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Why local realistic theories violate, nontrivially, the quantum mechanical EPR perfect correlations.

ABSTRACT

Using the Kochen-Specker contradiction, I prove that 'local realistic' theories predict nontrivial violations of the quantum mechanical EPR-type perfect anticorrelations. The proof invokes the same stochastic local realism conditions used in Bell arguments. For a class of theories called 'orthodox spin theories', the perfect anticorrelations used in the proof emerge from rotational symmetry. Therefore, an orthodox spin theorist must abandon either the spirit of relativity, as encoded by local realism, or the letter of relativity, which demands rotational invariance.

1 Introduction

2 Notation and Preliminary Assumptions

3 Local Realism Assumptions

4 Perfect Anticorrelations

5 The Proofs

5.1 Preliminaries 5.2 Local Realism Contradicts Fuzzy Correlations 5.3 From Fuzzy Correlations to 'Observable' Probabilities 5.4 Summary

6 Philosophical Implications 6.1 Comparison to Bell's Inequality 6.2 'Orthodox Spin' Theories 6.3 Local Realism and the Spirit of Relativity

7 Conclusion

I INTRODUCTION

Recent 'algebraic' nonlocality proofs show that any hidden-variable theory obeying certain local realism assumptions cannot reproduce the EPR-type perfect anticorrelations of quantum mechanics (QM). A perfect anticorrelation is a joint measurement result to which QM ascribes zero probability; and the QM perfect anticorrelations considered by Clifton et al. [1991a] and Elby [1990] emerge from fundamental conservation laws. By contrast, Bell inequalities show that local realism contradicts QM statistical predictions other than the perfect anticorrelations.

Elby's and Clifton's strict-correlations proofs cannot rule out local realistic theories that incorporate tiny derviations from QM's perfect anticorrelations.(1) In some such theories, angular momentum conservation and rotational invariance hold not as fundamental laws, but as 'approximate' symmetry principles such as charge-conjugation/parity (CP) invariance.

In this paper, I prove that local realistic theories must predict nontrivial ([unkeyable] 0.1%) violations of certain quantum mechanical perfect anticorrelations. I then discuss 'orthodox spin theories', in which failure of those perfect anticorrelations indicates violation of rotational invariance. Since rotational symmetry is a first principle of relativity, orthodox spin theorists must confront a tension between relativity and local realism. This tension is ironic, because local realism is motivated in part by relativity.

2 NOTATION AND PRELIMINARY ASSUMPTIONS

Consider the general class of theories in which measurement-result probabilities may depend on the 'hidden-variable' states of both the 'system' and the measuring apparatuses.

Let [lambda] denote the ontological (fully specified) state of a system. The system we'll consider consists of two correlated spin-1 particles.

Let [[mu].sub.Q] denote the ontological state of an apparatus set to measure physical quantity Q. These [lambda] and [mu] states may evolve either deterministically or stochastically in time.

In a hidden-variable framework, the quantum state [phi] is epistemic; a system in state [phi] actually occupies a more fully specified state [lambda]. Let [rho]([lambda]/[phi]) denote the probability density that a system described by quantum state [phi] occupies ontological state [lambda]. I assume that these state-occupation probability densities do not depend on whether the system is about to be measured. Call this condition Particle Locality:

Particle Locality: If a measuring-device setting is chosen outside the backward lightcone of a system, then the system's state is unaffected by that choice:

[rho]([lambda]/[phi]) = [rho]([lambda]/[phi],[[mu].sub.Q]) for all [[mu].sub.Q].

Particle Locality requires a system's state not to be instantaneously distributed by an event, specifically the preparation of a measuring device, that occurs spacelike separated from the system. (Of course, the apparatus may disturb the system after the system reaches the apparatus.) Violation of Particle Locality indicates a superluminal causal connection between the system and the measuring device (or else a pre-planned conspiracy). Such a connection is difficult to reconcile with relativity theory. Furthermore, Particle Locality is intuitive: how can a system 'know' which measurement we are about to perform?

Let [rho]([[mu].sub.Q]/[lambda],Q) be the probability density that an apparatus about to measure observable Q on a system in state [lambda] occupies microstate [[mu].sub.Q]. Similarly, [rho]([[mu].sub.Q],[[mu].sub.R]/[lambda],Q,R) is the joint probability density that apparatuses about to measure observables Q and R on a system in state [lambda] lie in microstates [[mu].sub.Q] and [[mu].sub.R], respectively.

Physically, these [rho] densities specify the distribution of hidden states underlying the quantum state of the system and of the measuring devices.

P(Q = q/[lambda],[[mu].sub.Q]) is the probability that a system in state [lambda], upon interacting with an apparatus in state [[mu].sub.Q], would yield measurement result Q = q, given that no other measurements occur. Similarly, P(Q = q,R = r/[lambda],[[mu].sub.Q],[m.sub.R]) is the joint probability that a system in state [lambda], upon interacting with apparatuses in states [[mu].sub.Q] and [[mu].sub.R], would yield Q = q and R = r, respectively.

Probability theory immediately gives

P(Q = q/[lambda]) = [integral of]P(Q = q/[lambda],[[mu].sub.Q]) [multiplied by] d[[mu].sub.Q] P(Q = q,R = r/[lambda]) = [integral of]P(Q = q,R = r/[lambda],[[mu].sub.Q],[[mu].sub.R]) [multiplied by] [rho]([[mu].sub.Q],[[mu].sub.R]/[lambda],Q,R) [multiplied by] d[[mu].sub.Q]d[[mu].sub.R], where the integrals range over all possible apparatus microstates.(2) P(Q = q/[lambda]) is the '[mu].averaged' probability that a system in state [lambda] would yield Q = q upon measurement.

According to a hidden-variable theory, the probability that a system in quantum state [phi] would yield Q = q upon measurement is found by averaging over the underlying [lambda] states:

P(Q = q/[phi] = integral of]P(Q = q/[lambda]) [multiplied by] [rho]([lambda]/[phi]) [multiplied by] d[lambda], where I've used Particle Locality. Obviously we also have

P(Q = q,R = r/[phi]) = [integral of]P(Q = q,R = r/[lambda]) [multiplied by] [rho]([lambda]/[phi]) [multiplied by] d[lambda].

These probabilities may disagree with the predictions of quantum mechanics (QM). Formally, in general P(Q = q/[phi]) [is not equal to] [P.sub.QM](Q = q/[phi]),

where [P.sub.QM](Q = q/[phi]) is the probability according to QM that a system in state [phi] would yield Q = q upon measurement; and P(Q = q/[phi]) is the probability according to the hidden-variable theory that a system in state [phi] would yield Q = q upon measurement.

In summary, I've defined three levels of measurement-result probabilities. The fundamental probabilities of the form P(Q = q/[lambda],[[mu].sub.Q]) depend on the system's state and also on the measuring apparatus's microstate. By averaging over apparatus microstates, we obtain probabilities of the form P(Q = q/[lambda]), which specify the statistical behavior of [lambda]-systems upon interacting with devices macroscopically set to measure Q. Finally, we average over the [lambda] states underlying the quantum state to obtain P(Q = q/[phi]). According to the hidden-variable theory, this [lambda]-averaged probability predicts the statistics we would 'observe' by measuring Q on many systems prepared in quantum state [phi], assuming the hidden states are uncontrollable.

3 LOCAL REALISM ASSUMPTIONS

In the proofs of Section 5, I consider a system comprised of two well-separated spin-1 particles. In QM, physical quantities associated with this system correspond to operators that 'live' in the product Hilbert space [H.sub.1] [cross product] [H.sub.2], where [H.sub.1] ([H.sub.2]) is the Hilbert space associated with particle 1 (2). Operators corresponding to particle 1 and 2 take the form Q [cross product] 1 and 1 [cross product] R, respectively, where 1 is the identity operator. We restrict attention to 'discrete' operators, those with a discrete (as opposed to continuous) set of eigenvalues.

Let {[r.sub.1], [r.sub.2], . . ., [r.sub.n]} be the possible results of measuring observable R. As a convenient shorthand, define

P(Q [cross product] 1 = q/[lambda],[[mu].sub.Q,[[mu].sub.R]) [equivalent][[sigma].sub.i]P(Q [cross product] 1 = q, 1 [cross product] R = [r.sub.i]/[lambda],[[mu].sub.Q],[[mu].sub.R]).

Physically, P(Q [cross product] 1 = q/[lambda],[[mu].sub.Q],[[mu].sub.R]) is the probability that Q-measurement of particle 1 (with an apparatus in microstate [[mu].sub.Q]) and R-measurement of particle 2 (with apparatus in microstate [[mu].sub.R]) would yield q for Q. Assume throughout that the measurements of particle 1 and 2 occur at spacelike separation.

My first two local [realism.sup.3] assumptions are the standard locality and completeness conditions from Jarrett [1984]:

Locality:

P(Q [cross product] 1 = q/[lambda],[[mu].sub.Q]) = P(Q [cross product] 1 = q/[lambda],[[mu].sub.Q],[[mu].sub.R])

Completeness:

[Mathematical Expression Omitted]

Locality demands that a measurement-result probability not depend on the setting or microstate of a distant apparatus. Locality violation would allow superluminal signalling if the hidden-variable states were sufficiently controllable, as Jarrett shows. For this and other reasons, violation of Locality almost certainly constitutes causal action at a distance (or else a pre-planned conspiracy between particles and apparatuses). Such nonlocal causality is difficult to reconcile with relativity theory. QM obeys Locality.

Completeness is often written

[Mathematical Expression Omitted]

which is equivalent to the above for nonzero P(Q[cross product]1 = q, 1[cross product]R = r|[lambda],[[mu].sub.Q],[[mu].sub.R]). Completeness requires that a measurement-result probability depend only on the ontological state of the particles and apparatuses, not on the result of a spacelike separated measurement. Put another way, the particle and apparatus states must be the Reichenbachian [1956] 'common cause' of the correlated measurement results.

Let me justify Completeness intuitively. Since particles 1 and 2 are correlated, we expect that measuring particle 2 might give us previously unknown information about particle 1, thereby changing our epistemic measurement-result probabilities associated with particle 1. But an ontological (objective) probability reflects the particle's actual physical state, not our knowledge of the state. So a change in a particle's ontological probabilities indicates a real physical change in the particle's ontological state. Therefore, we don't intuitively expect that obtaining the measurement result on particle 2 simultaneously 'influences', at a distance, the ontological probabilities associated with particle 1. Completeness rules out precisely this kind of influence. Such an influence constitutes a holistic noncausal connection between the two particles, as Redhead [1987], Elby [1992a], and others argue. This nonlocal, noncausal connection violates the spirit of relativity, as I'll show in Section 6. But Completeness violation does not contradict the formalism of relativity. For instance, although QM and quantum field theory violate Completeness, Lorentz invariant quantum field formalisms exist.

Deterministic theories necessarily obey Completeness, though the converse fails. See Elby [1990] for a proof.

In summary, Completeness allows measurement of particle 2 to reveal previously unknown information about particle 1, but prohibits the particle 2 measurement outcome from actually changing instantaneously the ontological propensities of particle 1.

Locality and Completeness taken together are equivalent to Factorizability:

Factorizability:

[Mathematical Expression Omitted]

which requires that probabilities of joint measurement outcomes factor into probabilities associated with the two particles separately.

I assume that the probability distributions of the apparatus microstates are similarly factorable:

Total Apparatus Factorability (TAF):

[Mathematical Expression Omitted]

TAF encodes two physical intuitions. First, the two measuring devices are ontologically separable; they are not holistically connected, and therefore it makes sense to specify the states of the two devices separately. Second, the probability that a measuring device occupies a certain microstate does not depend on the setting or microstate of a distant apparatus. If TAF fails, then by changing the state of one apparatus, we could alter the state, or at least the state-occupation probabilities, of the other apparatus. Such a dependence would indicate either a conspiracy or an instantaneous connection between the two devices. In short, the local realism intuitions underlying TAF are almost the same as those underlying Factorizability.

Factorizability, TAF, and Particle Locality (from Section 2) are the standard primary assumptions used to derive a stochastic Bell inequality; see Brown and Svetlichny [1990] and Clifton et al. [1991].

Bell derivations also rely on the following auxiliary condition.(4)

Spectrum Rule:

If z is not an eigenvalue of Q, then P(Q = z|[lambda]) = 0 unless [rho]([lambda]|[phi]) = 0 for all quantum states [phi].

The Spectrum Rule requires that a hidden-variable theory ascribe nonzero probability only to measurement outcomes permitted by QM. Violation of Spectrum Rule constitutes an immediate contradiction with QM's predictions.

Summary: in this section, I introduced two main conditions, Factorizability (i.e., Locality and Completeness) and TAF. In Section 2, I introduced Particle Locality. These local realism assumptions encode our intuition that no instantaneous connection exists between distant, spacelike separated events.

4 PERFECT ANTICORRELATIONS

Previous proofs show the inconsistency of local realism--Factorizability, Particle Locality, and TAF--with the EPR-type perfect anticorrelations of quantum mechanics. A perfect anticorrelation is a joint measurement result to which QM ascribes zero probability. Clifton et al. [1991a] and Elby [1990] show that local realistic theories cannot reproduce certain quantum mechanical perfect anticorrelations. These perfect anticorrelations are more fundamental than QM's general statistical predictions, for the following reason.

The perfect anticorrelations considered by Clifton and Elby emerge from fundamental conservation principles. For instance, when [phi] is the spin singlet state of two vector mesons, and [S.sub.n] is the n-component of spin, then [P.sub.QM]([S.sub.n][cross product]1 = + 1, 1[cross product][S.sub.n] = + 1|[phi]) = 0. This perfect anticorrelation reflects conservation of angular momentum, which in turn emerges from rotational invariance.

We know, however, that some conservation laws are approximate instead of absolute. A good example is charge-conjugation/parity (CP) invariance, originally considered fundamental, but now thought to be violated by 'weak nuclear' interactions. Perhaps rotational invariance, like CP invariance, is only approximate. In that case, the perfect anticorrelations predicted by QM for the spin singlet state would quite possibly fail. None the less, the failure might be tiny, so that an empirically adequate hidden-variable theory would almost reproduce those QM anticorrelations. Formally, if rotational invariance fails only minutely, we expect the following Near-Perfect Correlations condition to hold:

Near-Perfect Correlations:

[Mathematical Expression Omitted]

where [delta] is of order 1/1000, and where [P.sub.QM[ (Q[cross product]1 = q, 1[cross product] = r|[phi]) = 0 is any QM perfect anticorrelation stemming from conservation of angular momentun. Recall that the antecedent is a probability according to QM, while the consequent is a probability according to the hidden-variable theory. (Throughout this paper, '[unkeyable]' denotes entailment, while '[right arrow]' denotes material implication.)

Near-Perfect Correlations allows a theory to violate, minutely, certain perfect anticorrelations. A hidden-variable theorist could posit such violations for many reasons besides rotational non-invariance; see Section 6.2. But in some theories, failure of Near-Perfect Correlations indicates that rotational invariance fails utterly, and cannot be considered even approximate.

I now prove that for [delta]<1|(9 x 3 x 43) = 1|1161, Near-Perfect Correlations is inconsistent with the local realism assumptions discussed above.

5 THE PROOFS

Using the Kochen-Specker contradiction, I prove in theorem 1 that local realism is inconsistent with a 'Fuzzy Correlations' condition. I then show in theorem 2 that Near-Perfect Correlations implies Fuzzy Correlations, completing the proof that Near-Perfect Correlations contradicts local realism. Schematically,

Near-Perfect Correlations [unkeyable] Fuzzy Correlations [theorem 2]

Fuzzy Correlations [unkeyable] Failure of local realism [theorem 1] Therefore

Near-Perfect Correlations [unkeyable] Failure of local realism.

For presentational reasons, I reverse the order of the logic and prove theorem 1 first.

In the proof, I employ a [mu]-averaged version of factorizability:

[mu]-less Factorizability:

P(Q[cross product]1 = q, 1[cross product]R = r|[lambda]) = P(Q[cross product]1 = q|[lambda]).P(1 [cross product]R = r|[lambda]).

Interestingly, [mu]-less Factorizability follows from the local realism conditions already introduced:

Factorizability lemma:

Factorizability & TAF [unkeyable] [mu]-less Factorizability

I omit the easy proof. This lemma makes intuitive sense: if the fundamental [mu]-dependent probabilities associated with the two wings of the experiment are independent, as required by Factorizability & TAF, then clearly that independence remains after we average over [mu] states.

5.1 Preliminaries

In theorem 1, we consider two well-separated spin-1 particles in the spin singlet state

[psi] =

[-3.sup.-1/2](|[S.sub.x] = 0>[cross product]|[S.sub.x] = 0>-|[S.sub.y] = 0>[cross product]| [S.sub.y] = 0> + |[S.sub.z] = 0>[cross product]|[S.sub.z] = 0>).

For distinct nonzero numbers {a,b,c} and for orthogonal triad of directions [theta] = {x,y,z},

[H.sub.[theta]] = [aS.sub.x.sup.2] + ]bS.sub.y.sup2] + [cS.sub.z.sup.2]

is the spin-Hamiltonian operator, where [S.sub.n] is the n-component-of-spin operator. (The members of an orthogonal triad point at right angles to each other). In units of [unkeyable], the eigenvalues of [H.sub.[theta]] are {[h.sub.x] = b + c, [h.sub.y] = a + c.[h.sub.z] = a + b}. The eigenvalues of [S.sub.n] are {- 1,0 + 1}. Physically, [H.sub.[theta]] is the first-order energy perturbation experienced by an orthohelium atom in an electric field of orthorhombic symmetry; cf. Kochen and Specker [1967]. Recall that [H.sub.[theta]][cross product]1 and 1[cross product] [S.sub.n] are associated with the first and second particle, respectively.

5.2 Local Realism Contradicts Fuzzy Correlations

In theorem 1, I will consider 129 well-chosen quantum mechanical perfect anticorrelations of the spin singlet state [psi]. That's how many anticorrelations we need to complete a Kochen-Specker proof, as we'll see in Section 5.3. These anticorrelations all involve measuring a spin Hamiltonian on particle 1 and a spin component on particle 2. Formally, these 129 anticorrelations take the form [P.sub.QM]([H.sub.[theta]i] [cross product] 1 = [h.sub.i], 1 [cross product] [S.sub.i] = [s.sub.i]|[psi]) = 0, where [H.sub.[theta]i], [h.sub.i], [S.sub.i], and [s.sub.i] are the spin-Hamiltonian, spin-Hamiltonian measurement result, spin component, and spin-component measurement result corresponding to the i-th anticorrelation. My Fuzzy Correlations assumption, stated in terms of [mu]-averaged probabilities, is this:

Fuzzy Correlations: For a nonzero-measure set of [lambda] states underlying [psi], [P.sub.QM]([H.sub.[theta]i] [cross product] 1 = [h.sub.i], 1 [cross product] [S.sub.i] = [s.sub.i]|[psi]) = 0 [unkeyable] P([H.sub.[theta]i] [cross product] 1 = [h.sub.i], 1 [cross product] [S.sub.i] = [s.sub.i]|[lambda]) [less than] 1/9 for each of these 129 QM perfect anticorrelations.

Fuzzy Correlations requires some of the hidden states underlying [psi] to 'fuzzily' mirror 129 perfect anticorrelations of [psi]. Later, I prove that Near-Perfect Correlations implies Fuzzy Correlations when [delta] [less than] 1/1161.

Theorem 1:

Factorizability & TAF & Particle Locality & Spectrum Rule & Fuzzy Correlations[unkeyable]Kochen-Specker contradiction.

Proof:

Let [lambda] be a state such that the implication [P.sub.QM]([H.sub.[theta]i] [cross product] 1 = [h.sub.i], 1 [cross product] [S.sub.i] = [s.sub.i]|[psi]) = 0 [unkeyable] P([H.sub.[theta]i] [cross product] 1 = [h.sub.i], 1 [cross product] [S.sub.i] = [s.sub.i]|[lambda]) [less than] 1/9 holds for each of the 129 anticorrelations considered in this proof. By Fuzzy Correlations, a nonzero-measure set of [lambda] states satisfies this condition.

We'll now focus on three of these 129 perfect anticorrelations, namely those involving [H.sub.[theta]] [cross product] 1 for [theta] = {x,y,z}.

* Suppose P([H.sub.[theta]] [cross product] 1 = [h.sub.x]|[lambda]) [greater than or equal to] 1/3.

From QM, we have the following three perfect anticorrelations:

(a) [P.sub.QM]([H.sub.[theta]] [cross product] 1 = [h.sub.x], 1 [cross product] [S.sub.x] = [+ or -] 1|[psi]) = 0, (b) [P.sub.QM]([H.sub.[theta]] [cross product] 1 = [h.sub.x], 1 [cross product] [S.sub.y] = 0|[psi]) = 0, (c) [P.sub.QM]([H.sub.[theta]] [cross product] 1 = [h.sub.x], 1 [cross product] [S.sub.z] = 0|[psi]) = 0.

Applying Fuzzy Correlations to those three equalities yields

(a) P([H.sub.[theta]] [cross product] 1 = [h.sub.x], 1 [cross product] [S.sub.x] = [+ or -] 1|[lambda]) [less than] 1/9, (b) P([H.sub.[theta]] [cross product] 1 = [h.sub.x], 1 [cross product] [S.sub.y] = 0|[lambda]) [less than] 1/9, (c) P([H.sub.[theta]] [cross product] 1 = [h.sub.x], 1 [cross product] [S.sub.z] = 0|[lambda]) [less than] 1/9.

Applying [mu]-less Factorizability yields

(a) P([H.sub.[theta]] [cross product] 1 = [h.sub.x]|[lambda]) [multiplied by] P(1 [cross product] [S.sub.x] = [+ or -] 1|[lambda]) [less than] 1/9, (b) P([H.sub.[theta]] [cross product] 1 = [h.sub.x]|[lambda]) [multiplied by] P(1 [cross product] [S.sub.y] = 0|[lambda]) [less than] 1/9. (c) P([H.sub.[theta]] [cross product] 1 = [h.sub.x]|[lambda]) [multiplied by] P(1 [cross product] [S.sub.z] = 0|[lambda]) [less than] 1/9.

By supposition, P([H.sub.[theta]] [cross product] 1 = [h.sub.x]|[lambda] [greater than or equal to] 1/3. From simple algebra it follows that

(a) P(1 [cross product] [S.sub.x] = [+ or -] 1|[lambda]) [less than] 1/3, (b) P(1 [cross product] [S.sub.y] = 0|[lambda]) [less than] 1/3, (c) P(1 [cross product] [S.sub.z] = 0|[lambda]) [less than] 1/3.

According to the Spectrum Rule, the only three outcomes of measuring [S.sub.x] that have nonzero probability are {- 1,0, + 1}. By normalization we then have P(1 [cross product] [S.sub.x] = - 1|[lambda]) + P(1 [cross product] [S.sub.x] = 0|[lambda]) + P(1 [cross product] [S.sub.x] = + 1|[lambda]) = 1.

From this and inequality (a), algebra yields P(1 [cross product] [S.sub.x] = 0|[lambda]) [greater than] 1 - 2/3, that is (a [feet]) P(1 [cross product] [S.sub.x] = 0|[lambda]) [greater than] 1/3.

In summary, we have

(a [feet]) P(1 [cross product] [S.sub.x] = 0|[lambda]) [greater than] 1/3, (b) P(1 [cross product] [S.sub.y] = 0|[lambda]) [less than] 1/3, (c) P(1 [cross product] [S.sub.z] = 0|[lambda]) [less than] 1/3.

Now I define the following mathematical step function, which has no physical interpretation or importance:

K(x) = 0 if x [less than] 1/3 1 if x [greater than or equal to] 1/3.

Applying the K function to inequalities (a [feet]), (b), and (c) yields

(a [feet]) K(P(1 [cross product] [S.sub.x] = 0|[lambda])) = 1, (b) K(P(1 [cross product] [S.sub.y] = 0|[lambda])) = 0, (c) K(P(1 [cross product] [S.sub.z] = 0|[lambda])) = 0.

This conclusion followed from Factorizability, TAF, Fuzzy Correlations, Spectrum Rule, and the supposition that P([H.sub.[theta]] [cross product] 1 = [h.sub.x]|[lambda]) [greater than or equal to] 1/3. If we suppose instead that P([H.sub.[theta]] [cross product] 1 = [h.sub.y]|[lambda]) [greater than or equal to] 1/3, similar reasoning yields

(a) K(P(1 [cross product] [S.sub.x] = 0|[lambda])) = 0, (b [feet]) K(P(1 [cross product] [S.sub.y] = 0|[lambda])) = 1, (c) K(P(1 [cross product] [S.sub.z] = 0|[lambda])) = 0.

Or, if we suppose P([H.sub.[theta]] [cross product] 1 = [h.sub.z]|[lambda]) [greater than or equal to] 1/3, we get

(a) K(P(1 [cross product] [S.sub.x] = 0|[lambda])) = 0, (b) K(P(1 [cross product] [S.sub.y] = 0|[lambda])) = 0, (c) K(P(1 [cross product] [S.sub.z] = 0|[lambda])) = 1.

In summary, if P(H[theta] [cross product] 1 = [h.sub.x]|[lambda]) [greater than or equal to] 1/3, or if P(H[theta] [cross product] 1 = [h.sub.y]|[lambda]) [greater than or equal to] 1/3, or if P(H[theta] [cross product] 1 = [h.sub.z]|[lambda]) [greater than or equal to] 1/3, then the three values

{K(P(1[cross product] [S.sub.x] = 0|[lambda])), K(P(1 [cross product] [S.sub.y] = 0|[lambda])), K(P(1[cross product] [S.sub.z] = 0|[lambda]))}

are such that two of the values equal 0 while the third value equals 1. But by normalization and the Spectrum Rule,

P(H[theta] [cross product] 1 = [h.sub.x]|[lambda]) + P(H[theta] [cross product] 1 = [h.sub.z]|[lambda]) = 1,

from which it follows that at least one of those three spin-Hamiltonian measurement-result probabilities is greater than or equal to 1/3. Therefore, the three values

{K(P(1[cross product] [S.sub.x] = 0|[lambda])), K(P(1[cross product] [S.sub.y] = 0|[lambda])), K(P(1 [cross product] [S.sub.z] = 0|[lambda]))}

are indeed such that two of the values equal 0 while the third value equals 1.

This conclusion followed from considering three quantum mechanical perfect anticorrelations involving H[theta] [cross product] 1 for [theta] = {x,y,z}. Due to the spherical symmetry of the quantum spin singlet state [psi], the same argument applies to any orthogonal triad of directions [theta]' = {x', y', z'}. That is, for any orthogonal triad of directions {x', y', z'}, the above-listed local realism conditions along with Fuzzy Correlations imply that of the three values

{K(P(1 [cross product] [S.sub.x'] = 0|[lambda])), K(P(1 [cross product] [S.sub.y'] = 0|[lambda])), K(P(1 [cross product] [S.sub.z'] = 0|[lambda]))},

two values equal 0 while the third value equals 1.

To complete a Kochen-Specker proof, we must apply this argument to 43 well-chosen orthogonal triads corresponding to 3 x 43 = 129 anticorrelations. More on this in Section 5.3. Particle Locality implies that [lambda] is consistent with measurement of all these different observables.

Each point on the unit sphere is associated 1:1 with a unit vector (i.e., a direction) n. For each n in the 43 orthogonal triads, map to n the value K(P(1 [cross product] [S.sub.n] = 0|[lambda])). As just shown, this map is such that for any orthogonal triad, two points take on the value 0 while the third point takes on the value 1. But such a map is impossible, by Kochen and Specker's [1967].

Q.E.D.

In summary, theorem 1 shows that the local realism and auxiliary conditions needed to derive a stochastic Bell inequality--Factorizability, TAF, Particle Locality, and Spectrum Rule--are inconsistent with Fuzzy Correlations. Hence, in a local realistic theory, a nonzero-measure set of [lambda] states must encode a substantial (i.e., greater than 1/9) violation of at least one of the 129 perfect anticorrelations needed to complete the Kochen-Specker proof. I build upon this argument in the next subsection.

5.3 From Fuzzy Correlations to 'Observable' Probabilities

I now prove that for the appropriate choice of [delta], Near-Perfect Corrleations implies the Fuzzy Correlations condition used in theorem 1. To do so, I must first discuss some technicalia from Kochen and Specker's proof.

Essentially, Kochen and Specker 'color' a sphere with 43 well-chosen orthogonal triads of points. To see this, check the Kochen-Specker diagram in Redhead [1987]. Chapter 5. Each point in an orthogonal triad lies 90 [degrees] from the other two points. Those 43 orthogonal triads consist of only 117 points instead of 43 x 3 = 129, because a few points 'belong' to more than one triad. Kochen and Specker prove the impossibility of assigning values '0' and '1' to those 117 points such that for any orthogonal triad, two points have the value 0 while the third has the value 1.

To complete theorem 1, therefore, I invoked 43 different orthogonal triads [theta].sub.i], each corresponding to a different spin-Hamiltonian operator [H.sub[theta]i]. For each [H.sub.[theta]i], I assumed that Fuzzy Correlations holds with respect to the perfect anticorrelations between [H.sub.[theta]i] and three components of spin on the other particle; see Section 5.2. So in total I used 3 x 43 = 129 anticorrelations.

Streamlined Kochen-Specker-type arguments may show that we need fewer than 129 anticorrelations to reach a contradiction. To account for that possibility, let N denote the minimum number of anticorrelations needed to complete a Kochen-Specker-style proof.

We now have

Theorem 2: For [delta] [less than] 1/9 N,

Near-Perfect Correlations=>Fuzzy Correlations.

Proof:

By contradiction. Suppose Fuzzy Correlations fails. Then, for each [lambda] underlying [psi] (except [lambda's] belonging to a zero-measure subset), P([H.sub.[theta]i] [cross product] 1 = [h.sub.i], 1 [cross product] [S.sub.i] = [S.sub.i]|[lambda]) [greater than or equal to] 1/9 for at least one of the N anticorrelations used in the Kochen-Specker proof. (Recall from above that [H.sub.[theta]i], [h.sub.i], [S.sub.i], and [S.sub.i] correspond to the i-th anticorrelation, where i ranges from 1 to N.) In other words, all [lambda] states for which [rho]([lambda]|[psi]) [greater than] 0 belong to at elast one set {[lambda].sub.i]}, where {[lambda].sub.i]} denotes the set of states for which P([H.sub.[theta]i] [cross product] 1 = [h.sub.i], 1 [cross product] [S.sub.i] = [S.sub.i] = [S.sub.i]|[lambda]) [greater than or equal to] 1/9. Since all [lambda] states for which [rho]([lambda]|[psi]) [greater than] 0 fall into at least one of these {[lambda].sub.i]} sets, measure theory trivially implies

(*) m{[lambda].sub.1]} + m{[lambda].sub.2]} + m{[lambda].sub.3]} + ... + m{[lambda].sub.N]} [greater than or equal to] 1,

where m{[lambda].sub.i]} is the [rho]-measure of set {[lambda].sub.i]}. Formally,

m{[[lambda].sub.i]} [equivalent] {[[lambda].sub.i]}[integral of][rho]([lambda]|[psi]) [multiplied by] d[lambda],

where as indicated the integral ranges only over states in {[[lambda].sub.i]}.(5)

From (*), it follows that for at least one i, m{[[lambda].sub.i]} [greater than or equal to] 1/N. Consider that i. Recall that for each member of {[[lambda].sub.i]}, P([H.sub.[theta]i] [cross product] 1 = [h.sub.i], 1 [cross product] [S.sub.i] = [s.sub.i]|[lambda]) [greater than or equal to] 1/9 even though [P.sub.QM]([H.sub.[theta]i] [cross product] 1 = [h.sub.i], 1 [cross product] [S.sub.i] = [s.sub.i]|[psi]) = 0. So the relevant 'observable' probability according to the hidden-variable theory is

P([H.sub.[theta]i] [cross product] 1 = [h.sub.i], 1 [cross product] [S.sub.n] = [s.sub.i]|[psi]) = [integral of]d[lambda] [multiplied by] P([H.sub.[theta]i] [cross product] 1 = [h.sub.i], 1 [cross product] [S.sub.n] = [s.sub.i]|[lambda]) [multiplied by] [rho]([lambda]|[psi]) [greater than or equal to] (1/9) [multiplied by] (1/N),

since P([H.sub.[theta]i][cross product] 1 = [h.sub.i], 1 [cross product] [S.sub.i] = [s.sub.i]|[lambda]) [greater than or equal to] 1/9 for each [lambda] in a set of [rho]-measure m{[[lambda].sub.i]} [greater than or equal to] 1/N.

But according to Near-Perfect Correlations with [delta] [less than] 1/9N,

P([H.sub.[theta]i] [cross product] 1 = [h.sub.i], 1 [cross product] [S.sub.i] = [s.sub.i]|[psi]) [less than] (1/9) [multiplied by] (1/N).

This completes to proof by contradiction.

Q.E.D.

5.4 Summary

Theorem 2 shows that if P([H.sub.[theta]i] [cross product] 1 = [h.sub.i], 1 [cross product] [S.sub.i] = [s.sub.i]|[psi]) [less than] 1/9N for each of the N anticorrelations considered in theorem 1, then Fuzzy Correlations holds. But if Fuzzy Correlations holds, then the hidden-variable theory violates local realism (Factorizability & TAF & Particle Locality) or the Spectrum Rule, by theorem 1. Therefore, with N = 129 and 9 x 129 = 1161, we immediately have

Theorem 3: Near-Perfect Correlations (with [delta] [less than] 1/1161) & Factorizability & TAF & Particle Locality & Spectrum Rule [right arrow] Kochen-Specker contradiction.

Theorem 3 is the central technical result of this paper.

6 PHILOSOPHICAL IMPLICATIONS

6.1 Comparison to Bell's Inequality

As discussed in Section 4, Near-Perfect Correlations requires a hidden-variable theory to incorporate, at least approximately, the perfect anticorrelations of the quantum spin-1 singlet state. Bell inequalities show the inconsistency between local realism and QM statistical predictions other than the perfect anticorrelations. Theorem 3, however, shows the inconsistency between local realism and the EPR-type perfect anticorrelations. If you view these perfect anticorrelations as more fundamental than QM's general statistical predictions, then Theorem 3 improves on Bell's theorem by showing that a local realistic theory cannot reproduce even the conservation-motivated 'heart' of quantum mechanics.

I now argue that for an important class of theories, certain perfect anticorrelations emerge directly from a fundamental symmetry, rotational invariance. Rotational invariance is a first principle of relativity. Therefore, for that class of theories, either relativity fails or local realism fails. This is ironic, because relativity theory motivates local realism.

6.2 'Orthodox Spin' Theories

As discussed in Section 4, in quantum mechanics the perfect anticorrelations invoked in theorem 1 emerge from conservation of angular momentum, which in turn follows from rotational invariance. In some hidden-variable theories, those anticorrelations also emerge from rotational invariance. Call such constructions 'orthodox spin' theories.

Orthodox spin theory: Let T denote all the first principles of a theory other than rotational invariance. The theory is an orthodox spin theory iff (a) Spin 'observables' obey the Spectrum rule, and (b) T & (rotational invariance) [right arrow] (the perfect anticorrelations of theorem 1).

This definition does not presuppose that rotational invariance is a postulate of an orthodox spin theory, but does suppose rotational invariance to be consistent with T.

In some theories, of course, rotational invariance doesn't appear as a separate first principle, but instead gets 'built into' other postulates. If such a theory obeys the Spectrum Rule (for spin observables) and reproduces the perfect anticorrelations of theorem 1, then it's an orthodox spin theory.

We now have

Theorem 4: An orthodox spin theory either violates local realism or violates relativity.

which follows trivially from theorem 3, the definition of orthodox spin theories, and the fact that rotational invariance is a first principle of relativity.

Later, I'll discuss the dilemma theorem 4 poses for orthodox spin theorists. But, first, I explore which theories fit that description.

Any theory obeying these three conditions is an orthodox spin theory:

(A) Some particles display an intrinsic ('spin') angular momentum of magnitude h = h/2[pi], where h is Planck's constant. Measurement of a spin component yields [+ or -] h or 0.

(B) For those particles, the perfect anticorrelations invoked in theorem 1 follow, in part, from conservation of angular momentum.

(C) Conservation of angular momentum follows from rotational invariance. Condition (A) receives strong, though indirect support from Stern-Gerlach-type experiments. To claim those experiments support (A), we must assume a certain relation between a particle's spin and magnetic moment. (The copious direct evidence from particle accelerator experiments that spin-1 particles exists is 'evidence' only if we assume conservation of angular momentum--an assumption we can't make lightly in the present discussion!)

Condition (C) holds not only for quantum mechanics and quantum field theory, but also for classical mechanics, classical electrodynamics, and special relatively. Noether's theorem (cf. Ryder [1985]) shows that whenever equations of motion can be derived via variational calculus from a Lagrangian, symmetries of the Lagrangian lead to conserved quantities. I know of no present theory in which rotational invariance doesn't lead to a conserved 'angular momentum' quantity.

Condition (B) is perhaps the fishiest. In quantum mechanics, (B) holds because spin is quantized and particles exist in 'superposition' states of indefinite n-component of angular momentum, among other reasons. A general theory might not incorporate all these features, and hence (B) could fail. (B) could also fail because an undetected form of angular momentum or of spin-orbit coupling exists. Yet, (B) may hold for a large class of theories that propose small corrections to quantum mechanics without overhauling the whole theory.

(A), (B), and (C) are sufficient conditions for an orthodox spin theory. But they aren't necessary. Notably, David Bohm's construction violates (B) and (A) but is none the less an 'orthodox spin' theory. In his theory, particles don't 'have' intrinsic angular momentum. Stern-Gerlach experiments turn out the way they do because of an elaborate interaction, mediated by a 'quantum potential', between the particle and the measuring apparatus. None the less, in his theory, the EPR perfect anticorrelations emerge from rotational invariance of the relevant quantum potential. See Bohm et al. [1987].

In summary: although we have limited a prior motivation for singling out orthodox spin theories, such theories are plausible and important. Quantum mechanics itself, along with the best-developed hidden-variable construction (Bohm's), are orthodox spin theories. Therefore, no-go results such as theorem 4 deserve philosophical analysis.

In the next subsection, I show that orthodox spin theorists must renounce at least the spirit of relativity.

6.3 Local Realism and the Spirit of Relativity

As theorem 4 shows, an orthodox spin theorist must abandon either rotational invariance or local realism. Failure of local realism violates at least the spirit of relativity theory, as I now argue.

In my view, the spirit of relativity demands that the physical characteristics of a system (and its measuring device) can be affected only by events or states-of-affairs in the backward lightcone of that system (and measuring device). Therefore, by the spirit of relativity, neither putting the particle-2 measuring device into a certain state, nor obtaining a measurement result on particle 2, may instantaneously affect the ontological measurement-result probabilities associated with particle 1 and its measuring apparatus.

Assuming no 'conspiracies', failure of Locality, Particle Locality, or TAF almost certainly constitutes action at a distance, in violation of the spirit of relativity. And recall from Section 3 that when Completeness fails, obtaining a measurement outcome on particle 2 actually changes the propensities of particle 1, instantaneously at a distance. Therefore, violation of Completeness, though consistent with the relativistic formalism, constitutes a nonlocal connection that also violates the spirit of relativity.

So theorem 4 raises a dilemma for orthodox spin theorists. Either they must abandon local realism, thereby violating the spirit of relativity; or they must abandon rotational invariance, thereby contradicting the formalism of relativity. The irony is this: even though local realism encodes the spirit of relativity, local realism is logically inconsistent with relativity for orthodox spin theories.

Previous results point us in this direction, though not quite as strongly. Since Bell derivations don't use the perfect anticorrelations, Bell arguments do not engender as direct a tension between local realism and rotational invariance. The 'algebraic' proofs of Elby [1990] and Clifton et al. [1991a] show that local realism is inconsistent with the perfect anticorrelations of QM. Formally, they prove that Factorizability & TAF & Particle Locality contradict Near-Perfect Correlations for [delta] = 0.(6) Those arguments show that in a local realistic orthodox spin theory, rotational invariance fails. But if that failure is minute, then rotational invariance could still hold as an approximate symmetry. Theorems 3 and 4, however, suggest that a local realistic orthodox spin theory cannot incorporate rotational symmetry even approximately.

Clifton et al. [1991a], working along different lines, have also derived an 'imperfect correlations' algebraic proof. Their work can be used to show that Near-Perfect Correlations contradicts local realism, though they do not do so explicitly. The advantage of their proof, which does not invoke the Kochen-Specker contradiction, is its reliance on a very small number of anticorrelations. As a result, the [delta] that Clifton et al. would get in their Near-Perfect Correlations condition is about 200 times larger than mine.

Clifton et al. stress the experimental implications of their imperfect correlations proof. Specifically, they believe the predictions of QM are correct, so that experimental deviations from the perfect correlations stem from detector inefficiences. Only an ideal detector could confirm QM's perfect correlations. But an imperfect detector can verify Near-Perfect Correlations for large enough [delta]. Therefore, Clifton's work allows a practical perfect-correlations experiment to rule out local realism. Furthermore, Clifton et al. note, their experiment could improve slightly on Bell-type experiments by showing that a higher fraction (i.e., measure) of [lambda] states contradict one of the local realism conditions. See Clifton et al. [1991a] for details.

My focus, on the other hand, is more abstractly philosophical. Independent of whether an experiment can in practice verify my Near-Perfect Correlations assumption (with [delta] [less than] 1/1161), I'm interested in the dilemma raised by the logical contradiction between Near-Perfect Correlations and local realism. This contradiction forces a local realist to deny that certain quantum correlations hold even approximately. And this contradiction forces an orthodox spin theorist to renounce either the spirit of relativity theory as encoded by local realism, or relativity theory itself.

7 CONCLUSION

In this paper, I first discussed the intuitively compelling local realism assumptions needed to derive a Bell inequality. Those conditions were shown to contradict Near-Perfect Correlations, according to which certain quantum mechanical EPR-type perfect anticorrelations hold at least approximately. By demonstrating a direct and irreparable conflict between local realism and these perfect anticorrelations, the proof improves slightly on Bell inequalities and on algebraic strict-correlations derivations.

For an important and plausible class of constructions called orthodox spin theories, failure of Near-Perfect Correlations indicates nontrivial violation of rotational invariance. Orthodox spin theorists must either attack the spirit of relativity theory by renouncing local realism, or abandon relativity entirely by renouncing rotational invariance.

Often, philosophers and physicists characterize Bell derivations and other nonlocality 'no-go' theorems as underscoring a tension between QM's predictions and the spirit of relativity as expressed by local realism. It is surprising and ironic, therefore, that at least for orthodox spin theorists, a tension--indeed a contradiction--exists between local realism and relativity theory.

(1) Clifton et al. [1991a] also contains an 'imperfect correlations' proof which I'll discuss in Section 6. His arguement uses the Greenberger-Horne-Zeilinger perfect correlations (see Greenberger et al. [1989]). Clifton's proof, like mine in this paper, rules out local realistic theories incorporating small deviations from certain quantum mechanical perfect anticorrelations.

(2) Currently, P(Q = q,R = r|[lambda],[[mu].sub.Q],[[mu].sub.R]) is undefined when [[mu].sub.Q] and [[mu].sub.R] are incompatible. But we can define P(Q = q,R = r|[lambda],[[mu].sub.Q],[[mu].sub.R]) to equal an arbitrary constant when [[mu].sub.Q] and [[mu].sub.R] are incompatible. Our choice of constant has no effect on the value of P(Q = q,R = r|[lambda]) = [integral of]P(Q = q,R = r|[lambda],[[mu].sub.Q], [[mu].sub.R]) [multiplied by] [rho]([[mu].sub.Q],[[mu].sub.R]|[lambda],Q,R) [multiplied by] [d[mu].sub.Q][d[mu].sub.R], because [rho]([[mu].sub.Q],[[mu].sub.R]|[lambda],Q,R) = 0 for incompatible [[mu].sub.Q] and [[mu].sub.R].

Technically, I should define P(Q = q,R = r|[lambda]) [equivalent] [integral of]P(Q = q,R = r|[lambda],[[mu].sub.Q],[[mu].sub.R]) [multiplied by] d[rho]([[mu].sub.Q],[[mu].sub.R]|[lambda],Q,R). In light of my 'total apparatus factorability' (TAF) assumption in Section 3, the two definitions are equivalent.

(3) So-called 'local realism' conditions neither assume, imply, nor deny realism as a metaphysical outlook. A hard-core realist would say that local realism conditions refer to actual states of particles and their interactions with measuring devices. An antirealist would characterize local realism assumptions as constraints on our description of nature.

(4) To see this, follow the derivation in Section 4.3 of Redhead [1987]. Particularly, his equations 22 and 23 assume that a spin-1/2 particle has zero probability of yielding a spin result other than [+ or -] 1/2.

(5) In (*), equality holds only if the {[[lambda].sub.i]} sets are disjoint--that is, if no nonzero-measure [lambda] states belong to more than one {[[lambda].sub.i]}. Since the {[[lambda].sub.i]} sets in general overlap, (*) is in general an inequality.

(6) Actually, in Elby's and Clifton's strict-correlations proofs, TAF and Particle Locality can be replaced by weaker assumptions. See Clifton et al. [1991b] and Elby and Jones [1992].

Greenberger et al. [1990] and Mermin [1990] also derive contradictions between local realism conditions and certain perfect anticorrelations. Their proofs consider three correlated particles.

The most elegant 'imperfect correlations' proof could be obtained by using a Clifton-style argument on Greenberger et al. [1990] correlations.

REFERENCES

BOHM, D., HILEY, B and KALOYEROU, P. [1987]: 'An Ontological Basis for the Quantum Theory', Physics Reports, 144, pp. 321-75.

BROWN, H. R. and SVETLICHNY, G. [1990]: 'Nonlocality and Gleason's Lemma. Part 1. Deterministic Theories', Foundations of Physics, 20, pp. 1379-88.

CLIFTON, R. K., REDHEAD, M. and BUTTERFIELD, J. N. [1991a]: 'Generalization of the Greenberger-Horne-Zeilinger Algebraic Proof of Nonlocality', Foundations of Physics, 21, pp. 149-84.

CLIFTON, R. K., REDHEAD, M. and BUTTERFIELD, J. N. [1991b]: 'A Second Look at a Recent Algebraic Proof of Nonlocality', Foundations of Physics Letters, 4, pp. 395-403.

ELBY A. [1990]: 'Nonlocality and Gleason's Lemma. Part 2. Stochastic theories', Foundations of Physics, 20, pp. 1389-97.

ELBY, A. [1992]: 'Should we explain the EPR correlations causally?', Philosophy of Science, 59, pp. 16-25.

ELBY, A. and JONES, M. R. [1992]: 'Weakening the locality conditions in algebraic nonlocality proofs', Physics Letters A, 171, pp. 11-16.

GREENBERGER, D. M., HORNE, M. A. and ZEILINGER, A. [1989]: 'Going Beyond Bell's Theorem', in M. Kafatos (ed.), Bell's Theorem, Quantum Theory, and Conceptions of the Universe, pp. 69-76. Dordrecht: Kluwer Academic Publishers.

GREENBERGER, D. M., HORNE, M. A., SHIMONY, A. and ZEILINGER, A. [1990]: 'Bell's Theorem without Inequalities', American Journal of Physics, 58, pp. 1131-43.

JARRETT, J. P. [1984]: 'On the Physical Significance of the Locality Conditions in Bell Arguments', Nous, 18, pp. 569-80.

KOCHEN, S. and SPECKER, E. [1967]: 'The Problem of Hidden Variables in Quantum Mechanics', Journal of Mathematics and Mechanics, 17, pp. 59-87.

MERMIN, N. D. [1990]: 'Extreme Quantum Entanglement in a Superpositon of Macroscopically Distinct States', Physical Review Letters, 65, pp. 1838-40.

REDHEAD, M. L. G. [1987]: Incompleteness, Nonlocality, and Realism. Oxford: Clarendon Press.

REICHENBACH, H. [1956]: The Direction of Time. Berkeley: University of California Press.

RYDER, L. [1985]: Quantum Field Theory. Cambridge: Cambridge University Press.
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Date:Jun 1, 1993
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