Local Realism and Conditional Probability

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

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 Physics http://arxiv-org/abs/quant-ph/0410193) that to date, no experiment has provided a loophole-free refutation of Bell’s inequalities. He believes that this provides strong evidence for the principle of local realism, and argues that we should reject this principle only if we have extremely strong evidence. However, recent work by Malley and Fine (Non-commuting observables and local realism, http://arxiv-org/abs/quantph/0505016) appears to suggest that experiments refuting Bell’s inequalities could at most confirm that quantum mechanical quantities do not commute. They also suggest 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 and an argument by Bub from some years ago [Bub, The Interpretation of Quantum Mechanics, Chapter VI(Reidel, Dodrecht,1974)]. We also argue that the appearance of conflict between Santos on the one hand and Malley and Fine on the other is 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 us with great force and clarity that the case for experimental violations of the Bell inequalities is surprisingly weak. As he points out, there is to date no loophole-free experiment displaying any such violation. In a recent paper,(14) Santos has pushed his case further. In his view, local realism is a weighty enough principle that we should reject it only if we have compelling experimental evidence. The common wisdom that the evidence is 1

Department of Philosophy, University of Maryland, College Park, MD 20742, USA; e-mail: [email protected] 585 0015-9018/06/0400-0585/0 © 2006 Springer Science+Business Media, Inc.

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good enough rests on what he sees as mere subjective assessments of plausibility. In his view, the persistent failure to find a loophole-free experiment suggests that local realism may have the status of a fundamental principle comparable to the second law of thermodynamics. A recent pair of papers contrasts with Santos’s view in a remarkable way. According to Malley,(11) all that experimental violations of the inequalities could show is that quantum observables do not commute. According to a related and more recent paper by Malley and Fine,(13) it should be possible to refute local realism using elementary tests on single particles, without any issues about loopholes or inefficiencies. Putting all this together presents us with a puzzling situation. The majority opinion is that Bell’s inequalities have been well and truly refuted, thereby showing that local realism is false. However, members of this majority do not regard this as a trivial accomplishment. Santos agrees that there is nothing trivial here. He argues with considerable sensitivity to the experimental details that no experiment to date has managed to refute local 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 any experiments bearing on Bell’s inequality could at most tell us something we already take ourselves to know, or that even simple one-system experiments routinely violate local realism. In fact, a closer look suggests, contrary to first impressions, that these two conclusions come to much the same thing from Malley and Fine’s point of view. One might suspect that the phrase ‘local realism’ does not mean the same thing to all of the disputants. However, this is less important than it might seem. Whatever the differences between Santos and Malley–Fine over how to interpret the words ‘local realism’, there is a stark difference between the view that there is a deep issue here to which experiments are crucial, and the claim that the experiments tell us little if anything that we do not already know. Clearly this situation calls for some analysis. In what follows, we begin by reviewing the arguments of Malley and of Malley and Fine. We go on to discuss a similarity between their arguments and an argument made by one of us (Bub) some years ago. This raises some interesting questions of interpretation that bear on Santos’s concerns, and we turn to those in the final section of the paper. 2. MALLEY’S ARGUMENT First a matter of vocabulary. Malley couches his argument in terms of hidden variables; Santos prefers to talk of local realism. He notes that

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even though the failure of local realism would entail the impossibility of local hidden variables, if local realism is true, this still might not entail the existence of practically useful local hidden variables. In fact, Malley could agree with Santos on this point. The ‘hidden variables’ at issue have to do with what is possible in principle. As for what hidden variables are, Malley characterizes them by four conditions, which we will list in an equivalent but slightly different form. Suppose, we have a quantum system whose state is given by a density operator D. A hidden variable theory for this system is a classical probability space ! = (λ, σ ($), µ), consisting of a σ -algebra σ ($) of subsets of a set $ and a measure µ, satisfying four conditions.2 Malley states the conditions in full generality. However, since all the relevant arguments are in terms of projectors, we will restate the conditions accordingly. Unless otherwise noted, we assume that the Hilbert space is of dimension three or greater. HV(a): Each projector A is associated with a map λ(·) from $ to {0, 1}, and there is a subset a in σ ($) for which λ(a) = 1. We can read λ(a) = 1 as meaning that in the HV state λ, the projector A takes the value 1. HV(b): If A and B are commuting projectors, then λ(a ∩ b) = λ(a)λ(b). HV(c): For each projector, A, Tr(DA) = µ(a); that is, the HV theory correctly reproduces the quantum single-case probabilities, and HV(d): For each pair A, B of commuting projectors, Tr(DAB) = µ(a∩b); that is, the HV theory correctly reproduces the quantum joint probabilities. Write P r D (X) for the quantum probability in D that the projector X takes the value 1. If A and B are projectors, then what Malley calls the quantum conditional probability of A given B, written P r(A | B), is given by P r D (A | B) =

Tr(DBAB) Tr(DB)

(1)

This is the result of projecting the state onto the subspace associated with the 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), and HV(d) all hold, and Malley notes that HV(b) follows from the other three conditions.

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probability of A. Malley shows that if HV(a)–HV(d) are satisfied, then for any two projectors A and B, what he calls the conditional probability rule must hold. That is, we must have P r D (A | B) =

µ(a ∩ b) = µ(a | b). µ(b)

(2)

We will examine the proof of the conditional probability rule below. Meanwhile suppose the rule holds. Then, using the formula for P r D (A | B), we would have µ(a ∩ b) = µ(a | b)µ(b) =

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

(3)

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

(4)

but also µ(a ∩ b) = µ(a | b)µ(a) = 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 arbitrary projectors, it would follow: if HV(a)–HV(d) hold, all observables commute. This is a fascinating result. Here is the gist of the argument; for details, readers can turn to an earlier paper of Malley’s,(10) and to an expanded version of his 2004 paper,(12) in which he makes use of an Exercise from Beltrametti and Cassinelli’s The Logic of Quantum Mechanics [Ref. 1 p. 288]. The content of the Exercise is the following lemma, proved using Gleason’s theorem.(8) 3

C is skew-Hermitian iff C ∗ = −C. Suppose C is skew-Hermitian and that CC = 0. If $ 0. However, C ∗ C = C $= 0, then there is a |φ% such that C|φ% $= 0. But then &φ|C ∗ C|φ% = −CC = 0, and so &φ|C ∗ C|φ% = 0.

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Exercise. Let H be a Hilbert space of dimension greater than three, P(H) be the lattice of projectors on H, and let α be a measure on P(H). Let B be any projector such that α(B) $= 0. Then there exists a unique measure on P(H), denoted by Prα (· | B), such that for all projectors C ! B (i.e., for all projectors C such that CB = BC = C), Prα (C | B) = Prα (C)/Prα (B). Although the measure P r α (· | B) is defined by way of B, it’s a measure on the whole of P(H). Thus, the requirement that P r α (C|B) = P r α (C)/P r α (B) when C ! B determines a unique measure on the entire lattice, though P r α (·|B) will not in general be given by P r α (C)/P r α (B). Furthermore, within B, P r α (·|B) behaves like classical conditional probability. That is, classical conditional probability has the property that µ(c | b) =

µ(c) µ(b)

(7)

P r α (C) . P r α (B)

(8)

for c ⊆ b, which mirrors P r α (C | B) =

Suppose, then, that C ! B. Then Eq. (8) holds. By HV(c), we have µ(c) = P r α (C), µ(b) = P r α (B).

(9)

However, Malley shows that c ⊆ b, and so the laws of classical conditional probability ensure that in this case, µ(c | b) =

µ(c) . µ(b)

(10)

Combining Eqs. (8)–(10), we get P r α (C | B) = µ(c | b),

(11)

when C ! B. We would have the conditional probability rule if we could show that the identity P r α (C | B) = µ(c | b) holds for arbitrary C, B. How can we close the gap? Suppose, as a hidden variable theorist who accepts HV(a)–HV(d) must, that µ is a measure on σ ($) that faithfully reflects the quantum

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probabilities given by P r(·). Although the order of construction for a hidden variable theorist would be to begin with P(H) and define a measure on σ ($), it follows from HV(a) and HV(b) that any measure on σ ($) 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 σ ($), and that when two projectors A1 and A2 are orthogonal, their representatives a1 and a2 are disjoint. Now a probability measure α on P(H) is any function from projectors into [0, 1] with three properties: (i) α(0) = 0, (ii) α(I ) = 1, (iii) If the members of {A1 , A2 , . . . } are pairwise orthogonal, then α(A1 + A2 + · · · ) = α(A1 ) + α(A2 ) + · · · However, HV(a) and HV(b) require that the representative of the null operator 0 is ∅, the representative of the identity operator I is $, and since orthogonal projectors are represented by disjoint sets, any measure µ must satisfy µ(a1 ∪ a2 ∪ · · · ) = µ(a1 ) + µ(a2 ) + · · · Given a measure on σ ($), then, we induce a measure on P(H) simply by ‘reading back’ the measures of subsets that represent projectors onto the projectors themselves. The point is that if a hidden variable theory satisfying HV(a)–HV(d) is to be possible at all, then the orthogonality structure of P(H) would have to be reflected by disjointness of the corresponding sets on the phase space. From here to Malley’s result is conceptually straightforward.4 We have assumed (in effect for reductio) that µ faithfully captures the measure P r D (·) given by the density operator D. Now consider a projector B such that P r D (B) $= 0, and consider the measure µ(· | b) that we get by conditionalizing on b. It is well known that this measure is unique. As just pointed out, it must also induce a measure on P(H). Malley reminds us that this measure will agree with P r(· | B) for all C ! B. But, as the Exercise points out, requiring Eq. (8) for C ! B determines a unique measure 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 one measure that this could be, namely P r(· | B). Thus, the conditional probability rule follows, and by Eqs. (2)–(6), we conclude that all projectors must commute. This is an ingenious argument. We will inquire into its significance below. 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.

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3. MALLEY AND FINE Malley and Fine’s goal is to refine the earlier claims discussed in Sec. 2 and to defend the following theses about Bell-style no-hidden-variable arguments: (i) that the failure of local realism has nothing to do with entanglement or the violation of Bell’s inequalities, but (ii) occurs already for a single pair of noncommuting observables [Ref. 13, p. 1]: [Neither] entanglement nor the violation of the Bell inequalities are essential for these no-go results. Rather. . . the framework within which local realism is defined (and Bell-like results derived) must already fail given a single pair of noncommuting observables, regardless of whether the system is in an entangled state for 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 even for certain product composite states D = D1 ⊗ D2 where there is no entanglement and no violation of the Bell inequalities.

and (iii) that the ‘logical engine’ driving the no-go HV theorems is simply non-commutativity [Ref. 13 p. 2]: Our aim is to fully identify the logical engine driving the no-go theorems with the most basic, nonclassical feature of the quantum theory: that not all observables commute.

This is clearly significant in the context of a discussion of Santos’s work. Indeed, Malley and Fine claim [Ref. 13, p. 2] that strong no-localhidden variable theorems require neither ‘. . . careful pair production with spacelike 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(φ) 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 = Pφ = |φ%&φ|. This is a restatement of the result Malley discussed above. We can state Malley and Fine’s main technical claims this way: Claim 1 Given any two non-commuting projectors A, B, there exists a pure state φ such that (i) (AB − BA)φ $= 0, [read: A and B do not commute with respect to φ] but such that (ii) BKS (φ) contradicts (i), and is thus invalid. [That is: HV(a)–HV(d) imply, falsely, that A and B commute with respect to φ.]

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Claim 2 Given a pure state φ, there exists a pair of projectors A, B such that (i) (AB − BA)φ $= 0, but such that (ii) BKS(φ) contradicts (i) and is thus invalid. Although, Malley and Fine’s argument for these claims is perfectly correct, there is a different way of getting a weaker result that has intriguing similarities 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 conditional probability rule holds in a certain special case. What Bub maintained was that the Bell no-hidden variable result was trivial (a view he has since abandoned). 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 quantum transition probabilities (the probability of finding, e.g., spin-up in direction b starting with a system in the spin-up eigenstate of spin in direction a) as conditional probabilities, and it derives an inequality exactly parallel to Wigner’s. However, Bub pointed out, the quantum transition probabilities are not classical conditional probabilities, and he maintained that any argument which implicitly assumes that they are is suspect. This objection may seem to miss its mark. Bell–Wigner depends on joint probabilities for commuting observables, and insofar as conditional probabilities enter, they are conditional probabilities between commuting projectors. However, Bub pointed out that given the mirror-image correlations, the conditional probabilities between the two systems force us to treat transition probabilities for a single electron as conditional. Looking at how this works is instructive. To keep the notation easy to follow, we will use Roman letters for the left-hand system and corresponding Greek letters for the right-hand system. Thus, |a+% is the up-eigenstate of spin in direction a on the lefthand system, and |β−% is the down-eigenstate of spin in direction b on the right-hand system. We will also use a+, β−, etc., to pick out the sets that correspond to propositions ‘The spin of the right-hand particle in direction a is up,’ and so on. Finally, we will denote the corresponding projectors as Pa+ , etc. (shorthand for Pa+ ⊗ I , etc). Because of the mirror-image correlations, we have, using Malley’s notation: P r(Pb+ | Pa+ ) = P r(Pβ+ | Pa+ ).

(13)

(Here ‘P r’ denotes quantum probability.) However, because Pa+ and Pβ+ commute, we have P r(Pa+ | Pβ+ ) =

P r(Pa+ Pβ− ) . P r(Pa+ )

(14)

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Thus we get, P r(Pb+ | Pa+ ) =

P r(Pa+ Pβ− ) . P r(Pa+ )

(15)

For the hidden variable theory to reproduce the statistics correctly, we must have P r(Pa+ Pβ− ) µ(a + ∩ β−) = P r(Pa+ ) µ(a+)

(16)

but Eqs. (15) and (16) give us P r(Pa+ | Pb+ ) =

µ(a + ∩ β−) . µ(a+)

(17)

Now Pb+ and Pβ− are statistically equivalent in the singlet state. Not only do we have P r(Pb+ ) = P r(Pβ− ), we also have ⊥ ⊥ P r(Pb+ Pβ− ) = 0 = P r(Pb+ Pβ− ).

(18)

⊥ = (I − P ), and so P ⊥ = P .) That means we must have (Here Pb+ b+ b− b+

µ(b + ∩ β−) = 0 = µ(b − ∩ β+)

(19)

and hence, b+ and β− are statistically equivalent on the phase space. But on a classical probability space, whenever two sets x and y are statistically equivalent—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 P r(Pb+ | Pa+ ) =

µ(a + ∩ b+) = µ(b+ | a+), µ(a+)

(21)

i.e., the quantum transition probability P r(Pb+|Pa+ ) must equal the phase space conditional probability µ(b+|a+). Reasoning in the same way, we can also derive similar expressions for any third direction, and this leads immediately to the conditional probability version of the Bell–Wigner inequality: µ(c+ | a+) ! µ(b+ | a+) + µ(c+ | b+),

(22)

which, of course, is violated by the quantum transition probabilities P r(Pc+ | Pa+ ), P r(Pb+ | Pa+ ), P r(Pc+ | Pb+ ).

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Note that in this case the equality Tr(DBAB) = Tr(DABA)

(23)

is imposed by quantum mechanics itself, even though A and B do not commute. One would not take the above derivation of the inequality as showing that the ‘logical engine’ driving the result has anything to do with non-commutativity. This proves, by an argument that proceeds directly from Hilbert space probabilities to phase space probabilities (without invoking Gleason’s theorem) that for the hidden variable theory to reproduce the statistics of the singlet state, it must represent the quantum transition probabilities P r(Pc+ | Pa+ ), P r(Pb+ | Pa+ ), P r(Pc+ | Pb+ ) as conditional probabilities on the phase space, even though Pa+ , Pb+ , Pc+ do not commute pairwise. Bub objected to the Bell–Wigner proof because it implicitly required such relationships between quantum transition probabilities and classical conditional probabilities. However, since Bub also accepted the Kochen and Specker no-hidden variables proof, whose assumptions are at least as strong as Bell’s, his objection was not sustainable. Setting that aside, we can ask what lessons Bub’s argument might suggest for the present set of issues. Bub’s argument, though not intended to show that all transition probabilities would have to be represented as conditional probabilities, proves that for a particular state φ, namely the singlet state, we can find non-commuting projectors X, Y, Z such that P r(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]φ = 0 or [XY − Y X]φ = 0 or [Y Z − ZY ]φ = 0, which are false, but the argument goes through without the issue of commutativity arising. The argument may appear to depend on entangled states of pairs of systems, but in fact it can be generalized to apply to any quantum system whose Hilbert space is of dimension four or greater. Proceed in two stages. First, let H = H4 be any four-dimensional Hilbert space. Suppose that |(%&(| is a state on H4 . Let |)% be the singlet state on H2 ⊗ H2 and let u:H2 ⊗ H2 → H4 be a unitary map such that u(|)%&)|) = |(%&(|. Let P|a+% ⊗ I , P|b+% ⊗ I , and P|c+% ⊗ I be three non-commuting projectors on H2 ⊗ H2 , and let I ⊗ P|α−% , I ⊗ Pβ− , I ⊗ Pγ − be their mirror-image counterparts. Denote their images under u as Qa+ , Qb+ , Qc+ , Qα− , Qβ− , Qγ − . Then because u preserves all geometrical relations, an argument just like the one we went through above will show that if HV(a)–HV(d) hold,then P r ( (Qc+ | Qa+ ) =

µφ (qa+ ∩ qc+ ) , µ( (qa+ )

(24)

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P r ( (Qb+ | Qa+ ) =

µφ (qa+ ∩ qb+ ) , µ( (qa+ )

(25)

P r ( (Qc+ | Qb+ ) =

µφ (qb+ ∩ qc+ ) . µ( (qb+ )

(26)

Thus, for any pure state on H4 , we can find a small set of non-commuting projectors for which we can derive a simple no-go result via a special case of the conditional probability rule. Similarly, beginning with non-commuting projectors Q, R, S onto non-overlapping planes, there will be a state |(%&(| for which one can derive 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 as those 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 of the 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 for n " 4. This means that the result holds for all pure states on Hilbert spaces 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 argument to prove Malley’s original claim: that the conditional probability rule holds for all states and all pairs of projectors. So far, we have not found such a proof and we are not sure that one exists. It would also be gratifying 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, we have 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 would proceed as follows: (i) It would show that quantum theory requires a certain quantum transition probability P r(B | A) between non-commuting projectors A and B to equal a well-defined quantum probability ratio P r(AC)/P r(A) between commuting projectors A and C, where C 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 ⊥ ) and Pr(B ⊥ C) both equal zero.

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(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 and c are statistically equivalent on the phase space. (v) It would use the statistical equivalences of (iv), together with (i) and (ii) to conclude that P r(B | A) = µ(b | a).

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

However, there is another way one might consider using Bub’s original argument. Based on what we have seen, we can say that if e1 and e2 are 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 may simply be a ladder we climbed that can be thrown away once we reach our destination. Tracing out over the singlet state, the density matrix for e1 is DR =

1 (|+%&+| + |−%&−|). 2

(28)

Suppose we decide on an experiment that would refute hidden variables for one electron e1 from a pair in the singlet state. Since the reduced state for e1 is well-defined apart from e2 , quantum mechanics would predict exactly 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 same job that a careful Bell-type experiment could do. Invoking Santos’s understanding of the significance of the experiments, this seems to mean that manipulating a single electron in a singlet pair could refute local realism! This amounts to taking the modus tollens of Bub’s 1974 argument against Bell and turning it into a modus ponens. This is similar to the claim made by Malley and Fine in the quote above: the framework for local realism fails even if we consider a single qubit. In fact, in one way, it goes a bit further. What has just been said applies to the two-dimensional spin space of a single electron—a case in which it’s standardly assumed that a trivial hidden variable theory is possible. Malley and Fine agree that what they call the framework of local realism fails for this case, but they argue this point by invoking Busch’s extension(3) of Gleason’s theorem to two dimensions by appeal to POVMs. The argument under consideration here appeals only to the standard projection-based measurements.

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It may appear that we are relying on the singlet state, but that seems to be inessential. As noted, it is hard to see how the second electron could be relevant. The minimum we can assume about an electron—that it is in an equal-weight mixture like (28)—is already sufficient to account for whatever experiments on one member of a pair would supposedly refute local realism. This suggests that we could refute local realism by experiments on a single electron in a non-entangled state, as Malley and Fine suggest. Whether we think of the electron in state (28) as the result of tracing 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 in the state (28) could refute local realism, then it is hard to see how similar experiments on an electron in a pure state of spin could fail to be capable of the same feat. After all (28) is the least informative state we can assume. It is difficult to see how adding more information could make it harder to achieve the same refutation. All of this seems correct with no need to consider POVM’s. The idea that experiments on a single electron could upset local realism is quite astonishing. Is it really correct? That depends on what we mean by ‘local realism.’ Can we equate local realism with conditions HV(a)–HV(d). The identification is supported by results of Fine [Ref. 6, p. 293]. Fine, in particular, his Proposition 2. Necessary and also sufficient for the existence of a deterministic hidden-variables model is that the Bell/CH inequalities hold for the 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 on observation for their existence, but on which measurement results depend [Ref. 14, p. 3]. Locality, according to Santos [Ref. 14, p. 4], ‘is the belief that no influence may be transmitted with a speed greater than that of light.’ In his view, we have a sufficient condition for local realism if we require that correlations between measurements performed at different locations derive from events in the intersection of the past light cones. He operationalizes this by accepting Bell’s requirement that

5

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

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p(A, a; B, b) =

!

ρ(λ)P1 (λ; A, a)P2 (λ; B, b).

(29)

Here p(A, a; B, b) is the probability of getting result a for a measurement of 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 λ embodies all relevant information from the intersection of the backward light cones. Requiring Eq. (29) as a necessary condition for local realism amounts to accepting what is sometimes referred to as factorizability. Fine has long questioned whether failures of factorizability indicate non-local causation.6 We agree with Fine that the relationship between factorizability and local causality is not clear. We also point out that we are not attributing to Fine nor to Malley the claim that local experiments could establish non-local action at a distance. Rather, the claim would be that what has sometimes been called local realism would be refuted. Nonetheless, the line of thought we have been considering—a line of thought that we have derived from Bub’s argument of many years ago—would amount to a deflation of the importance of the Bell experiments. Since most people agree that persistent failures of factorizability are puzzling from a classical point of view, we need to consider just how successful such a deflation could be. The first point we want to suggest is that failures of factorizability are puzzling in a way that characterizing local realism in terms of HV(a)–HV(d) does not capture. Conditions HV(a)—HV(d), as Fine points out, are strong enough to capture 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 interpretive question. In the typical Bell-type experiments, the local observables are maximal on the local polarization or spin spaces. Thus, local contextualism—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 such a way that the functional relations among the observables are preserved. This requirement corresponds to HV(a) and HV(b). However, we seem clearly to be able to imagine a situation in which the Kochen and Specker requirement fails, and yet where it would seem perfectly appropriate to use the term ‘local realism.’ Suppose that quantum superpositions never persist when systems become space-like separated, and that for such systems, the only correct quantum mechanical predictions are ones that derive from states 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 hidden variable theory, in which there is one random variable for each locally maximal observable, but in which observables such as, for example, Sz2 for a spin-1 system are ‘split’ and correspond to different random variables for each locally maximal observable of which they are functions. Kochen and Specker’s trivial hidden variable theory is a familiar example of such a model. In a world like that, HV(a) and HV(b) would fail. But if spacelike separated systems can only exist in state of the form D1 ⊗ D2 —for short, if non-local entanglement is impossible—there would be no temptation to talk of failures of locality, nor would there be any reason to say that realism was under threat. In a perfectly straightforward sense, that world 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 should accept the following principle: Higher Hilbert Space Principle (HHS). Conclusions about a system in a given state should be the same whether or not that state results from tracing out over the state of a combined system. HHS seems reasonable. Our theory must be able to deal with more than one system at a time. However, HHS needs to be applied with a caveat: the conclusions we draw should depend on what states are possible. The argument above drew its conclusion about a single electron by assuming the existence of an entangled state for a space-like separated pair. If such states are possible, then the truth about a local system cannot be captured by a model satisfying HV(a)–HV(d). But Santos reminds us that this is a big ‘if’, and one for which we do not yet have airtight experimental evidence. The HHS can have strong implications for local systems only if a further claim that Santos regards with skepticism is true. Put another way, it could be, for all the experiments to date have definitely shown, that the quantum world is a world in which local contextualism holds, in which global contextualism fails (local observables, even though defined via the local 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 the failure of local realism. Understood in this way, what can we say about Santos’s suggestion that local realism is a fundamental principle that should only be rejected for the weightiest of reasons? Santos cites Einstein

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as a kindred spirit on this point, so it may be worth looking at what Einstein had to say. Einstein expresses his commitment to realism by saying that the concepts of physics refer to a world of things existing independent of the observer and arranged in space-time. He adds:(4) Further, it appears to be essential for this arrangement of the things introduced in physics that, at a specific time, these things claim an existence independent of one another, insofar as these things ‘lie in different parts of space.’ Without such an assumption of the mutually independent existence (the ‘being-thus’) of spatially distant 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 essential that we be able to individuate the systems we are trying to study. We think that’s plausible. What, exactly, it has to do with the Bell inequalities, however, is another question, and a rather difficult one at that. It may be (though we are not sure) that a contextual theory that required observables associated with one system to be individuated by reference to another system (a theory that violates what is sometimes called ‘separability’) would make physics impossible. However, as has often been observed, standard quantum mechanics is not such a theory. And as for whether violations of Bell-type inequalities would make physics impossible, the answer seems to be a clear no. The answer to the question ‘what would physics be like if the inequalities were violated?’ seems straightforward. It would be like quantum mechanics as most physicists already conceive of it. The very fact that Santos can tell us so clearly what would be required to observe a violation of the inequalities suggests that even if a loophole-free experiment one day confirms what Santos doubts and many others believe, physics will go on more or less as before. Nonetheless, Santos does us all a service by reminding us that there are important open questions here— even if the implications of those questions are not quite as startling as they 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’s Theorem,” http://arxiv.org/abs/quant-ph/9909073. 4. A. Einstein, “Quantenmechanik und Wirklichkeit,” Dialectica 2, 320–324 (1948). Quoted and translated in D. Howard, “Holism, separability and the metaphysical implications of the Bell experiments,” in Philosophical Consequences of Quantum Theory, J. T. Cushing and E. 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 and R. 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 quantum mechanics,” 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 commute simultaneously,” Phys. Rev. A 69, 022118 (2004). 12. J. D. Malley, “All quantum observables in a hidden-variables model must commute simultaneously,” 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).