhenever the number of the terms in the quantum state (in thepreferred basis) increases (this process is called‘splitting’).

The splitting-worlds reading of Everett faces a number of challenges.First, supporters of the Everett interpretation frequently motivatetheir interpretation by arguing that it postulates the existence ofneither a controversial wave collapse nor hidden variables, and itleaves the simple and elegant mathematical structure of quantummechanics intact. But, the splitting-worlds interpretation adds extrastructure to no-collapse orthodox quantum mechanics. Further, thisinterpretation marks a radical shift from orthodox quantummechanics. A scientific theory is not constituted only by itsmathematical formalism, but also by the ontology it postulates, theway it depicts the physical realm and the way it accounts for ourexperience. The many parallel worlds ontology of the splitting-worldsinterpretation and its account of our experience are radicallydifferent from the ontology of the intended interpretation of orthodoxquantum mechanics and its account for our experience. Second, relativestates are well defined in any basis, and the question arises as towhich basis should be preferred and the motivation for selecting oneparticular basis over others. Third, in the splitting-worldsinterpretation each of the worlds in the universe may split into twoor more worlds, and the problem is that (similarly to the collapse inorthodox collapse quantum mechanics) there are no clear criteria forwhen a splitting occurs and how long it takes. Fourth, there is thequestion of how the splitting-worlds interpretation accounts for thestatistical predictions of the orthodox theory. In the Everett-likeinterpretations in general, and in the splitting-worlds interpretationin particular, all the possible measurement outcomes in the EPR/Bexperiment are realized and may be observed. Thus, the questionarises as to the meaning of probabilities in this interpretation. Forexample, what is the meaning of the statement that in the state|ψ₁₀> (see section 5.3.2) the probability of theL-measurement apparatus pointing to the outcome ‘up’ in anearlierz-spin measurement on the L-particle is(approximately) ½? In the splitting-worlds interpretation theprobability of that outcome appears to be 1! Furthermore, settingaside the problem of interpretation, there is also the question ofwhether the splitting-worlds interpretation, and more generallyEverett-like interpretations, can account for the particular values ofthe quantum probabilities of measurement outcomes. Everett claimed toderive the Born probabilities in the context of hisinterpretation. But this derivation has been controversial. (Fordiscussions of the meaning of probabilities, or more precisely themeaning of the coefficients of the various terms in quantum states, inEverett-like interpretations, see Butterfield 1996, Lockwood 1996a,b,Saunders 1998, Vaidman 1998, Barnum et al. 2000, Bacciagaluppi 2002,Gill 2003, Hemmo and Pitowsky 2003, 2005, Wallace 2002, 2003, 2005a,b,Greaves 2004 and Saunders 2004, 2005.)

Other readings of Everett include the many-minds interpretation(Albert and Loewer 1988, Barrett 1999, chapter 7), theconsistent-histories approach (Gell-Mann and Hartle 1990), theEverett-like relational interpretation (Saunders 1995, Mermin 1998)and (what may be called) the many-structures interpretation (Wallace2005c). While these readings address more or less successfully theproblems of the preferred basis and splitting, except for themany-minds interpretation of Albert and Loewer the question of whetherthere could be a satisfactory interpretation of probabilities in thecontext of these theories and the adequacy of the derivation of theBorn probabilities are still a controversial issue (see Deutsch 1999,Wallace 2002, 2003, Lewis 2003, Graves 2004, Saunders 2004, Hemmo andPitowsky 2005, and Price 2006).

What kind of non-locality do Everett-like interpretations involve?Unfortunately, the answer to this question is not straightforward, asit depends on one's particular reading of the Everettinterpretation. Indeed, all the above readings of Everett seem to treatthe no-collapse wave function of the universe as a real physical entitythat reflects the non-separable state of the universe, and accordinglythey involve state non-separability. But, one may reasonably expectthat different readings depict different pictures of physical realityand accordingly might postulate different kinds of non-locality. Thus,any further analysis of the type of non-locality postulated by each ofthese readings requires a detailed study of their ontology (which weplan to conduct in future updates of this entry).

For example, the question of action at a distance in the EPR/Bexperiment may arise in the context of the splitting-worlds interpretation,but not in the context of Albert and Loewer's many-mindsinterpretation. Albert and Loewer's interpretation takes the bareno-collapse orthodox quantum mechanics to be the complete theory of thephysical realm. Accordingly, the L-apparatus in the state |ψ₁₁> does not display any definite outcome. Yet,in order to account for our experience of a classical-like world, whereat the end of measurements observers are typically in mental states ofperceiving definite outcomes, the many-minds interpretation appeals toa dualism of mind-body. Each observer is associated with a continuousinfinity of non-physical minds. And while the physical state of theworld evolves in a completely deterministic manner according to theSchrödinger evolution, and the pointers of the measurementapparatuses in the EPR/B experiment display no definite outcomes,states of minds evolve in a genuinely indeterministic fashion so as toyield an experience of perceiving definite measurement outcomes. Forexample, consider again, the state |ψ₁₀>. Whilein a firstz-spin L-measurement, this state evolvesdeterministically into the state |ψ₁₁>, minds ofobservers evolve indeterministically into either the state ofperceiving the outcomez-spin ‘up’ or the state ofperceiving the outcomez-spin ‘down’ with theusual Born-rule probabilities (approximately 50% chance for each ofthese outcomes). Since in this state the L-particle has no definitespin properties and the L-apparatus points to no definite measurementoutcome, and since in the laterz-spin measurement on theR-particle the R-particle does not come to possess any definite spinproperties and the R-apparatus points to no definite spin outcome, thequestion of whether there is action at a distance between the L-particle and theL-apparatus on the one hand and the R-particle and the R-apparatus onthe other does not arise.

6. Superluminal causation

In all the above interpretations of quantum mechanics, the failure offactorizability (i.e., the failure of the joint probability of themeasurement outcomes in the EPR/B experiment to factorize into theirsingle probabilities) involves non-separability, holism and/or sometype of action at a distance. As we shall see below,non-factorizability also implies superluminal causal dependenceaccording to certain accounts of causation.

First, as is not difficult to show, the failure of factorizabilityimplies superluminal causation according to various probabilisticaccounts of causation that satisfy Reichenbach's (1956, section 19)principle of the common cause (for a review of this principle, see theentry on Reichenbach's principle of the common cause).

Here is why. Reichenbach's principle may be formulated as follows:

PCC (Principle of the Common Cause). For anycorrelation between two (distinct) events which do not cause eachother, there is a common cause that screens them off from each other.Or formally: If distinct eventsx andy arecorrelated, i.e.,

(Correlation)
P(x &y) ≠P(x) ·P(y),

and they do not cause each other, then their common cause,CC(x,y), screens them off from each other, i.e.,

(Screening Off)

PCC(x,y)(x/y) =PCC(x,y)(x)	PCC(x,y)(y) ≠ 0
PCC(x,y)(y/x) =PCC(x,y)(y)	PCC(x,y)(x) ≠ 0.[29]

Accordingly,CC(x,y) rendersx andy probabilistically independent, and the joint probability ofx andy factorizes uponCC(x,y):

P_CC(x,y)(x &y) =P_CC(x,y)(x) ·P_CC(x,y)(y).

The above formulation of PCC is mainly intended to cover cases inwhichx andy have no partial, non-common causes. ButPCC can be generalized as follows:

PCC*. The joint probability of any distinct,correlated events,x andy, which are not causallyconnected to each other, factorizes upon the union of their partial(separate) causes and their common cause. That is, letCC(x,y) denote the common causes ofx andy, andPC(x) andPC(y) denote respectively their partial causes. Then,the joint probability ofx andyfactorizesupon the Union of their Causal Pasts (henceforth, FactorUCP),i.e., on the union ofPC(x),PC(y)andCC(x,y):

FactorUCP
P_{PC(x)PC(y)CC(x,y)} (x &y) =P_PC(x)CC(x,y) (x) ·P_PC(y)CC(x,y) (y).

Like PCC, the basic idea of FactorUCP is that the objectiveprobabilities of events that do not cause each other are determined bytheir causal pasts, and given these causal pasts they areprobabilistically independent of each other. As is not difficult tosee, factorizability is a special case of FactorUCP. That is, to obtainfactorizability from FactorUCP, substitute λ forCC(x,y),l forPC(x) andr forPC(y). FactorUCP and the assumption that theprobabilities of the measurement-outcomes in the EPR/B experiment aredetermined by the pair's state and the settings of themeasurement apparatuses jointly imply factorizability. Thus, given thislater assumption, the failure of factorizability implies superluminalcausation between the distant outcomes in the EPR/B experimentaccording to any account of causation that satisfies FactorUCP (forsome examples of such accounts, see Butterfield 1989 and Berkovitz1995a, 1995b, section 6.7, 1998b).^[30]

Superluminal causation between the distant outcomes also existsaccording to various counterfactual accounts of causation, includingaccounts that do not satisfy FactorUCP. In particular, in Lewis's(1986) influential account, counterfactual dependence between distinctevents implies causal dependence between them. And as Butterfield(1992b) and Berkovitz (1998b) demonstrate, the violation ofFactorizability involves a counterfactual dependence between thedistant measurement outcomes in the EPR/B experiment.

But the violation of factorizability does not imply superluminalcausation according to some other accounts of causation. Inparticular, in process accounts of causation there is no superluminalcausation in the EPR/B experiment. In such accounts, causal dependencebetween events is explicated in terms of continuous processes in spaceand time that transmit ‘marks’ or conserved quantitiesfrom the cause to the effect (see Salmon 1998, chapters 1, 12, 16 and18, Dowe 2000, the entry on causal processes, and referencestherein). Thus, recalling (sections 1, 2, 4 and 5) that none of theinterpretations of quantum mechanics and alternative quantum theoriespostulates any (direct) continuous process between the distantmeasurement events in the EPR/B experiment, there is no superluminalcausation between them according to process accounts of causation.

7. Superluminal signaling

Whether or not the non-locality predicted by quantum theories may beclassified as action at a distance or superluminal causation, thequestion arises as to whether this non-locality could be exploited toallow superluminal (i.e., faster-than-light) signaling of information.This question is of particular importance for those who interpretrelativity as prohibiting any such superluminal signaling. (We shallreturn to discuss this interpretation in section 10.)

Superluminal signaling would require that the state of nearbycontrollable physical objects (say, a keyboard in my computer)superluminally influence distant observable physical phenomena (e.g. apattern on a computer screen light years away). The influence may bedeterministic or indeterministic, but in any case it should cause adetectable change in the statistics of some distant physicalquantities.

It is commonly agreed that in quantum phenomena, superluminalsignaling is impossible in practice. Moreover, many believe that suchsignaling is excluded in principle by the so-called ‘no-signalingtheorem’ (for proofs of this theorem, see Eberhard 1978,Ghirardi, Rimini and Weber 1980, Jordan 1983, Shimony 1984, Redhead1987, pp. 113-116 and 118). It is thus frequently claimed with respectto EPR/B experiments that there is no such thing as a Bell telephone,namely a telephone that could exploit the violation of the Bellinequalities for superluminal signaling of information.^[31]

The no-signaling theorem demonstrates that orthodox quantummechanics excludes any possibility of superluminal signaling in theEPR/B experiment. According to this theory, no controllable physicalfactor in the L-wing, such as the setting of the L-measurementapparatus, can take advantage of the entanglement between the systemsin the L- and the R-wing to influence the statistics of the measurementoutcomes (or any other observable) in the R-wing. As we have seen insection 5.1.1, the orthodox theory is at best incomplete. Thus, thefact that it excludes superluminal signaling does not imply that otherquantum theories or interpretations of the orthodox theory also excludesuch signaling. Yet, if the orthodox theory is empirically adequate, asthe consensus has it, its statistical predictions obtain, andaccordingly superluminal signaling will be excluded as a matter offact; for if this theory is empirically adequate, any quantum theorywill have to reproduce its statistics, including the exclusion of anyactual superluminal signaling.

But the no-signaling theorem does not demonstrate that superluminalsignaling would be impossible if orthodox quantum mechanics were notempirically adequate. Furthermore, this theorem does not show thatsuperluminal signaling is in principle impossible in the quantum realmas depicted by other theories, whichactually reproduce thestatistics of orthodox quantum mechanics but do not prohibit in theorythe violation of this statistics. In sections 7.2-7.3, we shallconsider the in-principle possibility of superluminal signaling incertain collapse and no-collapse interpretations of quantum mechanics.But, first, we need to consider the necessary and sufficient conditionsfor superluminal signaling.

7.1 Necessary and sufficient conditions for superluminal signaling

To simplify things, in our discussion we shall focus onnon-factorizable models of the EPR/B experiment that satisfyλ-independence (i.e., the assumption that the distribution ofthe states λ is independent of the settings of the measurementapparatuses). Superluminal signaling in the EPR/B experiment would bepossible in theory just in case the value of some controllablephysical quantity in the nearby wing could influence the statistics ofmeasurement outcomes in the distant wing. And in non-factorizablemodels that satisfy λ-independence this could happen just incase the following conditions obtained:

Controllable probabilistic dependence. Theprobabilities of distant measurement outcomes depend on some nearbycontrollable physical quantity.

λ-distribution. There can be in theory an ensemble of particle pairs the states of which deviate from the quantum-equilibrium distribution; where the quantum-equilibrium distribution of pairs' states is the distribution that reproduces the predictions of orthodox quantum mechanics.

Four comments: (i) In controllable probabilistic dependence, the term'probabilities of measurement outcomes' refers to the modelprobabilities, i.e., the probabilities that the states λprescribe for measurement outcomes.
(ii) Our discussion in this entry focuses on models of the EPR/Bexperiment in which probabilities of measurement outcomes depend onlyon the pair's state λ and the settings of the measurementapparatuses to measure certain properties. In such models, parameterdependence (i.e., the dependence of the probability of the distantmeasurement outcome on the setting of the nearby measurementapparatus) is a necessary and sufficient condition for controllableprobabilistic dependence. But, recall (footnote 3) that in some modelsof the EPR/B experiment, in addition to the pair's state and thesetting of the L- (R-) measurement apparatus there are other localphysical quantities that may be relevant for the probability of the L-(R-) measurement outcome. In such models, parameter dependence is nota necessary condition for controllable probabilistic dependence. Someother physical quantities in the nearby wing may be relevant for theprobability of the distant measurement outcome. (That is, let αand β denote all the relevant local physical quantities, otherthan the settings of the measurement apparatuses, that may be relevantfor the probability of the L- and the R-outcome, respectively. Then,controllable probabilistic dependence would obtain if for some pairs'states λ, L-settingl, R-settingr and localphysical quantities α and β,P_{λl r α β}(y_r) ≠P_{λl rβ}(y_r) obtained.) For the relevanceof such models to the question of the in-principle possibility ofsuperluminal signalling in some current interpretations of quantummechanics, see sections 7.3 and 7.4.
(iii) The quantum-equilibrium distribution will not be the same in allmodels of the EPR/B experiment; for in general the states λwill not be the same in different models.
(iv) In models that actually violate both controllable probabilisticdependence and λ-distribution, the occurrence of controllableprobabilistic dependence would render the actual distribution ofλ states as non-equilbrium distribution. Thus, if controllableprobabilistic dependence occurred in such models, the actualdistribution of λ states would satisfyλ-distribution.

The argument for the necessity of controllable probabilisticdependence and λ-distribution is straightforward. Grantedλ-independence, if the probabilistic dependence of the distantoutcome on a nearby physical quantity is not controllable, there canbe no way to manipulate the statistics of the distant outcome so as todeviate from the statistical predictions of quantummechanics. Accordingly, superluminal transmission of information willbe impossible even in theory. And if λ-distribution does nothold, i.e., if the quantum-equilibrium distribution holds, controllableprobabilistic dependence will be of no use for superluminaltransmission of information. For, averaging over the modelprobabilities according to the quantum-equilbrium distribution, themodel will reproduce the statistics of orthodox quantummechanics. That is, the distribution of the λ-states will besuch that the probabilistic dependence of the distant outcome on thenearby controllable factor will be washed out: In some states thenearby controllable factor will raise the probability of the distantoutcome and in others it will decrease this probability, so that onaverage the overall statistics of the distant outcome will beindependent of the nearby controllable factor (i.e., the same as thestatistics of orthodox quantum mechanics). Accordingly, superluminalsignaling will be impossible.

The argument for the sufficiency of these conditions is alsostraightforward. If λ-distribution held, it would be possiblein theory to arrange ensembles of particle pairs in which controllableprobabilistic dependence would not be washed out, and accordingly thestatistics of distant outcomes would depend on the nearby controllablefactor. (For a proof that these conditions are sufficient forsuperluminal signaling in certain deterministic hidden variablestheories, see Valentini 2002.)

Note that the necessary and sufficient conditions for superluminalsignaling are different in models that do not exclude in theory theviolation of λ-independence. In such models controllableprobabilistic dependence is not a necessary condition for superluminalsignaling. The reasoning is as follows. Consider any empiricallyadequate model of the EPR/B experiment in which the pair's state andthe settings of the measurement apparatuses are the only relevantfactors for the probabilities of measurement outcomes, and thequantum-equilibrium distribution is λ-independent. In such amodel, parameter independence implies the failure of controllableprobabilistic dependence, yet the violation of λ-independencewould imply the possibility of superluminal signaling: Ifλ-independence failed, a change in the setting of the nearbymeasurement apparatus would cause a change in the distribution of thestates λ, and a change in this distribution would induce achange in the statistics of the distant (space-like separated)measurement outcome.

Leaving aside models that violate λ-independence, we now turnto consider the prospects of controllable probabilistic dependence andλ-distribution, starting with no-collapse interpretations.

7.2 No-collapse theories

7.2.1 Bohm's Theory

Bohm's theory involves parameter dependence and thus controllableprobabilistic dependence: The probabilities of distant outcomes dependon the setting of the nearby apparatus. In some pairs' statesλ, i.e., in some configurations of the positions of the particlepair, a change in the apparatus setting of the (earlier) sayL-measurement will induce an immediate change in the probability oftheR-outcome: e.g. the probability ofR-outcomez-spin ‘up’ will be 1 if the L-apparatus is setto measurez-spin and 0 if the L-apparatus is switched off(see section 5.3.1). Thus, the question of superluminal signalingturns on whether λ-distribution obtains.

Now, recall (section 5.3.1) that Bohm's theory reproduces the quantumstatistics by postulating the quantum-equilibrium distribution overthe positions of particles. If this distribution is not an accidentalfact about our universe, but rather obtains as a matter of law,superluminal signaling will be impossible in principle. Dürr,Goldstein and Zanghì (1992a,b, 1996, fn. 15) argue that, whilethe quantum-equilibrium distribution is not a matter a law, otherdistributions will be possible but atypical. Thus, they conclude thatalthough superluminal signaling is not impossible in theory, it mayoccur only in atypical worlds. On the other hand, Valentini (1991a,b,1992, 1996, 2002) and Valentini and Westman 2004) argue that there aregood reasons to think that our universe may well have started off in astate of quantum non-equilibrium and is now approaching gradually astate of equilibrium, so that even today some residual non-equilibriummust be present.^[32] Yet, even if such residual non-equilbrium existed, the question iswhether it would be possible to access any ensemble of systems in anon-equilbrium distribution.

7.2.2 Modal interpretations

The presence or absence of parameter independence (and accordingly thepresence or absence of controllable probabilistic dependence) in themodal interprtation is a matter of controversy, perhaps due in part tothe multiplicity of versions of this interpretation. Whether or notmodal interpretations involve parameter dependence would probablydepend on the dynamics of the possessed properties. At least some ofthe current modal interpretations seem to involve no parameterdependence. But, as the subject editor pointed out to the author, somethink that the no-go theorem for relativistic modal interpretation dueto Dickson and Clifton (1998) implies the existence of parameterdependence in all the interpretations to which this theorem isapplicable. Do modal interpretations satisfy λ-distribution?The prospects of this condition depend on whether the possessedproperties that the modal interpretation assigns in addition to theproperties prescribed by the orthodox interpretation, arecontrollable. If these properties were controllable at least intheory, λ-distribution would be possible. For example, if thepossessed spin properties that the particles have at the emission fromthe source in the EPR/B experiment were controllable, thenλ-distribution would be possible. The common view seems to bethat these properties are uncontrollable.

7.3 Collapse theories

7.3.1 Dynamical models for state-vector reduction

In the GRW/Pearle collapse models, wave functions represent the mostexhaustive, complete specification of states of individualsystems. Thus, pairs prepared with the same wave function have alwaysthe same λ state — a state that represents theirquantum-equilbrium distribution for the EPR/B experiment. Accordingly,λ-distribution fails. Do these models involve controllableprobabilistic dependence?

Recall (section 5.1.2) that there are several models of statereduction in the literature. One of these models is the so-callednon-linear Continuous Stochastic Localization (CSL) models (see Pearle1989, Ghirardi, Pearle and Rimini 1990, Butterfield et al. 1993, andGhirardi et al. 1993). Butterfield et al. (1993) argue that in thesemodels there is a probabilistic dependence of the outcome of theR-measurement on the process that leads to the (earlier) outcome ofthe L-measurement. In these models, the process leading to theL-outcome (eitherz-spin ‘up’ orz-spin‘down’) depends on the interaction between the L-particleand the L-apparatus (which results in an entangled state), and thespecific realization of the stochastic process that strives tocollapse this macroscopic superposition into a product state in whichthe L-apparatus displays a definite outcome. And the probability ofthe R-outcome depends on this process. For example, if this process isone that gives rise to az-spin ‘up’ (or rendersthat outcome more likely), the probability of R-outcomez-spin ‘up’ is 0 (more likely to be 0); and ifthis process is one that gives rise to az-spin‘down’ (or renders that outcome more likely), theprobability of R-outcomez-spin ‘down’ is 0 (morelikely to be 0). The question is whether there are controllablefactors that influence the probability of realizations of stochasticprocesses that lead to a specific L-outcome, so that it would bepossible to increase or decrease the probability of the R-outcome. Ifsuch factors existed, controllable probabilistic dependence would bepossible at least in theory. And if this kind of controllableprobabilistic dependence existed, λ-distribution would alsoobtain; for if such dependence existed, the actual distribution ofpairs' states (in which the pair always have the same state, thequantum-mechanical state) would cease to be the quantum-equilbriumdistribution.

7.4 The prospects ofcontrollable probabilistic dependence

In section 7.3.1, we discussed the question of the in-principlecontrollability of local measurement processes and in particular theprobability of their outcome, and the implications of suchcontrollability for the in-principle possibility of superluminalsignaling in the context of the CSL models. But this question is notspecific to the CSL model and (more generally) the dynamical modelsfor state-vector reduction. It seems likely to arise also in otherquantum theories that model measurements realistically. Here iswhy. Real measurements take time. And during that time, some physicalvariable, other than the state of the measured system and the settingof the measurement apparatus, might influence the chance (i.e., thesingle-case objective probability) of the measurement outcome. Inparticular, during the L-measurement in the EPR/B experiment, thechance of the L-outcomez-spin ‘up’(‘down’) might depend on the value of some physicalvariable in the L-wing, other than the state of the particle pair andthe setting of the L-measurement apparatus. If so, it will follow fromthe familiar perfect anti-correlation of the singlet state that thechance of R-outcomez-spin ‘up’(‘down’) will depend on the value of such variable (fordetails, see Kronz 1990a,b, Jones and Clifton 1993, pp. 304-305, andBerkovitz 1998a, section 4.3.4). Thus, if the value of such a variablewere controllable, controllable probabilistic dependence wouldobtain.

7.5 Superluminal signaling and action-at-a-distance

If superluminal signaling were possible in the EPR/B experiment in anyof the above theories, it would not require any continuous process inspacetime to mediate the influences between the two distantwings. Indeed, in all the current quantum theories in which theprobability of the R-outcome depends on some controllable physicalvariable in the L-wing, this dependence is not due to a continuousprocess. Rather, it is due to some type of ‘action’ or (touse Shimony's (1984) terminology) ‘passion’ at a distance,which is the ‘result’ of the holistic nature of thequantum realm, the non-separability of the state of entangled systems,or the non-separable nature of the evolution of the properties ofsystems.

8. The analysis of factorizability: implications for quantum non-locality

In sections 5-7, we considered the nature of quantum non-locality asdepicted by theories that violate factorizability, i.e., the assumptionthat the probability of joint measurement outcomes factorizes into thesingle probabilities of these outcomes. Recalling section 3,factorizability can be analyzed into a conjunction of two conditions:OI (outcome independence)—the probability of a distantmeasurement outcome in the EPR/B experiment is independent of thenearby measurement outcome; and PI (parameter independence)—theprobability of a distant measurement outcome in the EPR/B experimentis independent of the setting of the nearby measurementapparatus. Bohm's theory violates PI, whereas other mainstream quantumtheories satisfy this condition but violate OI. The question arises asto whether violations of PI involve a different kind of non-localitythan violations of OI. So far, our methodology was to study thenature of quantum non-locality by analyzing the way various quantumtheories account for the curious correlations in the EPR/Bexperiment. In this section, we shall focus on the question of whetherquantum non-locality can be studied in a more general way, namely byanalyzing the types of non-locality involved in violations of PI andin violations of OI, independently of how these violations arerealized.

8.1 Non-separability, holism and action at a distance

It is frequently argued or maintained that violations of OI involvestate non-separability and/or some type of holism, whereas violationsof PI involve action at a distance. For notable examples, Howard(1989) argues that spatiotemporal separability (see section 4.3)implies OI, and accordingly a violation of it implies spatiotemporalnon-separability; Teller (1989) argues that particularism (see section4.3) implies OI, and thus a violation of it implies relational holism;and Jarrett (1984, 1989) argues that a violation of PI involves sometype of action at a distance. These views are controversial,however.

First, as we have seen in section 5, in quantum theories theviolation of either of these conditions involves some type ofnon-separability and/or holism.

Second, the explicit attempts to derive OI from separability orparticularism seem to rely (implicitly) on some locality conditions.Maudlin (1998, p. 98) and Berkovitz (1998a, section 6.1) argue thatHoward's precise formulation of spatiotemporal separability embodiesboth separability and locality conditions, and Berkovitz (1998a,section 6.2) argues that Teller's derivation of OI from particularismimplicitly relies on locality conditions. Thus, the violation of OIper se does not imply non-separability or holism.

Third, a factorizable model, i.e., model that satisfies OI, may benon-separable (Berkovitz 1995b, section 6.5). Thus, OI cannot be simplyidentified with separability.

Fourth, Howard's spatiotemporal separability condition (see section4.3) requires that states of composite systems be determined by thestates of their subsystems. In particular, spatiotemporal separabilityrequires that joint probabilities of outcomes be determined as somefunction of the single probabilities of these outcomes. Winsberg andFine (2003) object that as a separability condition, OI arbitrarilyrestricts this function to be a product function. And they argue thaton a weakened formalization of separability, a violation of OI iscompatible with separability. Fogel (2004) agrees that Winsberg andFine's weakened formalization of separability is correct, but arguesthat, when supplemented by a certain ‘isotropy’ condition,OI implies this weakened separability condition. Fogel believes thathis suggested ‘isotropy’ condition is very plausible, but,as he acknowledges, this condition involves a nontrivial measurement context-independence.^[33]

Fifth, as the analysis in section 5 demonstrates, violations of OImight involve action at a distance. Also, while the minimal Bohmtheory violates PI and arguably some modal interpretations do not, thetype of action at a distance they postulate, namely action* at adistance (see section 5.2), is similar: In both cases, an earlierspin-measurement in (say) the L-wing does not induce any immediatechange in the intrinsic properties of the R-particle. TheL-measurement only causes an immediate change in the dispositions ofthe R-particle—a change that may influence the behavior of theR-particle in future spin-measurements in the R-wing. But, this changeof dispositions does not involve any change of local properties in theR-wing, as these dispositions are relational (rather than intrinsic)properties of the R-particle. Furthermore, the action at a distancepredicated by the minimal Bohm theory is weaker than the onepredicated by orthodox collapse quantum mechanics and the GRW/Pearlecollapse models; for in contrast to the minimal Bohm theory, in thesetheories the measurement on the L-particle induces a change in theintrinsic properties of the R-particle, independently of whether ornot the R-particle undergoes a measurement. Thus, if the R-particlecomes to possess (momentarily) a definite position, the EPR/Bexperiment as described by these theories involves action at adistance — a stronger kind of action than the action* at adistance predicated by the minimal Bohm theory.

8.2 Superluminal signaling

It was also argued, notably by Jarrett 1984 and 1989 and Shimony 1984,that in contrast to violations of OI, violations of PI may give rise(at least in principle) to superluminal signaling. Indeed, as is notdifficult to see from section 7.1, in theories that satisfyλ-independence there is an asymmetry between failures of PI andfailures of OI with respect to superluminal signaling: whereasλ-distribution and the failure of PI are sufficient conditionsfor the in-principle possibility of superluminal signaling,λ-distribution and the failure of OI are not. Thus, theprospects of superluminal signaling look better in parameter-dependenttheories, i.e., theories that violate PI. Yet, as we have seen insection 7.2.1, if the Bohmian quantum-equilbrium distribution obtains,then Bohm's theory, the paradigm of parameter dependent theories,prohibits superluminal signaling. And if this distribution is obtainedas a matter of law, then Bohm's theory prohibits superluminalsignaling even in theory. Furthermore, as we remarked in section 7.1,if the in-principle possibility of violating λ-independence isnot excluded, superluminal signaling may exist in theories thatsatisfy PI and violate OI. In fact, as section 7.4 seems to suggest,the possibility of superluminal signaling in theories that satisfy PIbut violate OI cannot be discounted even when λ-independence isimpossible.

8.3 Relativity

Jarrett (1984, 1989), Ballentine and Jarrett (1997) and Shimony (1984)hold that superluminal signaling is incompatible with relativitytheory. Accordingly, they conclude that violations of PI areincompatible with relativity theory, whereas violations of OI may becompatible with this theory. Furthermore, Sutherland (1985, 1989)argues that deterministic, relativistic parameter-dependent theories(i.e., relativistic, deterministic theories that violate PI) wouldplausibly require retro-causal influences, and in certain experimentalcircumstances this type of influences would give rise to causalparadoxes, i.e., inconsistent closed causal loops (where effectsundermine their very causes). And Arntzenius (1994) argues that allrelativistic parameter-dependent theories are impossible on pain ofcausal paradoxes. That is, he argues that in certain experimentalcircumstances any relativistic, parameter-dependent theory would giverise to closed causal loops in which violations of PI could notobtain.

It is noteworthy that the view that relativityper se isincompatible with superluminal signaling is disputable (for moredetails, see section 10). Anyway, recalling (section 8.2), ifλ-distribution is excluded as a matter of law, it will beimpossible even in theory to exploit the violation of PI to give riseto superluminal signaling, in which case the possibility ofrelativistic parameter-dependent theories could not be discounted onthe basis of superluminal signaling.

Furthermore, as mentioned in section 7.1 and 7.4, the in-principlepossibility of superluminal signaling in theories that satisfy PI andviolate OI cannot be excludeda priori. Thus, if relativitytheory excludes superluminal signaling, the argument from superluminalsignaling may also be applied to exclude the possibility of somerelativistic outcome-dependent theories.

Finally, Berkovitz (2002) argues that Arntzenius's argument for theimpossibility of relativistic theories that violate PI is based onassumptions about probabilities that are common in linear causalsituations but are unwarranted in causal loops, and that the realchallenge for these theories is that in such loops their predictivepower is undermined (for more details, see section 10.3).

8.4 Superluminal causation

In various counterfactual and probabilistic accounts of causationviolations of PI entail superluminal causation between the setting ofthe nearby measurement apparatus and the distant measurement outcome,whereas violations of OI entail superluminal causation between thedistant measurement outcomes (see Butterfield 1992b, 1994, Berkovitz1998b, section 2). Thus, it seems that theories that violate PIpostulate a different type of superluminal causation than theoriesthat violate OI. Yet, as Berkovitz (1998b, section 2.4) argues, theviolation of PI in Bohm's theorydoes involve some type ofoutcome dependence, which may be interpreted as a generalization ofthe violation of OI. In this theory, the specific R-measurementoutcome in the EPR/B experiment depends on the specific L-measurementoutcome: For any three different directionsx,y,z, if the probabilities ofx-spin ‘up’andy-spin ‘up’ are non-zero, the probability ofR-outcomez-spin ‘up’ will generally depend onwhether the L-outcome isx-spin ‘up’ ory-spin ‘up’. Yet, due to the determinism thatBohm's theory postulates, OI trivially obtains. Put it another way, OIdoes not reflect all the types of outcome independence that may existbetween distant outcomes. Accordingly, the fact that a theorysatisfies OI does not entail that it does not involve some other typeof outcome dependence. Indeed, in all the current quantum theoriesthat violate factorizability there are correlations between distantspecific measurement outcomes — correlations that may well beinterpreted as an indication of counterfactual superluminal causationbetween these outcomes.

8.5 On the origin and nature of parameter dependence

Parameter dependence (PI) postulates that in the EPR/B experiment theprobability of the later, distant measurement outcome depends on thesetting of the apparatus of the nearby, earlier measurement. It may betempting to assume that this dependence is due to a direct influenceof the nearby setting on the (probability of the) distant outcome. Buta little reflection on the failure of PI in Bohm's theory, which isthe paradigm for parameter dependence, demonstrates that the settingof the nearby apparatusper se has no influence on thedistant measurement outcome. Rather, it is because the setting of thenearby measurmenent apparatus influences the nearby measurementoutcome and the nearby outcome influences the distant outcome that thesetting of the nearby apparatus can have an influence on the distantoutcome. For, as is not difficult to see from the analysis of thenature of non-locality in the minimal Bohm theory (see section 5.3.1),the setting of the apparatus of the nearby (earlier) measurement inthe EPR/B experiment influences the outcome the nearby measurement,and this outcome influences the guiding field of the distant particleand accordingly the outcome of a measurement on that particle.

While the influence of the nearby setting on the nearby outcome isnecessary for parameter dependence, it is not sufficient for it. Inall the current quantum theories, the probabilities of joint outcomesin the EPR/B experiment depend on the settings of both measurementapparatuses: The probability that the L-outcome isl-spin‘up’ and the R-outcome isr-spin ‘up’and the probability that the L-outcome isl-spin‘up’ and the R-outcome isr-spin‘down’ both depend on (l −r),i.e., the distance between the anglesl andr. Intheories in which the sum of these joint probabilities is invariantwith respect to the value of (l −r),parameter independence obtains: for all pairs' states λ,L-settingl, and R-settingsr andr′,L-outcomex_l, and R-outcomesy_r andy_r′ :

(PI)
P_λ l r(x_l &y_r) +P_λ l r(x_l &¬y_r) =P_{λ l r′} (x_l &y_r′ ) +P_{λ l r′} (x_l & ¬y_r′ ).

Parameter dependence is a violation of this invariance condition.

9. Can there be ‘local’ quantum theories?

The focus of this entry has been on exploring the nature of thenon-local influences in the quantum realm as depicted by quantumtheories that violate factorizability, i.e., theories in which thejoint probability of the distant outcomes in the EPR/B experiment donot factorize into the product of the single probabilities of theseoutcomes. The motivation for this focus was that, granted plausibleassumptions, factorizability must fail (see section 2), and itsfailure implies some type of non-locality (see sections 2-8). But ifany of these plausible assumptions failed, it may be possible toaccount for the EPR/B experiment (and more generally for all otherquantum phenomena) without postulating any non-local influences. Letus then consider the main arguments for the view that quantumphenomena need not involve non-locality.

In arguments for the failure of factorizability, it is presupposedthat the distant measurement outcomes in the EPR/B experiment are realphysical events. Recall (section 5.3.3) that in Albert and Loewer's(1988) many-minds interpretation this is not the case. In thisinterpretation, definite measurement outcomes are (typically) notphysical events. In particular, the pointers of the measurementapparatuses in the EPR/B experiment do not display any definiteoutcomes. Measurement outcomes in the EPR/B experiment exist only as(non-physical) mental states in observers' minds (which arepostulated to be non-physical entities). So sacrificing some of ourmost fundamental presuppositions about the physical reality andassuming a controversial mind-body dualism, the many-mindsinterpretation of quantum mechanics does not postulate any action at adistance or superluminal causation between the distant wings of theEPR/B experiment. Yet, as quantum-mechanical states of systems areassumed to reflect their physical states, the many-minds theory doespostulate some type of non-locality, namely state non-separability andproperty and relational holism.

Another way to get around Bell's argument for non-locality in theEPR/B experiment is to construct a model of this experiment thatsatisfies factorizability but violates λ-independence (i.e.,the assumption that the distribution of all the possible pairs' statesin the EPR/B experiment is independent of the measured quantities). Insection 2, we mentioned two possible causal explanations for thefailure of λ-independence. The first is to postulate thatpairs' states and apparatus settings share a common cause, whichcorrelates certain types of pairs' states with certain types ofsettings (e.g. states of type λ₁ are correlated withsettings of typel andr, whereas states of typeλ₂ are correlated with settings of typel′ andr′, etc.). As we noted, thinkingabout all the various ways one can measure properties, thisexplanation seems conspiratorial. Furthermore, it runs counter to oneof the most fundamental presuppositions of empirical science, namelythat in experiments preparations of sources and settings ofmeasurement apparatuses are typically independent of each other. Thesecond possible explanation is to postulate causation from themeasurement events backward to the source at the emission time. (Foradvocates of this way out of non-locality, see Costa de Beauregard1977, 1979, 1985, Sutherland 1983, 1998, 2006 and Price 1984, 1994,1996, chapters 3, 8 and 9.) Maudlin (1994, p. 197-201) argues thattheories that postulate such causal mechanism areinconsistent. Berkovitz (2002, section 5) argues that Maudlin's lineof reasoning is based on unwarranted premises. Yet, as we shall see insection 10.3, this way out of non-locality faces somechallenges. Furthermore, while a violation of λ-independenceprovides a way out of Bell's theorem, it does not necessarily implylocality; for the violation of λ-independence is compatiblewith the failure of factorizability.

A third way around non-locality is to ‘exploit’ theinefficiency of measurement devices or (more generally) measurementset-ups. In any actual EPR/B experiment, many of the particle pairsemitted from the source fail to be detected, so that only a sample ofthe particle pairs is observed. Assuming that the observed samples arenot biased, it is now generally agreed that the statisticalpredictions of orthodox quantum mechanics have been vindicated (for areview of these experiments, see Redhead 1987, section 4.5). But ifthis assumption is abandoned, there are perfectly local causalexplanations for the actual experimental results (Clauser and Horne1974, Fine 1982b, 1989a). Many believe that this way out ofnon-locality isad hoc, at least in light of our currentknowledge. Moreover, this strategy would fail if the efficiency ofmeasurement devices exceeded a certain threshold (for more details,see Fine 1989a, Maudlin 1994, chapter 6, Larsson and Semitecolos 2000and Larsson 2002).

Finally, there are those who question the assumption thatfactorizability is a locality condition (Fine 1981, 1986, pp. 59-60,1989b, Cartwright 1989, chaps. 3 and 6, Chang and Cartwright 1993).Accordingly, they deny that non-factorizability implies non-locality.The main thrust of this line of reasoning is that the principle of thecommon cause is not generally valid. Some, notably Cartwright (1989)and Chang and Cartwright (1993), challenge the assumption that commoncauses always screen off the correlation between their effects, andaccordingly they question the idea that non-factorizability impliesnon-locality. Others, notably Fine, deny that correlations must havecausal explanation.

While these arguments challenge the view that the quantum realm asdepicted by non-factorizable models for the EPR/B experimentmust involve non-locality, they do not show that viablelocal, non-factorizable models of the EPR/B experiment (i.e., viablemodels which do not postulate any non-locality) are possible. Indeed,so far none of the attempts to construct local, non-factorisablemodels for EPR/B experiments has been successful.

10. Can quantum non-locality be reconciled with relativity?

The question of the compatibility of quantum mechanics with thespecial theory of relativity is very difficult to resolve. (Thequestion of the compatibility of quantum mechanics with the generaltheory of relativity is even more involved.) The answer to thisquestion depends on the interpretation of special relativity and thenature of the exact constraints it imposes on influences betweenevents.

A popular view has it that special relativity prohibits anysuperluminal influences, whereas theories that violate factorizabilityseem to involve such influences. Accordingly, it is held that quantummechanics is incompatible with relativity. Another common view has itthat special relativity prohibits only certain types of superluminalinfluence. Many believe that relativity prohibits superluminalsignaling of information. Some also believe that this theory prohibitssuperluminal transport of matter-energy and/oraction-at-a-distance. On the other hand, there is the view thatrelativityper se prohibits only superluminal influences thatare incompatible with the special-relativistic space-time, theso-called ‘Minkowski space-time,’ and that thisprohibition is compatible with certain types of superluminalinfluences and superluminal signaling (for a comprehensive discussionof this issue, see Maudlin, 1994, 1996, section 2).^[34]

It is commonly agreed that relativity requires that the descriptionsof physical reality (i.e., the states of systems, their properties,dynamical laws, etc.) in different coordinate systems should becompatible with each other. In particular, descriptions of the state ofsystems in different foliations of spacetime into parallel spacelikehyperplanes, which correspond to different inertial reference frames,are to be related to each other by the Lorentz transformations. If thisrequirement is to reflect the structure of the Minkowski spacetime,these transformations must hold at the level of individual processes,and not only at the level of ensembles of processes (i.e., at thestatistical level) or observed phenomena. Indeed, Bohm's theory,which is manifestly non-relativistic, satisfies the requirement thatthe Lorentz transformations obtain at the level of the observedphenomena.

However, satisfying the Lorentz transformations at the level ofindividual processes is not sufficient for compatibility withMinkowski spacetime; for the Lorentz transformations may also besatisfied at the level of individual processes in theories thatpostulate a preferred inertial reference frame (Bell 1976). Maudlin(1996, section 2) suggests that a theory is genuinely relativistic(both in spirit and letter) if it can be formulated without ascribingto spacetime any more, or different intrinsic structure than therelativistic metrics.^[35]The question of the compatibility ofrelativity with quantum mechanics may be presented as follows: Could aquantum theory that does not encounter the measurement problem berelativistic in that sense?

10.1 Collapse theories

The main problem in reconciling collapse theories with specialrelativity is that it seems very difficult to make state collapse(modeled as a real physical process) compatible with the structure ofthe Minkowski spacetime. In non-relativistic quantum mechanics, theearlier L-measurement in the EPR/B experiment induces a collapse of theentangled state of the particle pair and the L-measurement apparatus.Assuming (for the sake of simplicity) that measurement events occurinstantaneously, state collapse occurs along a single spacelikehyperplane that intersects the spacetime region of the L-measurementevent—the hyperplane that represents the (absolute) time of thecollapse. But this type of collapse dynamics would involve a preferredfoliation of spacetime, in violation of the spirit, if not the letterof the Minkowski spacetime.

The current dynamical collapse models are not genuinelyrelativistic, and attempts to generalize them to the specialrelativistic domain have encountered difficulties (see, forexample, the entry on collapse theories, Ghirardi 1996, Pearle 1996,and references therein). A more recent attempt to address thesedifficulties due to Tumulka (2004) seems more promising.

In an attempt to reconcile state collapse with special relativity,Fleming (1989, 1992, 1996) and Fleming and Bennett (1989) suggestedradical hyperplane dependence. In their theory, state collapse occursalong an infinite number of spacelike hyperplanes that intersect thespacetime region of the measurements. That is, in the EPR/B experimenta collapse occurs along all the hyperplanes of simultaneity thatintersect the spacetime region of the L-measurement. Similarly, acollapse occurs along all the hyperplanes of simultaneity thatintersect the distant (space-like separated) spacetime region of theR-measurement. Accordingly, the hyperplane-dependent theory does notpick out any reference frame as preferred, and the dynamics of thequantum states of systems and their properties can be reconciled withthe Minkowski spacetime. Further, since all the multiple collapses aresupposed to be real (Fleming 1992, p. 109), the predictions oforthodox quantum mechanics are reproduced in each referenceframe.

The hyperplane-dependent theory is genuinely relativistic. But thetheory does not offer any mechanism for state collapses, and it doesnot explain how the multiple collapses are related to each other andhow our experience is accounted for in light of this multiplicity.

Myrvold (2002b) argues that state collapses can be reconciled withMinkowski spacetime even without postulating multiple differentcollapses corresponding to different reference frames. That is, heargues with respect to the EPR/B experiment that the collapses inducedby the L- and the R-measurement are local events in the L- and theR-wing respectively, and that the supposedly different collapses(corresponding to different reference frames) postulated by thehyperplane-dependent theory are only different descriptions of thesame local collapse events. Focusing on the state of the particlepair, the main idea is that the collapse event in the L-wing ismodeled by a (one parameter) family of operators (the identityoperator before the L-measurement and a projection to the collapsedstate after the L-measurement), and it is local in the sense that itis a projection on the Hilbert space of the L-particle; and similarly,mutatis mutandis, for the R-particle. Yet, if the quantumstate of the particle pair represents their complete state (as thecase is in the orthodox theory and the GRW/Pearle collapse models),these collapse events seem non-local. While the collapse in the L-wingmay be said to be local in the above technical sense, it is bydefinition a change of local as well as distant (spacelike)properties. The operator that models the collapse in the L-wingtransforms the entangled state of the particle pair—a state inwhich the particles have no definite spins—into a product ofnon-entangled states in which both particles have definite spins, andaccordingly it causes a change of intrinsic properties in both the L-and the R-wing.

In any case, Myrvold's proposal demonstrates that even if statecollapses are not hyperplane dependent, they need not be incompatiblewith relativity theory.

10.2 No-collapse theories

Recall (section 5.3) that in no-collapse theories, quantum-mechanicalstates always evolve according to a unitary and linear equation ofmotion (the Schrödinger equation in the non-relativistic case),and accordingly they never collapse. Since the wave function has acovariant dynamics, the question of the compatibility with relativityturns on the dynamics of the additional properties —theso-called ‘hidden variables’— that no-collapsetheories typically postulate. In Albert and Loewer's many-minds theory(see section 5.3.3), the wave function has covariant dynamics, and noadditional physical properties are postulated. Accordingly, the theoryis genuinely relativistic. Yet, as the compatibility with relativityis achieved at the cost of postulating that outcomes of measurements(and, typically, any other perceived properties) are mental ratherthan physical properties, many find this way of reconciling quantummechanics with relativity unsatisfactory.

Other Everett-like interpretations attempt to reconcile quantummechanics with the special theory of relativity without postulatingsuch a controversial mind-body dualism. Similarly to the many-mindsinterpretation of Albert and Loewer, and contrary to Bohm's theory andmodal interpretations, on the face of it these interpretations do notpostulate the existence of ‘hidden variables.’ But(recalling section 5.3.3) these Everett-like interpretations face thechallenge of making sense of our experience and the probabilities ofoutcomes, and critics of these interpretations argue that thischallenge cannot be met without adding some extra structure to theEverett interpretation (see Albert and Loewer 1988, Albert 1992,pp. 114-5, Albert and Loewer 1996, Price 1996, pp. 226-227, andBarrett 1999, pp. 163-173); a structure that may render theseinterpretations incompatible with relativity. Supporters of theEverett interpretation disagree. Recently, Deutsch (1999), Wallace(2002, 2003, 2005a,b) and Greaves (2004) have suggested thatEverettians can make sense of the quantum-mechanical probabilities byappealing to decision-theoretical considerations. But this line ofreasoning has been disputed (see Barnumet al. 2000, Lewis2003b, Hemmo and Pitowsky 2005 and Price 2006).

Modal interpretations constitute another class of no-collapseinterpretations of quantum mechanics that were developed to reconcilequantum mechanics with relativity (and to solve the measurementproblem). Yet, as the no-go theorems by Dickson and Clifton (1998),Arntzenius (1998) and Myrvold (2002) demonstrate, the earlier versionsof the modal interpretation are not genuinely compatible withrelativity theory. Further, Earman and Ruetsche (2005) argue that aquantum-field version of the modal interpretation (which is set in thecontext of relativistic quantum-field theory), like the one proposedby Clifton (2000), would be subject to serious challenges. Berkovitzand Hemmo (2006a,b) develop a relational modal interpretation thatescapes all the above no-go theorems and to that extent seems toprovide better prospects for reconciling quantum mechanics withspecial relativity.

10.3 Quantum causal loops and relativity

Recall (section 8) that many believe that parameter-dependent theories(i.e., theories that violate parameter independence) are more difficultor even impossible to reconcile with relativity. Recall also that oneof the lines of argument for the impossibility of relativisticparameter-dependent theories is that such theories would give rise tocausal paradoxes. In our discussion, we focused on EPR/B experimentsin which the measurements are distant (spacelike separated). In arelativistic parameter-dependent theory, the setting of the nearbymeasurement apparatus in the EPR/B experiment would influence theprobability of the distant (spacelike separated) measurementoutcome. Sutherland (1985, 1989) argues that it is plausible tosuppose that the realization of parameter dependence would be the samein EPR/B experiments in which the measurements are not distant fromeach other (i.e., when the measurements are timelike separated). If so,relativistic parameter-dependent theories would involve backwardcausal influences. But, he argues, in deterministic, relativisticparameter-dependent theories these influences would give rise tocausal paradoxes, i.e., inconsistent closed causal loops.

Furthermore, Arntzenius (1994) argues that all relativisticparameter-dependent theories are impossible on pain of causalparadoxes. In his argument, he considers the probabilities ofmeasurement outcomes in a setup in which two EPR/B experiments arecausally connected to each other, so that the L-measurement outcome ofthe first EPR/B experiment determines the setting of the L-apparatusof the second EPR/B experiment and the R-measurement outcome of thesecond EPR/B experiment determines the setting of the R-apparatus ofthe first EPR/B experiment. And he argues that in this experiment,relativistic parameter-dependent theories (deterministic orindeterministic) would give rise to closed causal loops in whichparameter dependence would be impossible. Thus, he concludes thatrelativistic, parameter-dependent theories are impossible. (Stairs(1989) anticipates the argument that the above experimental setup maygive rise to causal paradoxes in relativistic, parameter-dependenttheories, but he stops short of arguing that such theories areimpossible.)

Berkovitz (1998b, section 3.2, 2002, section 4) argues thatArntzenius's line of reasoning fails because it is based on untenableassumptions about the nature of probabilities in closed causalloops—assumptions that are very natural in linear causalsituations (where effects do not cause their causes), but untenable incausal loops. (For an analysis of the nature of probabilities incausal loops, see Berkovitz 2001 and 2002, section 2.) Thus, heconcludes that the consistency of relativistic parameter-dependenttheories cannot be excluded on the grounds of causal paradoxes. Healso argues that the real challenge for relativisticparameter-dependent theories is concerned with their predictivepower. In the causal loops predicted by relativisticparameter-dependent theories in Arntzenius's suggested experiment,there is no known way to compute the frequency of events from theprobabilities that the theories prescribe. Accordingly, such theorieswould fail to predict any definite statistics of measurement outcomesfor that experiment. This lack of predictability may also present somenew opportunities. Due to this unpredictability, there may be anempirical way for arbitrating between these theories and quantumtheories that do not predicate the existence such causal loops inArntzenius's experiment.

Another attempt to demonstrate the impossibility of certainrelativistic quantum theories on the grounds of causal paradoxes isadvanced by Maudlin (1994, pp. 195-201). (Maudlin does not present hisargument in these terms, but the argument is in effect based on suchgrounds.) Recall (sections 2 and 9) that a way to try to reconcilequantum mechanics with relativity is to account for the curiouscorrelations between distant systems by local backward influencesrather than non-local influences. In particular, one may postulatethat the correlations between the distant measurement outcomes in theEPR/B experiment are due to local influences from the measurementevents backward to the state of the particle pair at the source. Insuch models of the EPR/B experiment, influences on events are alwaysconfined to events that occur in their past or future light cones, andno non-locality is postulated. Maudlin argues that theories thatpostulate such backward causation will be inconsistent. Moreparticularly, he argues that a plausible reading of Cramer's (1980,1986) transactional interpretation, and any other theory thatsimilarly attempts to account for the EPR/B correlations bypostulating causation from the measurement events backward to thesource, will be inconsistent.

Berkovitz (2002, section 5) argues that Maudlin's argument is, ineffect, that if such retro-causal theories were true, they wouldinvolve closed causal loops in which the probabilities of outcomesthat these theories assign will certainly deviate from the statisticsof these outcomes. And, similarly to Arntzenius's argument, Maudlin'sargument also rests on untenable assumptions about the nature ofprobabilities in causal loops (for a further discussion of Maudlin'sand Berkovitz's arguments and, more generally, the prospects ofCramer's theory, see Kastner 2004). Furthermore, Berkovitz (2002,sections 2 and 5.4) argues that, similarly to relativisticparameter-dependent theories, the main challenge for theories thatpostulate retro-causality is not causal paradoxes, but rather the factthat their predictive power may be undermined. That is, theprobabilities assigned by such theories may fail to predict thefrequency of events in the loops they predicate. In particular, thelocal retro-causal theories that Maudlin considers fail to assign anydefinite predictions for the frequency of measurement outcomes incertain experiments. Yet, some other theories that predicate theexistence of causal loops, such as Sutherland's (2006) localtime-symmetric Bohmian interpretation of quantum mechanics, seem notto suffer from this problem.

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• Donald, M. (1999),Progress in a many-minds interpretation of quantum theory.
• Larsson, J-A and Semitecolos, J. (2000),Strictdetector-efficiency bounds forn-site Clauser-Horneinequalities.
• Larsson, J. A. (2002),A Kochen-Specker inequality.
• Lewis, P. J. (2003b),Deutsch on Quantum Decision Theory.
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• Wallace, D. (2002),Quantum probability and decision theory, revisited, also available fromPhilSci Archive.
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• ––– (2005c),Everett and structure.

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Acknowledgments

For comments on earlier versions of this entry, I am very grateful toGuido Bacciagaluppi.