DOI: 10.1007/s10701-005-9031-yFoundations of Physics, Vol. 36, No. 4, April 2006 (© 2006)

Local Realism and Conditional Probability

Allen Stairs1 and Jeffrey Bub1

Received August 23, 2005 / Published online February 14, 2006

Emilio Santos has argued (Santos, Studies in History and Philosophy of Physicshttp://arxiv-org/abs/quant-ph/0410193) that to date, no experiment has provided aloophole-free refutation of Bell’s inequalities. He believes that this provides strongevidence for the principle of local realism, and argues that we should reject this prin-ciple only if we have extremely strong evidence. However, recent work by Malley andFine (Non-commuting observables and local realism, http://arxiv-org/abs/quant-ph/0505016) appears to suggest that experiments refuting Bell’s inequalities couldat most confirm that quantum mechanical quantities do not commute. They alsosuggest that experiments performed on a single system could refute local realism.In this paper, we develop a connection between the work of Malley and Fine andan argument by Bub from some years ago [Bub, The Interpretation of QuantumMechanics, Chapter VI(Reidel, Dodrecht,1974)]. We also argue that the appear-ance of conflict between Santos on the one hand and Malley and Fine on the otheris a result of differences in the way they understand local realism.

KEY WORDS: Bell’s inequality; local realism; hidden variables; nonlocality;Kochen–Specker theorem; quantum conditional probability.

PACS: 03.65.Ta.

1. INTRODUCTION

In a series of papers over a number of years, Santos has reminded uswith great force and clarity that the case for experimental violations ofthe Bell inequalities is surprisingly weak. As he points out, there is todate no loophole-free experiment displaying any such violation. In a recentpaper,(14) Santos has pushed his case further. In his view, local realism isa weighty enough principle that we should reject it only if we have com-pelling experimental evidence. The common wisdom that the evidence is

1 Department of Philosophy, University of Maryland, College Park, MD 20742, USA; e-mail:jbub@umd.edu

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586 Stairs and Bub

good enough rests on what he sees as mere subjective assessments of plau-sibility. In his view, the persistent failure to find a loophole-free experimentsuggests that local realism may have the status of a fundamental principlecomparable to the second law of thermodynamics.

A recent pair of papers contrasts with Santos’s view in a remark-able way. According to Malley,(11) all that experimental violations of theinequalities could show is that quantum observables do not commute.According to a related and more recent paper by Malley and Fine,(13) itshould be possible to refute local realism using elementary tests on singleparticles, without any issues about loopholes or inefficiencies.

Putting all this together presents us with a puzzling situation. Themajority opinion is that Bell’s inequalities have been well and truly refuted,thereby showing that local realism is false. However, members of thismajority do not regard this as a trivial accomplishment. Santos agrees thatthere is nothing trivial here. He argues with considerable sensitivity to theexperimental details that no experiment to date has managed to refutelocal realism, and he speculates that none ever will. On the other hand,the papers of Malley and of Malley and Fine inform us either that anyexperiments bearing on Bell’s inequality could at most tell us somethingwe already take ourselves to know, or that even simple one-system exper-iments routinely violate local realism. In fact, a closer look suggests, con-trary to first impressions, that these two conclusions come to much thesame thing from Malley and Fine’s point of view.

One might suspect that the phrase ‘local realism’ does not mean thesame thing to all of the disputants. However, this is less important thanit might seem. Whatever the differences between Santos and Malley–Fineover how to interpret the words ‘local realism’, there is a stark differencebetween the view that there is a deep issue here to which experiments arecrucial, and the claim that the experiments tell us little if anything that wedo not already know.

Clearly this situation calls for some analysis. In what follows, webegin by reviewing the arguments of Malley and of Malley and Fine. Wego on to discuss a similarity between their arguments and an argumentmade by one of us (Bub) some years ago. This raises some interestingquestions of interpretation that bear on Santos’s concerns, and we turn tothose in the final section of the paper.

2. MALLEY’S ARGUMENT

First a matter of vocabulary. Malley couches his argument in termsof hidden variables; Santos prefers to talk of local realism. He notes that

Local Realism and Conditional Probability 587

even though the failure of local realism would entail the impossibility oflocal hidden variables, if local realism is true, this still might not entail theexistence of practically useful local hidden variables. In fact, Malley couldagree with Santos on this point. The ‘hidden variables’ at issue have to dowith what is possible in principle. As for what hidden variables are, Malleycharacterizes them by four conditions, which we will list in an equivalentbut slightly different form.

Suppose, we have a quantum system whose state is given by a densityoperator D. A hidden variable theory for this system is a classical prob-ability space ! = (", # ($), µ), consisting of a # -algebra # ($) of subsetsof a set $ and a measure µ, satisfying four conditions.2 Malley states theconditions in full generality. However, since all the relevant arguments arein terms of projectors, we will restate the conditions accordingly. Unlessotherwise noted, we assume that the Hilbert space is of dimension threeor greater.

HV(a): Each projector A is associated with a map "(·) from $ to{0, 1}, and there is a subset a in # ($) for which "(a) = 1.We can read "(a) = 1 as meaning that in the HV state ", theprojector A takes the value 1.

HV(b): If A and B are commuting projectors, then "(a ! b) ="(a)"(b).

HV(c): For each projector, A, Tr(DA) = µ(a); that is, the HV the-ory correctly reproduces the quantum single-case probabili-ties, and

HV(d): For each pair A, B of commuting projectors, Tr(DAB) =µ(a!b); that is, the HV theory correctly reproduces the quan-tum joint probabilities.

Write PrD(X) for the quantum probability in D that the projectorX takes the value 1. If A and B are projectors, then what Malley callsthe quantum conditional probability of A given B, written Pr(A | B), isgiven by

PrD(A |B) = Tr(DBAB)

Tr(DB)(1)

This is the result of projecting the state onto the subspace associated withthe projector B, normalizing, and then using the new state to compute the

2 Malley says one or more of these conditions, but the proof assumes that HV(a), HV(c), andHV(d) all hold, and Malley notes that HV(b) follows from the other three conditions.

588 Stairs and Bub

probability of A. Malley shows that if HV(a)–HV(d) are satisfied, then forany two projectors A and B, what he calls the conditional probability rulemust hold. That is, we must have

PrD(A |B) = µ(a ! b)

µ(b)= µ(a |b). (2)

We will examine the proof of the conditional probability rule below.Meanwhile suppose the rule holds. Then, using the formula for PrD(A |B), we would have

µ(a ! b) = µ(a |b)µ(b) = Tr(DBAB)

Tr(DB)Tr(DB) = Tr(DBAB) (3)

but also

µ(a ! b) = µ(a |b)µ(a) = Tr(DABA)

Tr(DA)Tr(DA) = Tr(DABA) (4)

giving us

Tr(DBAB) = Tr(DABA). (5)

Since, this holds for every D, we would have

ABA = BAB. (6)

Using Eq. (6) and the idempotence of projectors, a little algebra gives us[AB"BA]2 = 0. However [AB"BA] is easily shown to be skew-Hermitian,and so it follows that [AB"BA] is also zero.3 Since A and B are arbitraryprojectors, it would follow: if HV(a)–HV(d) hold, all observables commute.

This is a fascinating result. Here is the gist of the argument; fordetails, readers can turn to an earlier paper of Malley’s,(10) and to anexpanded version of his 2004 paper,(12) in which he makes use of an Exer-cise from Beltrametti and Cassinelli’s The Logic of Quantum Mechanics[Ref. 1 p. 288]. The content of the Exercise is the following lemma, provedusing Gleason’s theorem.(8)

3 C is skew-Hermitian iff C# = "C. Suppose C is skew-Hermitian and that CC = 0. IfC $= 0, then there is a |%% such that C|%% $= 0. But then &%|C#C|%% $= 0. However, C#C ="CC = 0, and so &%|C#C|%% = 0.

Local Realism and Conditional Probability 589

Exercise. Let H be a Hilbert space of dimension greater than three, P(H)

be the lattice of projectors on H, and let & be a measure on P(H). Let B

be any projector such that &(B) $= 0. Then there exists a unique measure onP(H), denoted by Pr&(· | B), such that for all projectors C ! B (i.e., for allprojectors C such that CB = BC = C), Pr&(C |B) = Pr&(C)/Pr&(B).

Although the measure Pr&(· | B) is defined by way of B, it’s ameasure on the whole of P(H). Thus, the requirement that Pr&(C|B) =Pr&(C)/P r&(B) when C ! B determines a unique measure on the entirelattice, though Pr&(·|B) will not in general be given by Pr&(C)/P r&(B).Furthermore, within B, Pr&(·|B) behaves like classical conditional proba-bility. That is, classical conditional probability has the propertythat

µ(c |b) = µ(c)

µ(b)(7)

for c ' b, which mirrors

Pr&(C |B) = Pr&(C)

P r&(B). (8)

Suppose, then, that C ! B. Then Eq. (8) holds. By HV(c), we have

µ(c) = Pr&(C),

µ(b) = Pr&(B).(9)

However, Malley shows that c ' b, and so the laws of classical con-ditional probability ensure that in this case,

µ(c |b) = µ(c)

µ(b). (10)

Combining Eqs. (8)–(10), we get

Pr&(C |B) = µ(c | b), (11)

when C ! B. We would have the conditional probability rule if we couldshow that the identity Pr&(C |B) = µ(c |b) holds for arbitrary C, B. Howcan we close the gap?

Suppose, as a hidden variable theorist who accepts HV(a)–HV(d)must, that µ is a measure on # ($) that faithfully reflects the quantum

590 Stairs and Bub

probabilities given by Pr(·). Although the order of construction for ahidden variable theorist would be to begin with P(H) and define a mea-sure on # ($), it follows from HV(a) and HV(b) that any measure on # ($)

would have to induce a measure on P(H). The reason is that what HV(a)and HV(b) require is that each projector has a unique representative on# ($), and that when two projectors A1 and A2 are orthogonal, their rep-resentatives a1 and a2 are disjoint. Now a probability measure & on P(H)

is any function from projectors into [0, 1] with three properties:

(i) &(0) = 0,

(ii) &(I ) = 1,

(iii) If the members of {A1, A2, . . . } are pairwise orthogonal, then

&(A1 + A2 + · · · ) = &(A1) + &(A2) + · · ·

However, HV(a) and HV(b) require that the representative of the nulloperator 0 is (, the representative of the identity operator I is $, and sinceorthogonal projectors are represented by disjoint sets, any measure µ mustsatisfy µ(a1 ) a2 ) · · · ) = µ(a1) + µ(a2) + · · · Given a measure on # ($),then, we induce a measure on P(H) simply by ‘reading back’ the mea-sures of subsets that represent projectors onto the projectors themselves.The point is that if a hidden variable theory satisfying HV(a)–HV(d) is tobe possible at all, then the orthogonality structure of P(H) would have tobe reflected by disjointness of the corresponding sets on the phase space.

From here to Malley’s result is conceptually straightforward.4 Wehave assumed (in effect for reductio) that µ faithfully captures the mea-sure PrD(·) given by the density operator D. Now consider a projectorB such that PrD(B) $= 0, and consider the measure µ(· | b) that we getby conditionalizing on b. It is well known that this measure is unique. Asjust pointed out, it must also induce a measure on P(H). Malley remindsus that this measure will agree with Pr(· | B) for all C ! B. But, as theExercise points out, requiring Eq. (8) for C ! B determines a unique mea-sure on P(H). So the assumption that µ exists at all requires that µ(· |B)

induces some measure on P(H), and, by the Exercise, there is only onemeasure that this could be, namely Pr(· | B). Thus, the conditional prob-ability rule follows, and by Eqs. (2)–(6), we conclude that all projectorsmust commute.

This is an ingenious argument. We will inquire into its significancebelow. Meanwhile, we turn to the related results of Malley and Fine.

4 Thanks to Arthur Fine for clearing up a confusion we had about this.

Local Realism and Conditional Probability 591

3. MALLEY AND FINE

Malley and Fine’s goal is to refine the earlier claims discussed inSec. 2 and to defend the following theses about Bell-style no-hidden-vari-able arguments: (i) that the failure of local realism has nothing to do withentanglement or the violation of Bell’s inequalities, but (ii) occurs alreadyfor a single pair of noncommuting observables [Ref. 13, p. 1]:

[Neither] entanglement nor the violation of the Bell inequalities are essentialfor these no-go results. Rather. . . the framework within which local realism isdefined (and Bell-like results derived) must already fail given a single pair of non-commuting observables, regardless of whether the system is in an entangled statefor which the Bell inequalities are violated, or is even a composite system at all.It follows from the results below that the framework of local realism fails evenfor certain product composite states D = D1 *D2 where there is no entanglementand no violation of the Bell inequalities.

and (iii) that the ‘logical engine’ driving the no-go HV theorems is simplynon-commutativity [Ref. 13 p. 2]:

Our aim is to fully identify the logical engine driving the no-go theorems withthe most basic, nonclassical feature of the quantum theory: that not all observ-ables commute.

This is clearly significant in the context of a discussion of Santos’swork. Indeed, Malley and Fine claim [Ref. 13, p. 2] that strong no-local-hidden variable theorems require neither ‘. . . careful pair production withspacelike separated measurements, nor highly efficient detection.’

Malley and Fine refer to the conditions HV(a)–HV(d) as BKS (for‘Bell/Kochen and Specker’). Their arguments depend on

Lemma 1 If BKS(%) holds and neither Tr(DA) nor Tr(DB) is zero,then the relation

Tr(DABA) = Tr(DBAB) (12)

is valid for the system in state D = P% = |%%&%|.

This is a restatement of the result Malley discussed above. We can stateMalley and Fine’s main technical claims this way:

Claim 1 Given any two non-commuting projectors A, B, there exists apure state % such that (i) (AB " BA)% $= 0, [read: A and B do not com-mute with respect to %] but such that (ii) BKS (%) contradicts (i), and isthus invalid. [That is: HV(a)–HV(d) imply, falsely, that A and B commutewith respect to %.]

592 Stairs and Bub

Claim 2 Given a pure state %, there exists a pair of projectors A, B suchthat (i) (AB " BA)% $= 0, but such that (ii) BKS(%) contradicts (i) and isthus invalid.

Although, Malley and Fine’s argument for these claims is perfectly cor-rect, there is a different way of getting a weaker result that has intriguingsimilarities to Malley and Fine’s result. Some years ago one of us (Bub)[Ref. 2 Chapter 6] published an argument proving, in effect, that the condi-tional probability rule holds in a certain special case. What Bub maintainedwas that the Bell no-hidden variable result was trivial (a view he has sinceabandoned). He couched his case in terms of Wigner’s version(15) of Bell,and he reasoned this way: there is an argument that is formally like the Bell–Wigner argument, but it applies to a single electron. It treats the quantumtransition probabilities (the probability of finding, e.g., spin-up in directionb starting with a system in the spin-up eigenstate of spin in direction a)as conditional probabilities, and it derives an inequality exactly parallel toWigner’s. However, Bub pointed out, the quantum transition probabilitiesare not classical conditional probabilities, and he maintained that any argu-ment which implicitly assumes that they are is suspect.

This objection may seem to miss its mark. Bell–Wigner depends onjoint probabilities for commuting observables, and insofar as conditionalprobabilities enter, they are conditional probabilities between commutingprojectors. However, Bub pointed out that given the mirror-image corre-lations, the conditional probabilities between the two systems force us totreat transition probabilities for a single electron as conditional. Lookingat how this works is instructive.

To keep the notation easy to follow, we will use Roman letters forthe left-hand system and corresponding Greek letters for the right-handsystem. Thus, |a+% is the up-eigenstate of spin in direction a on the left-hand system, and |'"% is the down-eigenstate of spin in direction b on theright-hand system. We will also use a+, '", etc., to pick out the sets thatcorrespond to propositions ‘The spin of the right-hand particle in direc-tion a is up,’ and so on. Finally, we will denote the corresponding projec-tors as Pa+, etc. (shorthand for Pa+ *I , etc). Because of the mirror-imagecorrelations, we have, using Malley’s notation:

Pr(Pb+ |Pa+) = Pr(P'+ |Pa+). (13)

(Here ‘Pr’ denotes quantum probability.) However, because Pa+ and P'+commute, we have

Pr(Pa+ |P'+) =Pr(Pa+P'")

P r(Pa+). (14)

Local Realism and Conditional Probability 593

Thus we get,

Pr(Pb+ |Pa+) =Pr(Pa+P'")

P r(Pa+). (15)

For the hidden variable theory to reproduce the statistics correctly, wemust have

Pr(Pa+P'")

P r(Pa+)= µ(a + !'")

µ(a+)(16)

but Eqs. (15) and (16) give us

Pr(Pa+ |Pb+) = µ(a + !'")

µ(a+). (17)

Now Pb+ and P'" are statistically equivalent in the singlet state. Not onlydo we have Pr(Pb+) = Pr(P'"), we also have

Pr(Pb+P +'") = 0 = Pr(P +

b+P'"). (18)

(Here P +b+ = (I " Pb+), and so P +

b+ = Pb".) That means we must have

µ(b + !'") = 0 = µ(b " !'+) (19)

and hence, b+ and '" are statistically equivalent on the phase space. Buton a classical probability space, whenever two sets x and y are statisticallyequivalent—i.e., satisfy the analogue of Eq. (19)—we have

µ(x ! z) = µ(y ! z) (20)

for any set z. This means that from Eqs. (17) and (19), we get

Pr(Pb+ |Pa+) = µ(a + ! b+)

µ(a+)= µ(b+|a+), (21)

i.e., the quantum transition probability Pr(Pb+|Pa+) must equal the phasespace conditional probability µ(b+|a+). Reasoning in the same way, wecan also derive similar expressions for any third direction, and this leadsimmediately to the conditional probability version of the Bell–Wignerinequality:

µ(c+|a+) ! µ(b+|a+) + µ(c+|b+), (22)

which, of course, is violated by the quantum transition probabilitiesPr(Pc+ |Pa+), Pr(Pb+ |Pa+), Pr(Pc+ |Pb+).

594 Stairs and Bub

Note that in this case the equality

Tr(DBAB) = Tr(DABA) (23)

is imposed by quantum mechanics itself, even though A and B do notcommute. One would not take the above derivation of the inequality asshowing that the ‘logical engine’ driving the result has anything to do withnon-commutativity.

This proves, by an argument that proceeds directly from Hilbert spaceprobabilities to phase space probabilities (without invoking Gleason’s the-orem) that for the hidden variable theory to reproduce the statistics ofthe singlet state, it must represent the quantum transition probabilitiesPr(Pc+ | Pa+), Pr(Pb+ | Pa+), Pr(Pc+ | Pb+) as conditional probabilitieson the phase space, even though Pa+, Pb+, Pc+ do not commute pairwise.

Bub objected to the Bell–Wigner proof because it implicitly requiredsuch relationships between quantum transition probabilities and classicalconditional probabilities. However, since Bub also accepted the Kochen andSpecker no-hidden variables proof, whose assumptions are at least as strongas Bell’s, his objection was not sustainable. Setting that aside, we can askwhat lessons Bub’s argument might suggest for the present set of issues.

Bub’s argument, though not intended to show that all transitionprobabilities would have to be represented as conditional probabilities,proves that for a particular state %, namely the singlet state, we can findnon-commuting projectors X, Y, Z such that Pr(Z |X), P r(Y |X), P r(Z |Y )

would have to be represented in a hidden variable theory as µ(z | x),

µ(y | x), µ(z | y). This does not entail that [XZ " ZX]% = 0 or [XY "YX]% = 0 or [YZ " ZY ]% = 0, which are false, but the argument goesthrough without the issue of commutativity arising.

The argument may appear to depend on entangled states of pairs ofsystems, but in fact it can be generalized to apply to any quantum sys-tem whose Hilbert space is of dimension four or greater. Proceed in twostages. First, let H = H4 be any four-dimensional Hilbert space. Supposethat |(%&(| is a state on H4. Let |)% be the singlet state on H2 *H2 andlet u:H2 * H2 ,H4 be a unitary map such that u(|)%&)|) = |(%&(|. LetP|a+% * I , P|b+% * I , and P|c+% * I be three non-commuting projectors onH2 * H2, and let I * P|&"%, I * P'", I * P*"be their mirror-image coun-terparts. Denote their images under u as Qa+, Qb+, Qc+, Q&", Q'", Q*".Then because u preserves all geometrical relations, an argument just likethe one we went through above will show that if HV(a)–HV(d) hold,then

Pr((Qc+ |Qa+) =µ%(qa+ ! qc+)

µ((qa+), (24)

Local Realism and Conditional Probability 595

Pr((Qb+ |Qa+) =µ%(qa+ ! qb+)

µ((qa+), (25)

Pr((Qc+ |Qb+) =µ%(qb+ ! qc+)

µ((qb+). (26)

Thus, for any pure state on H4, we can find a small set of non-commutingprojectors for which we can derive a simple no-go result via a special caseof the conditional probability rule.

Similarly, beginning with non-commuting projectors Q, R, S ontonon-overlapping planes, there will be a state |(%&(| for which one canderive a similar no-go result. We simply find three projectors P|a+% * I ,P|b+%*I , P|c+%*I on H2 * H2 whose geometrical relations are the same asthose between Q, R, and S. Then we choose u such that u(P|a+% *I ) = Q,u(P|b+% * I ) = R, u(P|c+% * I ) = S. The state |(%&(| will be the image ofthe singlet state |)%&)| under u.

So much for the first stage of the generalization. For the second stage,we observe that H2 * H2 can be imbedded in any Hilbert space Hn forn " 4. This means that the result holds for all pure states on Hilbertspaces of dimension greater than or equal to four.

4. DOES ENTANGLEMENT MATTER?

It would be satisfying to have a way of using the Bub-style argu-ment to prove Malley’s original claim: that the conditional probability ruleholds for all states and all pairs of projectors. So far, we have not foundsuch a proof and we are not sure that one exists. It would also be grat-ifying to be able to extend the argument of the previous section to H3,since it could then be applied to the case of a spin-1 particle. So far, wehave not found a construction that works and are not sure that one exists.In either case, a proof along the lines of what we have seen above wouldproceed as follows:

(i) It would show that quantum theory requires a certain quantumtransition probability Pr(B |A) between non-commuting projec-tors A and B to equal a well-defined quantum probability ratioPr(AC)/P r(A) between commuting projectors A and C, whereC also commutes with B.

(ii) It would use HV(a)–HV(d) to conclude that Pr(B | A) = µ(a !c)/µ(c).

(iii) It would show that the quantum probabilities Pr(BC+) andPr(B+C) both equal zero.

596 Stairs and Bub

(iv) It would use HV(a)–HV(d) to conclude that µ(a ! c-) = 0 =µ(a- ! c) on the phase space, and hence to conclude that a andc are statistically equivalent on the phase space.

(v) It would use the statistical equivalences of (iv), together with(i) and (ii) to conclude that Pr(B |A) = µ(b |a).

(vi) Finally, it would show that the argument can be made to work forseveral projectors, proving that Pr(C | A) = µ(c | a), Pr(B | A) =µ(b |a), Pr(C |B) = µ(c |b).

However, there is another way one might consider using Bub’s origi-nal argument. Based on what we have seen, we can say that if e1 and e2are two electrons in the singlet state, then we already know that HV(a)–HV(d) would require:

Tr(DPa+Pb+Pa+) = Tr(DPb+Pa+Pb+) (27)

for D the singlet state. But looked at another way, the singlet state maysimply be a ladder we climbed that can be thrown away once we reach ourdestination. Tracing out over the singlet state, the density matrix for e1 is

DR = 12(|+%&+| + |"%&"|). (28)

Suppose we decide on an experiment that would refute hidden variablesfor one electron e1 from a pair in the singlet state. Since the reduced statefor e1 is well-defined apart from e2, quantum mechanics would predictexactly the same behavior for a lone electron in the equal-weight mixture(28). This presents us with an odd situation: by following Bub’s argument,experiments on one electron in a singlet pair could potentially do the samejob that a careful Bell-type experiment could do. Invoking Santos’s under-standing of the significance of the experiments, this seems to mean thatmanipulating a single electron in a singlet pair could refute local realism!This amounts to taking the modus tollens of Bub’s 1974 argument againstBell and turning it into a modus ponens. This is similar to the claim madeby Malley and Fine in the quote above: the framework for local realismfails even if we consider a single qubit. In fact, in one way, it goes a bitfurther. What has just been said applies to the two-dimensional spin spaceof a single electron—a case in which it’s standardly assumed that a triv-ial hidden variable theory is possible. Malley and Fine agree that whatthey call the framework of local realism fails for this case, but they arguethis point by invoking Busch’s extension(3) of Gleason’s theorem to twodimensions by appeal to POVMs. The argument under consideration hereappeals only to the standard projection-based measurements.

Local Realism and Conditional Probability 597

It may appear that we are relying on the singlet state, but that seemsto be inessential. As noted, it is hard to see how the second electron couldbe relevant. The minimum we can assume about an electron—that it isin an equal-weight mixture like (28)—is already sufficient to account forwhatever experiments on one member of a pair would supposedly refutelocal realism. This suggests that we could refute local realism by experi-ments on a single electron in a non-entangled state, as Malley and Finesuggest. Whether we think of the electron in state (28) as the result oftracing out the state over a larger Hilbert space or as a stand-alone state,the predictions are the same. Furthermore, if experiments on an electron inthe state (28) could refute local realism, then it is hard to see how similarexperiments on an electron in a pure state of spin could fail to be capa-ble of the same feat. After all (28) is the least informative state we canassume. It is difficult to see how adding more information could make itharder to achieve the same refutation. All of this seems correct with noneed to consider POVM’s.

The idea that experiments on a single electron could upset local real-ism is quite astonishing. Is it really correct?

That depends on what we mean by ‘local realism.’ Can we equatelocal realism with conditions HV(a)–HV(d). The identification is sup-ported by results of Fine [Ref. 6, p. 293]. Fine, in particular, his

Proposition 2. Necessary and also sufficient for the existence of a deter-ministic hidden-variables model is that the Bell/CH inequalities hold forthe probabilities of the experiment.5

HV(a)–HV(d) characterize a deterministic hidden-variables model.Santos has something more general in mind. He understands realism

as the claim that physical bodies have properties that do not depend onobservation for their existence, but on which measurement results depend[Ref. 14, p. 3]. Locality, according to Santos [Ref. 14, p. 4], ‘is the beliefthat no influence may be transmitted with a speed greater than that oflight.’ In his view, we have a sufficient condition for local realism ifwe require that correlations between measurements performed at differentlocations derive from events in the intersection of the past light cones. Heoperationalizes this by accepting Bell’s requirement that

5 We note that the experiments that Fine considers are ones in which local contextualismsimply is not an issue. For example, if the two systems are spin-1/2 systems, then therelevant local observables are locally maximal; the Kochen and Specker functional rela-tions condition is trivial for H2.

598 Stairs and Bub

p(A, a; B, b) =!

+(")P1("; A, a)P2("; B, b). (29)

Here p(A, a; B, b) is the probability of getting result a for a measurementof A one system and result b for a measurement of B on the other system,the assumption being that the two measurements are space-like separated.The variable " embodies all relevant information from the intersection ofthe backward light cones.

Requiring Eq. (29) as a necessary condition for local realism amountsto accepting what is sometimes referred to as factorizability. Fine has longquestioned whether failures of factorizability indicate non-local causation.6

We agree with Fine that the relationship between factorizability and localcausality is not clear. We also point out that we are not attributing to Finenor to Malley the claim that local experiments could establish non-localaction at a distance. Rather, the claim would be that what has sometimesbeen called local realism would be refuted. Nonetheless, the line of thoughtwe have been considering—a line of thought that we have derived from Bub’sargument of many years ago—would amount to a deflation of the impor-tance of the Bell experiments. Since most people agree that persistent fail-ures of factorizability are puzzling from a classical point of view, we needto consider just how successful such a deflation could be. The first point wewant to suggest is that failures of factorizability are puzzling in a way thatcharacterizing local realism in terms of HV(a)–HV(d) does not capture.

Conditions HV(a)—HV(d), as Fine points out, are strong enough tocapture not just Bell’s constraints but those of Kochen and Specker.(9)

However, many people would resist identifying Kochen–Specker and Bell.Although Fine’s 1982 result is mathematically correct, it raises an inter-pretive question. In the typical Bell-type experiments, the local observablesare maximal on the local polarization or spin spaces. Thus, local contex-tualism—which HV(a)–HV(d) do not permit—is not an issue. In general,the Kochen and Specker proof requires each observable, local or global,to be represented by a single random variable on the phase space, in sucha way that the functional relations among the observables are preserved.This requirement corresponds to HV(a) and HV(b). However, we seemclearly to be able to imagine a situation in which the Kochen and Speckerrequirement fails, and yet where it would seem perfectly appropriate to usethe term ‘local realism.’ Suppose that quantum superpositions never per-sist when systems become space-like separated, and that for such systems,the only correct quantum mechanical predictions are ones that derive fromstates of the form D1 * D2. Suppose, however, that the ‘metaphysically

6 See, e.g., Refs. 5 and 7.

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correct’ account of individual systems is given by a contextual hiddenvariable theory, in which there is one random variable for each locallymaximal observable, but in which observables such as, for example, S2

z fora spin-1 system are ‘split’ and correspond to different random variablesfor each locally maximal observable of which they are functions. Kochenand Specker’s trivial hidden variable theory is a familiar example of sucha model. In a world like that, HV(a) and HV(b) would fail. But if space-like separated systems can only exist in state of the form D1 * D2—forshort, if non-local entanglement is impossible—there would be no tempta-tion to talk of failures of locality, nor would there be any reason to saythat realism was under threat. In a perfectly straightforward sense, thatworld would satisfy local realism.

Perhaps the best way to understand the situation is in two parts.First, if we are serious about making sense of quantum theory, we shouldaccept the following principle:

Higher Hilbert Space Principle (HHS). Conclusions about a system in agiven state should be the same whether or not that state results from tracingout over the state of a combined system.

HHS seems reasonable. Our theory must be able to deal with morethan one system at a time. However, HHS needs to be applied with acaveat: the conclusions we draw should depend on what states are possible.The argument above drew its conclusion about a single electron by assum-ing the existence of an entangled state for a space-like separated pair. Ifsuch states are possible, then the truth about a local system cannot be cap-tured by a model satisfying HV(a)–HV(d). But Santos reminds us that thisis a big ‘if ’, and one for which we do not yet have airtight experimentalevidence. The HHS can have strong implications for local systems only if afurther claim that Santos regards with skepticism is true. Put another way,it could be, for all the experiments to date have definitely shown, that thequantum world is a world in which local contextualism holds, in whichglobal contextualism fails (local observables, even though defined via thelocal context, are well-defined, with no reference needed to other systems),and in which non-local entanglement never occurs.

5. SANTOS AND LOCAL REALISM

Santos takes the failure of the Bell inequalities to amount to thefailure of local realism. Understood in this way, what can we say aboutSantos’s suggestion that local realism is a fundamental principle thatshould only be rejected for the weightiest of reasons? Santos cites Einstein

600 Stairs and Bub

as a kindred spirit on this point, so it may be worth looking at what Ein-stein had to say. Einstein expresses his commitment to realism by sayingthat the concepts of physics refer to a world of things existing indepen-dent of the observer and arranged in space-time. He adds:(4)

Further, it appears to be essential for this arrangement of the things introduced inphysics that, at a specific time, these things claim an existence independent of oneanother, insofar as these things ‘lie in different parts of space.’ Without such anassumption of the mutually independent existence (the ‘being-thus’) of spatially dis-tant things. . . physical thought in the sense familiar to us would not be possible.

Einstein’s point seems to be that in order to do physics, it is essen-tial that we be able to individuate the systems we are trying to study. Wethink that’s plausible. What, exactly, it has to do with the Bell inequal-ities, however, is another question, and a rather difficult one at that. Itmay be (though we are not sure) that a contextual theory that requiredobservables associated with one system to be individuated by reference toanother system (a theory that violates what is sometimes called ‘separabil-ity’) would make physics impossible. However, as has often been observed,standard quantum mechanics is not such a theory. And as for whether vio-lations of Bell-type inequalities would make physics impossible, the answerseems to be a clear no. The answer to the question ‘what would physicsbe like if the inequalities were violated?’ seems straightforward. It wouldbe like quantum mechanics as most physicists already conceive of it. Thevery fact that Santos can tell us so clearly what would be required toobserve a violation of the inequalities suggests that even if a loophole-freeexperiment one day confirms what Santos doubts and many others believe,physics will go on more or less as before. Nonetheless, Santos does us alla service by reminding us that there are important open questions here—even if the implications of those questions are not quite as startling asthey might seem.

REFERENCES

1. E. G. Beltrametti and G. Cassinelli, The Logic of Quantum Mechanics (Addison-Wesley,Reading, MA, 1981).

2. J. Bub, The Interpretation of Quantum Mechanics, Chapter VI. (Reidel, Dodrecht,1974).

3. P. Busch, “Quantum States and Generalized Observables: A Simple Proof of Gleason’sTheorem,” http://arxiv.org/abs/quant-ph/9909073.

4. A. Einstein, “Quantenmechanik und Wirklichkeit,” Dialectica 2, 320–324 (1948). Quotedand translated in D. Howard, “Holism, separability and the metaphysical implications ofthe Bell experiments,” in Philosophical Consequences of Quantum Theory, J. T. Cushing andE. McMullin, eds. (University of Notre Dame Press, Notre Dame, 1989), p. 233.

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5. A. Fine, “Correlations and physical locality,” in PSA 1980 vol. 2, P. Asquith andR. Giere, eds. (Philosophy of Science Association, East Lansing, MI, 1982), pp. 535–562.

6. A. Fine, “Hidden variables, joint probability and the Bell inequalities,” Phys. Rev. Lett.48, 291–295 (1982).

7. A. Fine, “Antinomies of entanglement; the puzzling case of the tangled statistics,”J. Philos. 79, 733-747 (1982).

8. A. M. Gleason, “Measures on the closed subspaces of Hilbert space,” J. Math. Mech.6, 885–893 (1957).

9. S. Kochen and E. P. Specker, “The problem of hidden variables in quantummechanics,” J. Math. Mech. 17, 59–87(1967).

10. J. D. Malley, “Quantum conditional probability and hidden-variables models,” Phys.Rev A 58, 812 (1998).

11. J. D. Malley, “All quantum observables in a hidden-variables model must commutesimultaneously,” Phys. Rev. A 69, 022118 (2004).

12. J. D. Malley, “All quantum observables in a hidden-variables model must commutesimultaneously,” http://arxiv.org/abs/quant-ph/0402126.

13. J. D. Malley and A. Fine, “Noncommuting observables and local realism,” http://arxiv.org/abs/quant-ph/0505016.

14. E. Santos, “Bell’s theorem and the experiments: increasing support to local realism?,”fothcoming in Studies in History and Philosophy of Physics; available at http://arxiv.org/abs/quant-ph/0410193 (references here are to this version).

15. E. P. Wigner, Am. J. Phys. 38, 1005–1009 (1970).