Quantum Mechanics and Separability

 

The insensible perceptions are as eminently useful in pneumatology as are the insensible corpuscles in Physics, and it is equally unreasonable to reject the one or the other under the pretext that they are out of reach of our senses. Nothing is accomplished all at once, and it is one of my great maxims, and one of the most verified, that nature makes no jumps...

                                                                                                        Leibniz

 

If we have to go on with these damned quantum jumps, then I'm sorry that I ever got involved.

                                                                                                   Schrodinger

 

It is well known that quantum mechanics entails the violation of Bell's inequality, whereas from the assumption of separability of the test electrons, counterfactual definiteness, and free independent choices for the measurement angles (and assuming no super-determinism, time-symmetric causation, etc.) it follows by simple algebra that Bell's inequality cannot be violated. This is why, assuming quantum mechanics is valid (as all the evidence indicates), we evidently need to give up one of our deeply held beliefs about separability, etc. Nevertheless, one occasionally encounters individuals who deny this, and claim instead that, for all we know, actual electrons may be deterministic and separable. For example, they claim that the electrons in the classic EPR (Bohm) experiment can give results consistent with quantum mechanics and yet still be deterministic and separable… or as least separable. Of course, there are some well-known possibilities, such as time-symmetric causation, super-determinism, etc., but these individuals typically reject all such exotic notions. They claim that ordinary common sense is sufficient to show that the phenomena of quantum mechanics are compatible with classical separability and causality. Is there any merit in such claims?

 

Consider, for example, Bohm's version of the EPR experiment, with two entangled electrons emitted in opposite directions. At some distance away are two experimenters with spin measuring devices, each of which can be freely oriented at any desired angle to measure the spin of the respective electron at that chosen angle. If the two experimenters happen to measure the spins at the same angle, they always get opposite results (one UP and one DOWN). Consequently, if we posit that the electrons are deterministic and separable, each individual electron must be prepared to give a particular result for any particular measurement angle. But this implies a certain algebraic inequality on the correlations of the results when measured at unequal angles. Quantum mechanics predicts that this inequality is violated, and this has been confirmed by experiment. Therefore, the electrons are not deterministic and separable – barring such concepts as time-symmetrical causation, super-determinism, etc.

 

It should be stressed that Bell's inequality is not based on the assumption of hidden variables, it's based solely on the hypothesis that electrons are deterministic and separable. From this premise, the paper of Einstein, Podolsky, and Rosen showed that quantum mechanics must be incomplete, meaning that there must be hidden variables.  Bohr did not disagree with this conclusion, he merely rejected the premise, i.e., he says the electrons are not separable. Thus the argument between Bohr and Einstein was not over the validity of the reasoning but over the correctness of the premises. Bell’s inequality shows that, under certain additional assumptions (e.g., no time-symmetrical causation, no super-determinism, etc.), the premise of separability cannot be correct.

 

The derivation of Bell's inequality assumes that quantum mechanics correctly predicts that each electron exhibits quantum spin, yielding either UP or DOWN with equal probability, regardless of the angle of measurement, and that entangled electrons invariably give opposite results when measured at the same angle. From these premises, it follows that the predictions of quantum mechanics for unequal angles are false. Therefore, if quantum mechanics is completely correct, electrons cannot be deterministic and separable. That was the consensus in 1930, and it remains the consensus today, reinforced by the experimental demonstrations of the violations of Bell's inequality.

 

People sometimes think that the lack of separability implies "action at a distance", but that's a misunderstanding. Everyone agrees that quantum mechanics does not entail any action at a distance, because no information or energy propagates faster than light. Nevertheless, the entangled parts of a quantum system are not separable, and this is precisely what the violations of Bell's inequality demonstrate. It’s true that in the classical context the only way things could not be separable would be by action at a distance, but the peculiar feature of quantum mechanics is that things can be non-separable without implying any action at a distance. The non-separability is subtle, but it represents a profoundly non-classical aspect of the world.

 

Bell's inequality is derived from the assumptions of determinism and separability. So, if someone claims that electrons are deterministic and separable, if they also reject the well-known alternative accounts involving time-symmetrical causation, super-determinism, etc., then the demonstrated violations of Bell's inequality prove them wrong. They may counter by saying "No, Bell's inequality is based on hidden variables, which I reject", but this is a non-sequitur, because "hidden variables theory" is really just a historical term for theories that satisfy determinism and separability. Hidden variables follow from these premises, as explain in the EPR paper, which (remember) Bohr did not dispute.

 

Now, it’s well known that no energy or information propagates faster than light according to quantum mechanics, and yet some people still have the vague idea that quantum entanglement implies action at a distance.  This may be partly due to lack of clarity about the technical meaning of the word “action” in physics, but also partly due to the fact that the non-separability of quantum mechanics is subtle, involving distant correlations but not communication. People often use sloppy language, saying things like “one electron is affected by the measurement of the other”, but they are really referring to the existence of correlations, not to any action at a distance. Even Bell himself, who evidently yearned for a return to a Lorentzian world view, admitted that instantaneous action at a distance is inconsistent with the well-established Lorentz invariance.

 

In discussion of the foundations of quantum mechanics, vague terminology is a big problem. People often disagree about the meanings of English words like locality, realism, causation, influence, determinism, etc., whereas there is actually broad agreement on many of the technical concepts. For example, everyone agrees that no information or energy propagates faster than light, but not everyone agrees to call this "localism". Everyone would have agreed to call this "localism" prior to quantum mechanics, but with the recognition of the non-classical correlations of quantum mechanics, as exemplified by violations of Bell's inequality, came a perceived need for a more sophisticated concept of non-locality, recognizing that the responses of entangled entities to (presumed) freely-chosen interactions at spacelike-separated events exhibit correlations that defy explanation under traditional notions of causation. When someone says they are in the localist camp, they might just mean they believe no energy or information propagates faster than light, in which case everyone agrees, or they might mean that they deny the existence of correlations that defy explanation under traditional notions of causation, in which case very few people would agree. It all depends on how people define these terms, so it's essential to be clear and explicit. Simply saying "I'm in the localist camp" doesn't make one’s meaning clear, and saying that electrons are deterministic and separable tends to suggest a lack of recognition of the violations of Bell's inequality.

 

Basic quantum mechanics entails the non-classical correlations that are highlighted by the violation of Bell’s inequality, and several people got Nobel prizes for this, including Bohr, Heisenberg, Schrodinger, Dirac, and Born. Understanding why the quantum correlations are so difficult to fit into any prior conceptual framework requires some careful thought. Consider two entangled electrons, heading toward two experimenters at distant locations. Each experimenter has a spin measuring device, which he can orient at will to measure the incoming electron’s spin (Up or Down) along any axis he chooses. For simplicity, let’s agree that they will measure only along one of three directions, 0, 120, or 240 degrees. Each experimenter finds that, whichever direction he measures, half the electrons are spin UP and half are spin DOWN. Each experimenter keeps a record of his measurements, noting the angle he selected and the result (UP or DOWN) that he got for each incoming electron. When the experimenters get together later to compare their results, they make an astounding discovery: Every time the two experimenters happened to measure a pair of entangled electrons along the same direction, they ALWAYS got opposite results (one UP and one DOWN), and whenever they measured in different directions they got the same result (both UP or both DOWN) 3/4 of the time. Barring super-determinism, backward causation, or some other non-classical premise, these results are impossible to reconcile with the assumption that the electrons are deterministic and separable.

 

It’s worth stressing again that determinism and separability imply hidden variables (although the converse is not true). To understand this, note that the two experimenters can be arbitrarily far apart, and they can each freely choose which direction to measure, and yet if they happen to measure in the same direction they invariably get opposite results. In other words, if experimenter A finds UP for one electron at his detector at a given angle, then experimenter B must find DOWN for the corresponding electron at his detector at that same angle. Now, if we believe the electrons are separable, it follows that B must find DOWN for that electron at that angle, whether A measures at that angle or not. This is a key point. The only way to avoid hidden variables is to give up separability. If we don’t wish to do this, we would be forced to conclude that each electron must be prepared to give a definite result for any selected measurement angle, and, from this, Bell's inequality follows. We know Bell's inequality is violated, so the premise of separability is wrong (barring the unconventional alternatives).

 

As mentioned above, Bell's inequality is not based on the assumption of hidden variables, it's based on the hypothesis that electrons are deterministic and separable. To be more explicit: We begin with the assumption that the electrons are deterministic and separable. Combining this with the perfect anti-correlation predicted by quantum mechanics for pairs of entangled electrons when measured along the same angle, it follows that the electrons must be prepared to respond in a definite way to a spin measurement at each angle. Thus from these premises we must conclude local hidden variables. (This was the EPR argument.) But then the quantum mechanical correlations at unequal angles prove that no local hidden variables can work. So, although Bell's theorem does indeed rule out local hidden variables, it is not based on the assumption of local hidden variables, it is based only on the assumption of deterministic and separable electrons, and makes use of the EPR argument to show that this implies hidden variables, which is then shown to imply certain inequalities which are violated by quantum mechanics (and by experiment).

 

Sometimes people claim, that, with a deterministic and separable context, they can account for the perfect anti-correlation without hidden variables, but these claims are never substantiated. It’s amusing to review some of the attempts that have been made on these lines. As an example, when challenged to explain how the concept of deterministic and separable electrons can account for the perfect anti-correlation without hidden variables, one individual quickly discarded determinism (formerly one of his two unshakable pillars!), and then offered three possible answers to the question:

 

His first answer was “Maybe those electrons are physically determined by the emission process.” Needless to say, this is the “hidden variable” explanation, which can indeed account for the perfect anti-correlation at equal angles, but it is provably incompatible with the correlations at unequal angles, so it is ruled out. His second answer was “Maybe they are intrinsically stochastic and have aspects that are not determined until a measurement.” But this hypothesis doesn’t yield perfect anti-correlation for the separate spin measurements at equal angles, assuming locality/separability. It’s fine to say the spin is “not determined until it is measured”, but it actually is determined given that the other electron’s spin in measured (at the same angle), and if we insist on separability/locality, that other measurement can’t cause this electron to suddenly have a determined spin, unless we go back to the previous answer, i.e., the pair was pre-programmed to give opposite spins, so we’re back to hidden variables, which can’t account for the correlation at unequal angles. His third answer was “Maybe it is all determined by events pre-dating the emission.” Again, he is saying the electrons are “determined”, so he is back to hidden variables. Yes, this can account for the perfect anti-correlation at equal angles, but not for the correlations at unequal angles.

 

He concluded: “I do not think that these distinctions make any mathematical or scientific sense. You can believe in any of these as you please, and positivists like myself will be indifferent.” But we can’t believe in any of these as we please, because they all fail to account for the facts. The only way he has offered to account for the perfect anti-correlation at equal angles is by the electrons being “physically determined”, either by the emission process or something pre-dating the emission. That is the hidden variable explanation, which is ruled out by the correlations at unequal angles. So, he has not provided any viable explanation for the claim that the electrons are deterministic and separable – or even just separable. Barring the well-known unconventional alternatives, there is no way to account for both the perfect anti-correlation at equal angles and the 3/4 correlations at unequal angles. Bell’s paper on Bertleson’s Socks explains the situation very clearly:

 

Let us summarize the logic that leads to the impasse. The EPRB correlations are such that the result of the experiment on one side immediately foretells that on the other, whenever the analyzers happen to be parallel. If we do not accept the intervention on one side as a causal influence on the other, we seem obliged to admit that the results on both sides are determined in advance anyway, independently of the intervention on the other side, by signals from the source and by the local magnet setting. But this has implications for non-parallel settings which conflict with those of quantum mechanics.

 

Here he is explaining why, if we rule out communication, the perfect anti-correlation at equal angles obliges us to admit that the results are determined in advance, by common cause, i.e., by extra variables. This is not an assumption, it is the only remaining causal option, deduced from the assumption of separability combined with the perfect anti-correlation at equal angles.

 

On the next few pages of the Bertleson paper, following the above overview of the argument, Bell describes the details in a general way, and introduces the extra variables (lambda) by noting that, although the joint probability doesn't factor, there is one remaining possibility, namely, we can hope that it might be made factorable (without communication) by introducing sufficiently many causal factors such that the residual fluctuations will be independent (factorable). This is not introduced as an assumption, it is introduced as the only conceivable way that we can render the joint probability factorable. Of course, it turns out that even if we do this, we arrive at a contradiction (the correlations at unequal angles), so even this last desperate escape route is blocked. Thus the entangled particles are not separable. “Hidden variables” is not assumed, it is an intermediate deduction (via the EPR argument), and no one has ever offered a refutation of this argument (as distinct from a disagreement with the premises).

 

The correlations between entangled particles in quantum mechanics can be said to “violate causality” in the sense that distant correlations arise with no local cause, i.e., no common cause (hidden variables) and no transfer of energy or information between the separate events.

 

Sometimes people appeal to the authority of Bohr on the question of whether entangled electrons in quantum mechanics are separable. But, according to Bohr, “The peculiar individuality of the quantum effects presents us, as regards the comprehension of well-defined evidence, with a novel situation unforeseen in classical physics and irreconcilable with conventional ideas...”, and he emphasized (in his final response to EPR) that in the application of the quantum mechanical formalism it is necessary to consider the whole experimental arrangement, not just the separate parts. So an appeal to the authority of Bohr (whose quasi-mystical concept of complementarity has always been hopelessly obscure) does not provide any support for the claim that entangled electrons are separable. Quite the contrary.


Rather than claiming that "electrons are deterministic and separable" (which is inconsistent with quantum mechanics), one should say, instead, that the results of space-like separated measurements of entangled electrons exhibit correlations that defy any deterministic and separable account, but that these correlations do not imply that any information or energy propagates faster than light.

 

Occasionally people appeal to positivism to claim that there is no meaningful sense in which separability is violated in quantum mechanics. It was recognized long ago that positivism is nothing but half-baked solipsism, which enables those who espouse it to believe whatever they like. As Bell said, “Solipsism cannot be refuted. But if such a theory were taken seriously, it would hardly be possible to take anything else seriously." After conceding that the concept of determinism has no positivistic content, the positivist may say that separability means “causality is confined to the light cone”, but the concept of “causality” has no more positivistic meaning than the concept of “determinism”. In fact, causality has always been the poster child for concepts that have no positivistic content. A more meaningful concept is the proposition that no energy or information propagates faster than light. Everyone agrees with this, although not everyone agrees that this captures the entire content of what should be called “locality”. The word “non-separable” is often taken to encompass things like the non-classical correlations implicit in quantum mechanics (for which the joint probabilities do not factor into separate probabilities). With these definitions we would say the electrons are local but not separable.

 

Sometimes people argue that the non-classical correlations implicit in quantum mechanics are unobservable, but that is clearly untrue. They are quite observable, and not just at the atomic level. Bohr certainly did not deny this, nor did he deny that the observable facts require a thorough revision of the classical concept of causality. He wrote in response to the EPR paper

 

The apparent contradiction [described by EPR] in fact discloses only an essential inadequacy of the customary viewpoint of natural philosophy for a rational account of physical phenomena... Indeed the finite interaction between object and measuring agencies conditioned by the very existence of the quantum of action entails the necessity of the final renunciation of the classical ideal of causality and a radical revision of our attitude towards the problem of physical reality.

 

Thus Bohr finds it necessary to renounce the classical ideal of causality in order to account for the phenomena of quantum mechanics, in particular, the non-classical correlations for spacelike-separated measurements. The specific aspect of the classical ideal of causality that he sacrifices is separability.

 

As explained in detail above, the derivation of Bell's inequality does not assume hidden variables. The fact that some authors prefer to split off the first part of Bell's derivation separately, and refer only to the second part as "Bell's theorem" is purely a matter of convention. The point is that Bell's inequalities follow from the assumptions of separability, determinism, and perfect anti-correlation at equal angles (and in fact the more sophisticated versions dispense with determinism and apply to stochastic models as well).

 

Positivism can be useful by prompting us to give operational meanings to our terms, and to carefully examine the epistemological foundations of our concepts. But, carried to its logical conclusion, positivism becomes solipsism, so a physicist needs to wield positivism with tact if he is to avoid degenerating into irrelevancy. Ironically, most of the physical concepts (space, time, causation, etc.) that the typical positivist enthusiast fully accepts without question actually have no positivistic justification, since the only evidence we ever have is our own raw sense impressions, which do not give unambiguous warrant to any of those high-level concepts. In fact, one of the forefathers of logical positivism, David Hume, famously took "causation" as a prime example of a concept with no positivistic meaning or justification. If one were willing to dispense with such high-level concepts as space, time, multiplicity, otherness, separability, and causation, then one could legitimately dismiss claims that quantum mechanics implies non-separability of entangled particles, by simply denying that the concept of separability has any positivistic meaning. But if we wish to retain those high-level concepts, then positivism does not justify ignoring the observable but apparently acausal joint correlations of measurements of entangled particles.

 

Barring solipsistic arguments, the non-classical correlations in quantum mechanics are observable facts for free measurements of spacelike-separated entangled particles, and the joint probabilities do not factor into separate individual probabilities, and there is no common cause or causal communication to account for those correlations. This is what people mean when they say that quantum mechanics does not obey separability, even though no energy or information propagates faster than light. The wave function in quantum mechanics is inherently non-separable in space and time, because its domain for a system of N particle is 3N-dimensional configuration space, not 3-dimensional space. This is why Bohr and Heisenberg acknowledged that quantum mechanics is fundamentally irreconcilable with any causal account in the context of space and time.

 

Needless to say, there is no positivistic support for concepts like the imagined spatio-temporal structure of the objective world. In fact, there is no positivistic support for the objective world. Positivism is ultimately self-defeating. Every adolescent passes through a phase of enthusiasm for positivism, as it begins by unfolding some surprising insights, but before long a person realizes that, if carried to its logical conclusion, it degenerates into solipsism. Only those who can't see it through to its conclusion can remain deceived by it for long.

 

Einstein argued for strict locality, separability, and even causality for the descriptions of events in space and time – although he recognized that causality may be ultimately undecidable. Einstein was against spookiness, whereas Bohr embraced spookiness. The EPR argument essentially showed that we can't have both separability and completeness. Since EPR insisted on separability, they concluded that quantum mechanics must be incomplete. Bohr said no, quantum mechanics is complete but not separable. He argued that we must simply accept the spooky quantum correlations with no causal account possible in ordinary space and time. Bohr may well have been correct about this, but the positions of Bohr and Einstein in this debate have often been mis-understood.

 

Prior to 1927 Einstein tried to find ways to falsify quantum mechanics, but after that date he accepted the correctness of it. He simply maintained that it was incomplete, because on principle he didn't think we should relinquish causal and separable relations in space and time. Einstein did not demand that a theory be deterministic, and many times he explicitly stated this (and discounted the one off-hand and droll remark about dice). As Pauli explained in a letter to Born, “Einstein does not consider the concept of determinism to be as fundamental as it is frequently held to be (as he told me emphatically many times)... he disputes that he uses as a criterion for the admissibility of a theory the question: 'Is it rigorously deterministic?'" Einstein sought a causal foundation for physics, but he acknowledged that ultimately it will never be possible to decide with certainty whether the world is causal or not (because he understood the limitations of epistemology). Bohr did not dispute the fact that quantum mechanics must either be incomplete (hidden variables) or violate separability. The only disagreement was that Bohr said quantum mechanics is complete (no hidden variables) and accepted that it is not separable (in space and time), whereas EPR said it must be separable (in space and time) so it must be incomplete (hidden variables). If we accept non-separability, we’re done (Bohr’s position). If we insist on separability (as Einstein did), then we are forced to hidden variables, in which case the second part of Bell’s Theorem leads to Bell’s inequality, which contradicts quantum mechanics.

 

It has sometimes been alleged that Bell only belatedly tried to fudge his theorem by pretending to be able to deduce hidden variables from other hypotheses, but this is a completely spurious charge, since in Bell's earliest paper (1964) he did indeed note how we can "deduce hidden variables" from the hypothesis of separability and the perfect anti-correlation at equal angles.

 

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