Bell's "Kinky" Square Polytope

Recall Bell's description of the entanglement process, with its assumption that first one measurement is made, and the other measurement is made later.

If measurement of the component σ₁ • a, where a is some unit vector, yields the value + 1 then, according to quantum mechanics, measurement of σ₂ • a must yield the value — 1 and vice versa... Since we can predict in advance the result of measuring any chosen component of σ₂, by previously measuring the same component of σ₁, it follows that the result of any such measurement must actually be predetermined.

Since the collapse of the two-particle wave function is indeterminate, nothing is pre-determined, although σ₂ is indeed determined to have opposite sign (to conserve spin momentum) once σ₁ is measured. Here Bell is describing the "following" measurement to be in the same direction as the "previous" measurement. In Bell's description, Bob is measuring "the same component" as Alice, meaning that he measures at the same angle as Alice.

If Bob should measure in a different spin direction from Alice (a different spin component), his measurements will lose their perfect correlation, slowly at first for a small angle. As the angle between their measurements increases, the correlation falls off as the squared cosine of the angle. Oddly, Bell's inequality for local hidden variables predicts a linear falloff with angle. We shall try to understand how Bell came up with a linear angle dependence for what he called his ad hoc model and later his "inequality." It is this linear dependence that leads to Bell's polytope being a square.

Supporters of the Copenhagen Interpretation claim that the properties of particles (like angular or linear momentum) do not exist until they are measured. It was Pascual Jordan who claimed the measurement creates the value of a property. This is true when the preparation of the state is in an unknown linear combination (superposition) of quantum states.

In our case, the entangled particles have been prepared in a superposition of states, but both of them have total spin zero.

ψ₁₂ = (1/√2) [ ψ₊ (1) ψ_- (2) - ψ_- (1) ψ₊ (2) ]

So whichever of these two states is created by the preparation, it will put the two particles in opposite spin states, randomly + - or - + , but still supporting Bell's view, that they will be perfectly (anti-)correlated when measured at exactly the same angle (measuring the same spin component).

Wolfgang Pauli called it a "measurement of the first kind" when a system is prepared in a state and if measured again, will be certainly found in the same state. (This is the basis for the quantum zeno effect.)

Since our two electrons have been prepared with one spin up and the other down, what could possibly cause them to change, for example, to both spins in the same direction, or as Copenhagen claims, simply to have both spins no longer definite until the next measurement?

As long as nothing interferes with either entangled particle as they travel to the distant detectors, they will be found to be still perfectly correlated, if (and only if) they are measured at the same angle. Otherwise, the correlations should fall off as the cosine (or perhaps the square of the cosine?) of the angle difference.

We can illustrate the straight-line predictions of Bell's inequalities for local hidden variables, the cosine curves predicted by quantum mechanics and conservation of angular momentum, and the odd "kinks" at angles 0°, 90°, 180°, and 270°, with what is called a "Popescu-Rorhlich box."
This square box is also called the Bell polytope.

It shows Bell’s local hidden variables prediction as four straight lines of the inner square. The circular region of quantum mechanics correlations are found outside Bell's straight lines, "violating" his inequalities. Quantum mechanics and Bell's inequalities meet at the corners, where Bell's predictions show a distinctly non-physical right-angle that Bell called a "kink."

All experimental results have been found to lie along the curved quantum predictions called the "Tsirelson bound."

In 1976, Bell gave us this diagram of the "kinks" in his local hidden variables inequality. He says,

Unlike the quantum correlation, which is stationary in θ at θ = 0, at the hidden variable correlation must have a kink there

("Einstein-Podolsky-Rosen Experiments," republished in Speakable and Unspeakable in Quantum Mechanics," 1987, p. 85)

Bell provides us no physical insight into the "kinky" square shape of his "local hidden variables" inequality.

In his famous 1981 article on "Bertlmann's Socks," Bell explains that the predictions for his "ad hoc" model are linear in the angle difference |a - b|, and he notes the fact that his inequality only agrees with the quantum predictions at the corners of the square of linear predictions above, and not at intermediate angles.

To account then for the Einstein-Podolsky-Rosen-Bohm correlations we have only to assume that the two particles emitted by the source have oppositely directed magnetic axes. Then if the magnetic axis of one particle is more nearly along (than against) one Stern-Gerlach field, the magnetic axes of the other particle will be more nearly against (than along) a parallel Stern- Gerlach field. So when one particle is deflected up, the other is deflected down, and vice versa. There is nothing whatever problematic or mind-boggling about these correlations, with parallel Stern-Gerlach analyzers, from the Einsteinian point of view.
So far so good. But now go a little further than before, and consider non-parallel Stern-Gerlach magnets. Let the first be rotated away from some standard position, about the particle line of flight, by an angle a. Let the second be rotated likewise by an angle b. Then if the magnetic axis of either particle separately is randomly oriented, but if the axes of the particles of a given pair are always oppositely oriented, a short calculation gives for the probabilities of the various possible results, in the ad hoc model,...
P(up, down) = P(down, up) = 1/2 - |a-b|/2π
where ‘up’ and ‘down’ are defined with respect to the magnetic fields of the two magnets. However, a quantum mechanical calculation gives
P(up, down) = P(down, up) = 1/2 - 1/2(sin(a - b)/2)² [= 1/2(cos(a - b)/2)²]
Thus the ad hoc model does what is required of it (i.e., reproduces quantum mechanical results) only at (a — b) = 0, (a - b) = π/2 and (a — b) = π, but not at intermediate angles.

("Bertlmann's Socks and the Nature of Reality," republished in Speakable and Unspeakable in Quantum Mechanics," 1987, p. 146)

What was Bell's "short calculation" that gives "the probabilities of possible results" in his ad hoc model as linearly proportional to the angle |a-b|??

And what exactly was Bell's "quantum mechanical calculation" that gives us probabilities proportional to the cosine of the angle |a-b| squared?

Bell does not give us any underlying physical reasons for the linear dependence on angle. He clearly knows that his linear "inequality" is a strong challenge to the curved cosine prediction of quantum mechanics.

And Bell's odd prediction of sharp corners or "kinks" where his straight lines turn ninety degrees (it is only at these corners where his linear inequality agrees with the curving quantum mechanics), surely should have prompted Bell to give us a deeper explanation of his theorem?

When John Clauser wrote to Bell suggesting an experimental test of his inequality, Bell replied

"In view of the general success of quantum mechanics, it is very hard for me to doubt the outcome of such experiments. However, I would prefer these experiments, in which the crucial concepts are very directly tested, to have been done and the results on record."

And he added

"Moreover, there is always the chance of an unexpected result, which would shake the world."

The Age of Entanglement, by Louisa Gilder, p. 256.

Clauser later recalled to Gilder

"Being a young student in this age of revolutionary thinking, I naturally wanted to 'shake the world' ."

ibid..

The dependence on the square of the cosine is the so-called "law of Malus" for crossed polarizers as pointed out by Abner Shimony in his Stanford Encyclopedia article on Bell's Theorem.

Paul Dirac taught his "principle of superposition" with crossed polarizers in his 1930 textbook The Principles of Quantum Mechanics.

Can Perfect Correlations Be Explained by Conservation Laws?

We find that David Bohm and John Bell implicitly and Eugene Wigner explicitly, used conservation of angular momentum (or particle spin) to tell us that if one spin-1/2 electron is measured up, the other must be down. Just as Albert Einstein implicitly used conservation of linear momentum in his development of the EPR Paradox.

David Bohm and Yakir Aharonov wrote in 1957,

We consider a molecule of total spin zero consisting of two atoms, each of spin one-half. The wave function of the system is therefore

ψ = (1/√2) [ ψ₊ (1) ψ_- (2) - ψ_- (1) ψ₊ (2) ]

where ψ₊ (1) refers to the wave function of the atomic state in which one particle (A) has spin +ℏ/2, etc. The two atoms are then separated by a method that does not influence the total spin. After they have separated enough so that they cease to interact, any desired component of the spin of the first particle (A) is measured. Then, because the total spin is still zero, it can immediately be concluded that the same component of the spin of the other particle (B) is opposite to that of A.

("Discussion of Experimental Proof for the Paradox of Einstein, Rosen, and Podolsky,” Physical Review, vol.108, no.4. p.1070, 1957)

Eugene Wigner wrote in 1963

If a measurement of the momentum of one of the particles is carried out — the possibility of this is never questioned — and gives the result p, the state vector of the other particle suddenly becomes a (slightly damped) plane wave with the momentum -p. This statement is synonymous with the statement that a measurement of the momentum of the second particle would give the result -p, as follows from the conservation law for linear momentum. The same conclusion can be arrived at also by a formal calculation of the possible results of a joint measurement of the momenta of the two particles.

Writing a few years after Bohm, and one year before Bell, Wigner explicitly describes Einstein's conservation of momentum example as well as the conservation of angular momentum (spin) that explains perfect correlations between angular momentum (spin) components measured in the same direction
One can go even further: instead of measuring the linear momentum of one particle, one can measure its angular momentum about a fixed axis. If this measurement yields the value mℏ, the state vector of the other particle suddenly becomes a cylindrical wave for which the same component of the angular momentum is -mℏ. This statement is again synonymous with the statement that a measurement of the said component of the angular momentum of the second particle certainly would give the value -mℏ. This can be inferred again from the conservation law of the angular momentum (which is zero for the two particles together) or by means of a formal analysis.
(The Problem of Measurement, Eugene Wigner, in Wheeler and Zurek, Quantum Measurement p,340)

John Bell wrote in 1964,

With the example advocated by Bohm and Aharonov, the EPR argument is the following. Consider a pair of spin one-half particles formed somehow in the singlet spin state and moving freely in opposite directions. Measurements can be made, say by Stern-Gerlach magnets, on selected components of the spins σ₁ and σ₂. If measurement of the component σ₁ • a, where a is some unit vector, yields the value + 1 then, according to quantum mechanics, measurement of σ₂ • a must yield the value — 1 and vice versa. Now we make the hypothesis, and it seems one at least worth considering, that if the two measurements are made at places remote from one another the orientation of one magnet does not influence the result obtained with the other.
"pre-determination" is too strong a term. The first measurement just "determines" the later measurement. We shall see that the second measurement is synchronous with the "first" in a "special" frame
Since we can predict in advance the result of measuring any chosen component of σ₂, by previously measuring the same component of σ₁, it follows that the result of any such measurement must actually be predetermined.
Since the initial quantum mechanical wave function does not determine the result of an individual measurement, this predetermination implies the possibility of a more complete specification of the state.

("On the Einstein-Podolsky-Rosen Paradox," Physics, 1.3, p.195.)

Just like Bohm and Wigner, Bell is implicitly using the conservation of total spin.

Albert Einstein made the same argument in 1933, shortly before EPR, though with conservation of linear momentum, asking Leon Rosenfeld,

Suppose two particles are set in motion towards each other with the same, very large, momentum, and they interact with each other for a very short time when they pass at known positions. Consider now an observer who gets hold of one of the particles, far away from the region of interaction, and measures its momentum: then, from the conditions of the experiment, he will obviously be able to deduce the momentum of the other particle. If, however, he chooses to measure the position of the first particle, he will be able tell where the other particle is.

(Niels Bohr, His Life and Work as seen by His Friends and Colleagues, 1967, S. Rozental, pp.128-129)

Supporters of the Copenhagen Interpretation claim that the properties of the particles (like angular or linear momentum) do not exist until they are measured. It was Pascual Jordan who claimed the measurement creates the value of a property. This is true when the preparation of the state is in an unknown linear combination (superposition) of quantum states.

And in our case, quantum mechanics describes the entangled particles as prepared in a superposition of two-particle states, but note that both of the states have total spin zero.

ψ₁₂ = (1/√2) [ ψ₊(1) ψ_-(2) - ψ_-(1) ψ₊(2)] (1)

Now this initial entangled state is spherically symmetric and rotationally invariant. It has no preferred spin direction that could "pre-determine" the directions that will be found by Alice and Bob, as Bell described.

The preferred direction is created by Alice's measurement, or by Bob's should he measure first in the "special frame" in which Alice and Bob are "at rest" and equidistant from the location of the initial entanglement.

Let's assume that Alice measures first and gets spin +1/2. The prepared state has been projected (randomly) into ψ₊(1) ψ_-(2).

But most important, Alice's measurement establishes the angle of her spin measurement - the angle of her Stern-Gerlach magnet in the x,y plane. Werner Heisenberg says it is her free choice to measure the x-component. As the Copenhagen Interpretation describes this , Alice brings this x-component property into existence. (This was Pascual Jordan's contribution to the interpretation.)

There was no x- or y-component in the rotationally invariant prepared entanglement.

Paul Dirac pointed out that the actual value for the property depend's on what he calls "Nature's choice." The initial prepared state (1) might equally have collapsed into ψ_-(2). This is the source of the quantum randomness which is critically important for quantum encryption.

Whichever of the two states is projected by Alice's measurement, it breaks the original symmetry, and puts the two particles in opposite spin states, randomly + - or - +, supporting the views of Bohm, Wigner, and Bell, that particles will be perfectly (anti-)correlated when measured.

In our example, since Alice measured the x-component of spin as +1/2, Bob will necessarily (and because of conservation of angular momentum) measure the x-component as -1/2.

As we saw above, Wolfgang Pauli called it a "measurement of the first kind" when a system is prepared in a state, so that when measured, it will certainly be found in the same state.

As long as nothing interferes with either entangled particle as they travel to the distant detectors (though perhaps decoherence?), they will be found to be perfectly correlated if (and only if) they are measured at the same angle (in our case, the x-component). Otherwise. the correlations should fall off as the square of the cosine of the angle difference. It is strange that Bell accepted an inequality that predicts correlations fall off with angle as a non-physical straight-line function with "kinks."

In any case, conservation laws tell us that when either particle is measured, we know instantly those properties of the other particle, including its location equidistant from, but on the opposite side of, the entangling interaction, and all other conserved properties such as spin.

But this is not "action-at-a-distance." It's just "knowledge-at-a-distance."

A more recent (2005) study showing that correlations in Bell tests is the result of conservation of angular momentum is "Correlation functions, Bell's inequalities and the fundamental conservation laws," by C. S. Unnikrishnan of the Tata Institute in India. He also discusses the odd "kinks" in Bell's linear predictions of correlations compared to the conservation law curve.

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