Einstein, Bohm, Bell, and the Derivation of Bell’s Inequality: Entanglement and Quantum Non-locality (2024)

By 1935, the Copenhagen interpretation had become the orthodoxy. It was already the default way in which physicists were meant to think about quantum mechanics. Einstein referred to this as ‘Talmudic’; a ‘religious’ philosophy that is to be interpreted only through its qualified priests, who insist on its essential truth, and who will countenance no rivals.1

The philosopher Karl Popper called it a schism:2

If he was to continue his challenge, Einstein needed to find a way to render Bohr’s disturbance (or clumsiness) defence either irrelevant or inadmissible. Despite its seeming impossibility, this meant imagining a physical situation in which it is indeed possible, in principle, to acquire knowledge of the physical state of a quantum system without disturbing it in any way. Working with two young theorists, Boris Podolsky and Nathan Rosen, Einstein devised a new challenge that was extraordinarily cunning. It seemed that they had found a way to do the impossible.

Entangled States

Imagine a situation in which two quantum particles interact or are formed together in some physical process, and then move apart. These particles may be photons, for example, emitted in rapid succession from an atom, or they could be electrons or atoms. For convenience, we’ll label these particles as 1 and 2. For our purposes we just need to suppose that, as a result of the operation of some law of conservation, the two particles are each produced in quantum states that are obliged to be orthogonal.

At this stage it really doesn’t matter what these states are, so let’s just imagine that the particles are photons in states of left- and right-circular polarization. Suppose the law of conservation says that if photon 1 is found to be in a state of left-circular polarization, $|L1〉$⁠, then photon 2 must be in a state of right-circular polarization, $|R2〉$⁠, as judged from the perspective of the source (rather than a detector). Similarly, if particle 1 is found to be in the state $|R1〉$⁠, then particle 2 must be in the state $|L2〉$⁠. The reason for this particular choice of combination will become apparent later in this chapter.

The two photons form composite states, $|L1R2〉=|L1〉|R2〉$ and $|R1L2〉=|R1|L2〉$⁠, which we can presume are both equally probable. We don’t know which composite state we’re going to get from any specific individual physical event that creates it, but we know from our discussion of the Pauli principle in Chapter 8 that the correct way to proceed in these circ*mstances is to form these into a normalized superposition (cf. Eq. (8.16)),

where I’ve used the Greek letter ζ (zeta) to indicate the composite pair state (from the Greek word $ζευ´γος$⁠, meaning ‘pair’).

The photons move a long distance apart, each eventually passing through a linear polarization analyser (such as a calcite crystal) before being detected, amplified, and counted. Both polarization analysers are aligned along common vertical/horizontal axes, so the possible measurement eigenstates are

We know what to do next. To analyse this situation we must expand the initial composite state $|ζ12〉$ in the basis of the measurement eigenstates:

I think we know by now where this is heading. So, let’s assume a joint measurement operator $M^12$ with eigenvalues $1v1v$ (the detectors register vertical polarization for photons 1 and 2, respectively), $1v1h$⁠, $1h1v$⁠, and $1h1h$⁠. From Eqs. (11.30) and (11.31), we know we can write the expectation value as

where $Pvv=|〈v1v2〉|ζ12〉|2$ is the probability of observing the combination vertical/vertical, and so on for $Pvh$⁠, $Phv$⁠, and $Phh$⁠.

It will prove useful in what follows to define a generalized correlation function, $C12$⁠, which summarizes the extent of the correlation between the two photons. This is based on the expectation value given by Eq. (12.4), in which we (arbitrarily) assign $1v=+1$ and $1h=−1$⁠:

Having set up these general expressions—which will prove useful very soon—we can now go on to deduce the individual projection amplitudes using the entries in Table 11.2:

Similarly,

and

So from (12.3)we have

From Eq. (12.4), the expectation value reduces to

And from Eq. (12.5) we see that $C12=1$⁠, which simply means that the outcomes are perfectly correlated.

By the time we make measurements on them, we presume that photons 1 and 2 have moved a long distance apart. We can, in principle, make a measurement on either photon to discover its state. Of course, for each measurement we will only ever see one outcome. We must therefore presume that the composite state $|ζ12〉$ collapses to deliver only one outcome: either$|v1〉|v2〉$or$|h1〉|h2〉$⁠, such that in a series of repeated measurements on identically prepared systems we will get either vertical/vertical or horizontal/horizontal 50 percent of the time.

Now suppose we make a measurement on photon 1 and discover that it is vertically polarized. Following von Neumann’s logic, this must mean that $|ζ12〉$ collapsed to leave photon 2 in a vertically polarized state. Likewise, if we discover that photon 1 is horizontally polarized, this must mean that $|ζ12〉$ collapsed to leave photon 2 in a horizontally polarized state. Based on (12.10), there are no other possible outcomes.

In the review article in which Schrödinger introduced his cat paradox, he also stated:3

Such a pair of particles are now said to be entangled.

We have no way of knowing in advance if photon 1 will be measured to be vertically or horizontally polarized. But this really doesn’t matter, for once we know the state of photon 1, we also know the state of photon 2 with certainty, even though we may not have measured it. In other words, we can discover the state of photon 2 with certainty without disturbing it in any way. All we have to assume is that any measurement we make on photon 1 in no way affects or disturbs photon 2, which could be an arbitrarily long distance away (say halfway across the universe). We conclude that the state of photon 2 (and by inference, the state of photon 1) must surely have been determined all along.

A Reasonable Definition of Reality

In their 1935 paper, which was titled ‘Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?’, Einstein, Podolsky, and Rosen (EPR) offered a philosophically loaded ‘definition’ of physical reality:4

We can see what the authors were trying to do. If the wavefunction is interpreted realistically, then it ought to account for the reality of the physical quantities—such as the states of photons 1 and 2—that it purports to describe. It clearly doesn’t. There is nothing in the formulation that describes what these states are before we make a measurement on photon 1, so quantum mechanics cannot be complete.

The alternative is to accept that the reality of the state of photon 2 is determined by the nature of a measurement we choose to make on a completely different particle an arbitrarily long distance away. Whatever we think might be going on, a realistic interpretation of the wavefunction implies some kind of ‘spooky action at a distance’ at odds with the special theory of relativity. EPR argued that: ‘No reasonable definition of reality could be expected to permit this.’5

Details of this latest challenge were reported in The New York Times before the EPR paper was published, in a news article headlined ‘Einstein Attacks Quantum Theory’. This provided a non-technical summary of the main arguments, with extensive quotations from Podolsky who, it seems, had been the principal author of the paper.

There is much in the language and nature of the arguments employed in the paper that Einstein appears later to have regretted, especially the reality criterion. He deplored The New York Times article and the publicity surrounding it. All the more disappointing, perhaps, as the main challenge presented by EPR does not require this (or any) criterion, though it does rest on the presumption that, however reality is defined, it is presumed to be local, meaning that—no matter how they might have been formed—as photons 1 and 2 move apart, they are assumed to exist completely independently of each other. This is sometimes referred to as ‘Einstein separability’.

This new challenge sent shockwaves through the small community of quantum physicists. It hit Bohr like a ‘bolt from the blue’.6 Pauli was furious. Dirac exclaimed: ‘Now we have to start all over again, because Einstein proved that it does not work.’7

Bohr’s response, when it came a short time later, inevitably targeted the reality criterion as the principal weakness. He argued that the stipulation ‘without in any way disturbing a system’ is essentially ambiguous, since the quantum system is influenced by the very conditions which define its future behaviour. In other words, the composite state $|ζ12〉$ is deliberately set up with coded information based on what we already know from previous experience. This allows us to predict $Pvv$ and $Phh$⁠. The measurements then simply update our knowledge. All is well, provided we don’t ask how nature manages this particular conjuring trick.

Einstein was, at least, successful in pushing Bohr to give up his clumsiness defense, and to adopt a more firmly anti-realist position. Those in the physics community who cared about these things seemed to accept that Bohr’s response had put the record straight. But not everybody was satisfied.

Hidden Variables

In his debate with Bohr and Schrödinger, Einstein had hinted at a statistical interpretation. In his opinion, quantum probabilities, derived as the modulus-squares of the projection amplitudes, actually represent statistical probabilities, averaged over large numbers of physically real particles. We resort to probabilities because we’re ignorant of the states of the physically real quantum things.

Einstein toyed with just such an approach in May 1927. This was a modification of quantum mechanics that combined classical wave and particle descriptions, with the wavefunction taking the role of a ‘guiding field’ (in German, a Führungsfeld), guiding or ‘piloting’ the physically real particles. In this kind of scheme, the wavefunction is responsible for all the wave-like effects, such as diffraction and interference, but the particles maintain their integrity as localized, physically real entities. Instead of waves or particles, as the Copenhagen interpretation demands, Einstein’s adaptation of quantum mechanics was constructed from waves and particles.

But Einstein lost his enthusiasm for this approach within a matter of weeks of formulating it. It hadn’t come out as he’d hoped. The wavefunction had taken on a significance much greater than merely statistical. Einstein thought the problem was that distant particles were exerting some kind of strange force on one another, which he really didn’t like. But the real problem was that the guiding field is capable of exerting spooky non-local influences. He withdrew a paper he had written on the approach before it could be published. It survives in the Einstein Archives as a handwritten manuscript.8

This experience probably led Einstein to conclude that his initial belief—that quantum mechanics could be completed through a more direct fusion of classical wave and particle concepts—was misguided. He subsequently expressed the opinion that a complete theory could only emerge from a much more radical revision of the entire theoretical structure. He felt that quantum mechanics would eventually be replaced by an elusive grand unified field theory, the search for which took up most of his intellectual energy in the last decades of his life.

This early attempt by Einstein at completing quantum mechanics is known generally as a hidden variables formulation, or just a ‘hidden variables theory’. It is based on the idea that there is some aspect of the physics that governs what we see in an experiment, but which makes no appearance in the representation. There are, of course, many precedents for this kind of approach in the history of science. As I’ve already explained, Boltzmann formulated a statistical theory of thermodynamics based on the ‘hidden’ motions of real atoms and molecules. Likewise, in Einstein’s abortive attempt to rethink quantum mechanics, it is the positions and motions of real particles, guided by the wavefunction, that are hidden.

However, in Mathematical Foundations, von Neumann presented a proof which appeared to demonstrate that all hidden variable extensions of quantum mechanics are impossible.9 This seemed to be the end of the matter. If hidden variables are impossible, why bother even to speculate about them?

And, indeed, silence prevailed for nearly twenty years. The dogmatic Copenhagen view prevailed, seeping into the mathematical formalism and becoming the quantum physicists’ conscious or unconscious default interpretation. The physics community moved on and just got on with it, content to ‘shut up and calculate’.10

Then David Bohm broke the silence.

Enter Bohm

In February 1951, Bohm published a textbook, simply called Quantum Theory, in which he followed the party line and dismissed the challenge posed by EPR’s ‘bolt from the blue’, much as Bohr had done. But even as he was writing the book he was already having misgivings. He felt that something had gone seriously wrong.

Einstein welcomed the book, and invited Bohm to meet with him in Princeton sometime in the spring of 1951. The doubts over the interpretation of quantum theory that had begun to creep into Bohm’s mind now crystallized into a sharply defined problem. ‘This encounter had a strong effect on the direction of my research,’ Bohm later wrote, ‘Because I then became seriously interested in whether a deterministic extension of quantum theory could be found’.11 The Copenhagen interpretation had transformed what was really just a method of calculation into an explanation of reality, and Bohm was more committed to the preconceptions of causality and determinism than perhaps he had realized.

In Quantum Theory, Bohm asserted that ‘no theory of mechanically determined hidden variables can lead to all of the results of the quantum theory’.12 Bohm went on to develop a derivative of the EPR thought experiment which he published in a couple of papers in 1952 and which he elaborated in 1957 with physicist Yakir Aharonov.13 This is based on the idea of fragmenting a diatomic molecule (such as hydrogen, H₂) into two spin-aligned atoms.

Through their efforts, Bohm and Aharonov brought the EPR experiment down from the lofty heights of pure thought and into the practical world of the physics laboratory. In fact, the purpose of their 1957 paper was to claim that experiments capable of measuring correlations between distant entangled particles had already been carried out. For those few physicists paying attention, Bohm’s assertion and the notion of a practical test suggested some mind-blowing possibilities.

Enter Bell

John Bell was paying attention. In 1964, he had an insight that was completely to transform questions about the representation of reality at the quantum level. After reviewing and dismissing von Neumann’s ‘impossibility proof’ as flawed and irrelevant, he derived what was to become known as Bell’s inequality. ‘Probably I got that equation into my head and out on to paper within about one weekend,’ he later explained. ‘But in the previous weeks I had been thinking intensely all around these questions. And in the previous years it had been at the back of my head continually.’14

The Ingredients

The Recipe

So far in our discussion of the system of entangled photons, we’ve assumed that the two polarizing analysers used to discover the linear polarization states of photons 1 and 2 are orientated so that they are aligned on common vertical/horizontal axes. But what if we now rotate one (or both) of these analysers? In Step (1), we elaborate the projection amplitudes, projection probabilities, expectation value, and correlation function for this situation.

We take a curious diversion in Step (2). Bell was constantly on the lookout for ‘everyday’ examples of pairs of objects that are spatially separated but whose properties are correlated, as these provide accessible analogues for the EPR experiment. He found a perfect example in the dress sense of one of his colleagues at CERN, Reinhold Bertlmann. So, in this step I will introduce you to Bertlmann’s socks. In a paper published in 1981, Bell made use of a series of hypothetical experiments involving prolonged washing of these socks at different temperatures to develop some numerical relationships involving the outcomes, and we consider these in Step (3). These relationships are generalized—in Step (4)—to experiments on pairs of socks, from which Bell’s inequality can be derived.

We return to quantum mechanics in Step (5), which details the derivation of Bell’s theorem.

Step (1): Measurements with Different Analyser Orientations

We’ll start by considering the situation in which the two polarization analysers are both orientated at different angles relative to the (arbitrary) laboratory vertical axis. We suppose that the analyser for photon 1 is orientated at an angle α measured clockwise from the vertical axis. This defines the $v′/h′$ axes (as before, see Fig. 11.3). We suppose that the analyser for photon 2 is orientated at an angle β, defining another set of axes which we denote as $v″/h″$⁠. As before, the possible measurement eigenstates are

where vertical^′ indicates that photon 1 emerges from the vertical channel of analyser 1, and vertical^′′ indicates that photon 2 emerges from the vertical channel of analyser 2, and so on. We can use the entries in Table 11.2 to deduce the projection amplitude $〈v1′v2′′|ζ12〉$ as follows:

where $φ=β−α$⁠, and we have made use of the complex exponential form for $cosφ$⁠. We note in passing that when $φ=0°$⁠, $〈v1′v2′′|ζ12〉=〈v1v2|ζ12〉$ and from Eq. (12.12) we recover (12.6).

It follows that

We learn that for this particular entangled system the projection amplitudes depend (rather neatly!) only on the differences between the orientation angles of the polarization analysers.

The corresponding projection probabilities are then

And the expectation value for the joint measurements is, therefore,

The correlation function for the joint measurement is

where we have made use of the trigonometric identity $cos2A−sin2A=cos2A$⁠.

Notice how the correlation now varies as we change the difference angle between the analyser orientations, from $C12(φ)=+1$ (perfect correlation) when $φ=0°$⁠, to $C12(φ)=0$ (no correlation) when $φ=45°$⁠, to $C12(φ)=−1$ (perfect anti-correlation) when $φ=90°$⁠.

Step (2): Bertlmann’s Socks

The photons are correlated, but is this really so mysterious? In a paper titled ‘Bertlmann’s Socks and the Nature of Reality’, published in 1981, Bell wrote:15

This situation is illustrated in Fig. 12.1.*

Let’s suppose that Dr Bertlmann is a physicist who is very interested in the physical properties of his socks. Imagine that he has secured a contract from a consumer research organization to study how his socks stand up to the rigours of prolonged washing at different temperatures. Being a theoretical physicist, he knows that he can discover some simple relationships between the numbers of socks that pass (+ result) or fail (− result) such tests without actually having to perform them using real socks and real washing machines. This makes his study inexpensive and therefore attractive to his sponsors.

Step (3): Washing Socks and Outcome Spaces

Bertlmann decides to subject his left socks (which we label collectively as socks 1) to three different tests:

Of course, we are assuming that the properties of the socks are uniform—there is no physical difference (perhaps other than colour) between socks in a large collection. We can define the result or outcome ‘spaces’ for these three tests as follows:

Think of it this way. When Bertlmann discovers that a sock successfully passes the α test, he places a tick in the $α+$ space; if it fails he places a tick in the $α−$ space. Likewise for all the other spaces.

Bertlmann now defines three experiments. In experiment 1, he determines how many socks pass test α and fail test β. This number is denoted as $n(α+,β−)$⁠. You can think of this as the count of the number of ticks that lie in the overlapping $α+$ and $β−$ outcome spaces. In experiment 2 he determines the number of socks passing β and failing γ, $n(β+,γ−)$⁠. And in experiment 3 he determines the number passing α and failing γ, $n(α+,γ−)$⁠. These three experiments map to the outcome spaces as follows:

where the shaded areas indicate the overlapping outcome spaces in each experiment.

We can see from this that $n(α+,β−)$ is the sum of two subspaces:

Similarly,

So, if we now add $n(α+,β−)$ and $n(β+,γ−)$⁠, from (12.19)and (12.20), we get

Or

Step (4): Generalize for Experiments on Pairs of Socks

Astute readers will have already spotted the flaw in Bertlmann’s reasoning. We really have no way of knowing if any given sock will pass one test and at the same time fail another. If we try to perform a sequence of tests on any individual sock, then we can’t be sure that surviving the rigours of 1000 washing cycles will leave the sock in its pristine state, ready for a subsequent test. And if failing a test means that the sock is destroyed, then it is obviously unavailable for any further test.

But then Bertlmann remembers that his socks always come in pairs. Aside from differences in colour, if the socks in each pair are assumed to have otherwise identical physical properties, then we can safely assume that the result of a test performed on sock 2 implies that the same result would have been obtained for sock 1, even though we haven’t performed this test on it directly. He must further assume that whatever test he chooses to perform on sock 2 in no way affects the outcome of any other test he might perform on sock 1, but this seems so obviously valid that he doesn’t give it a second thought.

Now the three different sets of experiments are carried out on three samples containing the same total number of pairs of socks, N. In experiment 1, for each pair, sock 1 is subjected to test α and sock 2 is subjected to test β. If sock 2 fails test β, this implies that sock 1 would also have failed test β had it been performed on 1. The number of pairs of socks for which sock 1 passes test α and sock 2 fails test β, which Bertlmann denotes as $N+−(α,β)$⁠, must be equal to the (theoretical) number of socks 1 which pass test α and fail test β, i.e. $N+−(α,β)=n(α+,β−)$⁠. The same logic follows for experiments 2 and 3.

Bertlmann can now generalize this result for any batch of pairs of socks. By dividing each of these numbers by the total number of pairs of socks N he arrives at the relative frequencies with which each joint result is obtained. He identifies these relative frequencies as probabilities for obtaining the results for experiments yet to be performed on any batch of pairs of socks that, statistically, have the same properties, i.e. $P+−(α,β)=N+−(α,β)/N$⁠, and so on. From Eq. (12.22) he is led to the inescapable conclusion

This is Bell’s inequality.

As we can see, this has nothing whatsoever to do with quantum mechanics or hidden variables. It is simply a logical conclusion derived from the relationships between independent sets of numbers and their related probabilities. Any pair of socks which pass α and fail γ will contribute to the probability $P+−(α,γ)$⁠. But such a pair will also either pass or fail β, contributing either to $P+−(β,γ)$ or $P+−(α,β)$⁠. Thus, there is simply no way that $P+−(α,γ)$ can exceed the sum $P+−(α,β)+P+−(β,γ)$⁠.

Step (5): Bell’s Theorem: Quantum Non-locality

This has, no doubt, been an illuminating diversion, but we need to get back to quantum mechanics. Actually, this is really quite straightforward. Instead of experimenting with pairs of socks, we experiment with pairs of photons. Instead of washing at different temperatures, we perform experiments with different orientations of the polarization analysers.

Our three experiments are now:

Analysers 1 and 2 refer to the polarization analysers through which photons 1 and 2, respectively, will pass. Again, the orientation angles $α,β,γ$ are measured relative to the (arbitrary) laboratory vertical axis.

We can follow precisely the same logic, replacing pass/fail (⁠$+/−$⁠) results with detection from the vertical/horizontal channels of the analysers (⁠$v/h$⁠). We retain the assumption that the relative orientation of analyser 2 can in no way affect the outcome of the measurement performed on photon 1. In essence, this means that the photons are assumed to be ‘Einstein separable’, just like Bertlmann’s socks. We invoke the existence of some kind of hidden variable which governs the linear polarization properties of the two photons, such that their properties are predetermined before they pass through the analysers and are detected. The photons are said to be locally real, meaning that their properties are determined all along and photon 1 cannot be influenced by whatever we choose to do to photon 2, and vice versa.

If we accept this, then from Eq. (12.23), we have

In these experiments we’re free to set whatever orientation angles we like for the analysers. So, let’s set $α=0°$⁠, $β=22½°$⁠, and $γ=45°$⁠, such that $φ1=22½°$⁠, $φ2=22½°$⁠, and $φ3=45°$⁠. From (12.25)we get

The conclusion is inescapable. For this particular configuration of the analysers, quantum mechanics predicts that Bell’s inequality should be violated.

Bell’s inequality is quite general. It does not depend on what kind of hidden variable theory we might devise, so long as it is locally real. This generality allowed Bell to formulate a ‘no-go’ theorem:16

A complementary no-go theorem was devised in 1967 by Simon Kochen and Ernst Specker.17

The Aspect Experiments

The real repercussions of Bell’s 1966 papers were felt through the work of a small group of theoreticians and experimentalists who had read the papers and had become obsessed with the problem that they posed, and the first direct tests of Bell’s inequality were performed in 1972, by Stuart Freedman and John Clauser. These experiments produced the violations of Bell’s inequality predicted by quantum mechanics but, because of some further assumptions that were necessary in order to extrapolate the data, only a weaker form of the inequality was tested.

Other results followed, but the first comprehensive experiments designed specifically to test the general form of Bell’s inequality were those performed by Alain Aspect and his colleagues Philippe Grangier, Gérard Roger, and Jean Dalibard, at the Institut d’Optique Théoretique et Appliquée, Université Paris-Sud in Orsay, in 1981 and 1982.18,19

Aspect and his colleagues settled on excited calcium atoms as the source of entangled photons. In the lowest energy ‘ground’ electronic state of the calcium atom, the outermost 4s orbital is filled with two spin-paired electrons (4s²). If one of these electrons absorbs a photon of the right wavelength, then the electron is excited to a higher-energy 4p orbital. In this process, the photon that is absorbed imparts a quantum of angular momentum, and this appears as orbital angular momentum of the excited electron, the value of L increasing by 1. If there is no change in the spin orientations of the two electrons, the excited state is still a singlet state, S is equal to 0, and, since $L=1$⁠, there is only one possible value for $J$⁠: $J=1$⁠. This excited state is labeled 4s¹4p¹(¹P₁) (cf. the discussion of the electronic states of helium in Chapter 8).

Now suppose that it is possible to excite a second electron (the one ‘left behind’ in the 4s orbital) also into this same excited 4p orbital, but in a way that maintains the alignment of the electron spins. In other words, we create a doubly excited state in which the electron spins remain paired (4p²). This gives rise to three different electronic states corresponding to the three different ways of combining the angular momentum vectors. In one of these the orbital angular momentum vectors of the individual electrons cancel, $L=0$ and, since $S=0$⁠, we have $J=0$⁠. This particular doubly excited state is labeled 4p²(¹S₀).

This doubly excited state undergoes a rapid cascade emission through the intermediate 4s¹4p¹(¹P₁) state to return to the ground state (see Fig. 12.2). Two photons are emitted. Because the quantum number $J$ changes from $0→1→0$ in the cascade, the net angular momentum of the photon pair must be zero: they are emitted in opposite states of circular polarization, either $|L1R2〉$ or $|R1L2〉$ (as judged from the perspective of the source), described by the superposition in Eq. (12.1). The photons are entangled.

In fact, the photons have wavelengths in the visible region. Photon 1, from the 4p²(¹S₀) → 4s¹4p¹(¹P₁) transition, has a wavelength of 551.3 nm (green) and photon 2, from the 4s¹4p¹(¹P₁) → 4s²(¹S₀) transition, has a wavelength of 422.7 nm (blue).

Aspect and his colleagues used two high-power lasers to produce the excited calcium atoms, which were formed in an atomic ‘beam’, produced by passing gaseous calcium from a high-temperature oven through a tiny hole into a vacuum chamber. Subsequent collimation of the atoms entering the sample chamber provided a well-defined beam of atoms. The low density of atoms at the point of intersection with the laser beams ensured that the calcium atoms did not collide with each other or with the walls of the chamber before absorbing and subsequently emitting photons.

The physicists monitored the light emitted in opposite directions from the atomic beam source, using coloured filters to isolate the green photons (photons 1) on the left and the blue photons (photons 2) on the right. The photons were then passed into an arrangement consisting of two polarization analysers, four photomultipliers to amplify the signals from the detected photons, and electronic devices designed to detect and record coincident signals from the photomultipliers.

Each polarization analyser was mounted on a platform which allowed it to be rotated about its optical axis. Experiments could therefore be performed for different relative orientations of the two analysers, placed about 13 metres apart. The electronics were set to look for coincidences in the arrival and detection of photons 1 and 2 within a time window of just 20 ns. Any kind of ‘spooky’ signal passed between the photons, ‘informing’ photon 2 of the polarization state of photon 1, for example, would therefore need to travel the 13 metres between the detectors within this time window. In fact, it takes about twice this amount of time for a signal moving at the speed of light to cover this distance. The measurements were therefore ‘space-like’ separated.

Aspect and his colleagues measured the correlation $C12(φ)$ for seven different sets of analyser orientations. Their results are shown in Fig. 12.3. We saw from Eq. (12.16) that the quantum-mechanical prediction for $C12(φ)$ is simply $cos2φ$⁠, and this is the curve plotted through the data points, after corrections for experimental imperfections.* As anticipated, the predictions demonstrate that perfect correlation and perfect anti-correlation were not quite realized in these experiments. However, it is quite clear that the measured values of $C12(φ)$ agree well with the predictions.

The physicists also tested a generalized version of Bell’s inequality based on four different analyser configurations (with angles $α,β,γ,δ$⁠), applicable to non-ideal experiments in which perfect correlation or anti-correlation can’t be achieved. I won’t derive this here, but it is

The orientations were $φ1=β−α=22½°$⁠, $φ2=β−γ=−22½°$⁠, $φ3=δ−γ=22½°$⁠, and $φ4=δ−α=67½°$⁠. We can use the general form for $C12(φ)$ given in Eq. (12.16) to predict that

And we see that, once again, quantum mechanics predicts that the inequality is violated. Aspect and his colleagues obtained the result 2.697 ± 0.015, a violation of the inequality by 83% of the theoretical maximum (⁠$22=2.828$⁠).

You will note that nowhere in the above has it been necessary to introduce a specific local hidden variable theory to compare and contrast with the predictions of quantum mechanics. This might make you somewhat suspicious. If so, you can allay your suspicions by taking a quick look at Appendix 8, which summarizes a very simple (but intuitive) local hidden variables theory. This predicts results which do not violate Bell’s inequality.

Closing the Loopholes

Of course, this was not the end of the matter. For those physicists with deeply held realist convictions, there just had to be something else going on. More questions were asked: ‘What if the hidden variables are somehow influenced by the way the experiment is set up?’ This was just the first in a series of ‘loopholes’, invoked in attempts to argue that these results didn’t necessarily rule out all possible local hidden variable theories.

Aspect himself had anticipated this first loophole, and he and his colleagues performed further experiments to close it off. The experimental arrangement was modified to include devices which could randomly switch the paths of the photons, directing each of them towards analysers orientated at different angles. This prevented the photons from ‘knowing’ in advance along which path they would be traveling, and hence through which analyser they would eventually pass. This is equivalent to changing the relative orientations of the two analysers while the photons are in flight. It made no difference. Bell’s inequality was still violated.20

The problem can’t be made to go away simply by increasing the distance between the source of the entangled particles and the detectors. Experiments have been performed with detectors located in Bellevue and Bernex, two small Swiss villages outside Geneva almost 11 kilometers apart.21 Subsequent experiments placed detectors in La Palma and Tenerife in the Canary Islands, 144 kilometers apart. Bell’s inequality was still violated.22

Okay, but what if the hidden variables are still somehow sensitive even to random choices in the experimental setup, simply because these choices are made on the same timescale? In experiments reported in 2018, the experimental settings were determined by the colours of photons detected from distant quasars, the active nuclei of distant galaxies. The random choice of settings was therefore already made nearly 8 billion years before the experiment was performed, as this is how long it took for the trigger photons to reach the Earth. Bell’s inequality was still violated.23

There are other loopholes, and these too have been closed off in experiments involving both entangled photons and ions. Experiments involving entangled triplets of photons performed in 2000 ruled out all manner of locally realistic hidden variable theories without recourse to Bell’s inequality.24

If we want to adopt a realistic interpretation, then it seems we must accept that this reality is determinedly non-local.

Leggett’s Inequality: Crypto Non-local Hidden Variables

But can we still meet reality halfway? In these experiments, we assume that the properties of the entangled particles are governed by some, possibly very complex, set of hidden variables. These possess unique values that determine the quantum states of the particles and their subsequent interactions with the measuring devices. We further assume that the particles are formed with a statistical distribution of these variables determined only by the physics and not by the way the experiment is set up.

Local hidden variable theories are characterized by two further assumptions. In the first, we assume (as did EPR) that the outcome of the measurement on particle 1 can in no way affect the outcome of the measurement on 2, and vice versa. In the second, we assume that the setting of the device we use to make the measurement on 1 can in no way affect the outcome of the measurement on 2, and vice versa.

The experimental violations of Bell’s inequality show that one or other (or both) of these assumptions is invalid. But they don’t tell us which.

In a paper published in 2003, Anthony Leggett chose to drop the setting assumption. This means that the behaviour of the particles and the outcomes of subsequent measurements is assumed to be influenced by the way the measuring devices are set up. This is still all very spooky and highly counter-intuitive:25

By keeping the outcome assumption, we define a class of non-local hidden variable theories in which the individual particles possess defined properties before the act of measurement. What is actually measured will of course depend on the settings, and changing these settings will somehow affect the behaviour of distant particles (hence, ‘non-local’). Leggett referred to this broad class of theories as ‘crypto’ non-local hidden variable theories.

He went on to show that dropping the setting assumption is in itself still insufficient to reproduce all the results of quantum mechanics. Just as Bell had done in 1964, he derived an inequality that is valid for all classes of crypto non-local hidden variable theories but which is predicted to be violated by quantum mechanics. At stake then was the rather simple question of whether quantum particles have the properties we assign to them before the act of measurement. Put another way, here was an opportunity to test whether quantum particles have what we might want to consider as ‘real’ properties before they are measured.

The results of experiments designed to test Leggett’s inequality were reported in 2007 and, once again, the answer is pretty unequivocal. For a specific arrangement of the settings in these experiments, Leggett’s inequality demands a result which is less than or equal to 3.779. Quantum mechanics predicts 3.879, a violation of less than 3%. The experimental result was 3.8521, with an error of ± 0.0227. Leggett’s inequality was violated.26 Several variations of experiments to test Leggett’s inequality have been performed more recently. All confirm this result.

The debate continues. But we must acknowledge that in any realistic interpretation in which the wavefunction is assumed to represent the real physical state of a quantum system, the wavefunction must be non-local.

Notes

Further Reading