In a previous post on Bell’s Inequalities, I argued that the inequalities could be derived from two premises:

Premise 1: *One can devise a test that will give one of two discrete results.* For simplicity we label these (+) and (-).

Premise 2: *We can carry out such a test under three different sets of conditions, which we label A, B and C.*

Since a violation of a mathematical relationship falsifies it, and since tests on entangled particles are alleged to comply with these two premises yet the inequalities were violated, either one of these premises were violated, or a new mathematical relationship is required. In this post I shall present one argument that the experiments that involve rotating polarizing detectors, with the classic experiment of Aspect *et al.* (*Phys. Rev. Lett.* **49**, 91-94, 1982) as an example, did not, as claimed, show violations of the inequality.

Before proceeding, this argument does not in any way deny entanglement, nor does it say anything about locality/non-locality. I am merely arguing there is a logic mistake in what is generally considered. To proceed, to clarify my definition of entanglement:

*An entangled pair of particles is such that certain properties are connected by some rule such that when you know the value of a discrete property of one particle, you know the value of the other particle, even though you have not measured it. *

Thus if one particle was found to have a spin clockwise, the spin of its entangled partner MUST be either clockwise or anticlockwise, depending on the rule *or the particles are not entangled*. That there are only two discrete values for properties such as spin or polarization means you can apply Bell’s Inequality, which for purposes of illustration, we shall test in the form

A(+) B(-) + B(+)C(-) ≧ A(+)C(-)

The Aspect experiment tested this by making a pair of entangled photons, and the way the experiment was set up, the rule was, each photon in an entangled pair would have the same polarization. The reason why they had entangled polarization was that the photons were generated from an excited 4P spin-paired state of calcium, which in sequence decayed to the spin-paired 4S state. The polarization arose from the fact that when decaying from a P state to an S state, each electron loses one quantum of action associated with angular momentum, and since angular momentum must be conserved, the photon associated with each decay must carry it away, and that is observed as polarization. There is nothing magic here; all allowed emissions of photons from electronic states involve a change of angular momentum, and usually of one quantum of action.

What Aspect did was to assign a position for the polarization detectors such that A was assigned to be vertical for the + measurement, say, 12 o’clock, and horizontal, say 3 o’clock for the – measurement. The position B was to rotate the detectors clockwise by 22.5^{o}, and for C, to rotate by 45^{o}. There is nothing magic about these rotations, but they are chosen to maximise the chances of seeing the effects. So you do the experiment and what happens? All detectors count the same number of photons. The reason is, the calcium atoms have no particular orientation so pairs of photons are emitted in all polarizations. What has happened is the first detector has detected half the entangled pairs, and the second the other half. We want only the second photon of the entangled pair detected by the first detector, so, instead at the (-) detector we only count photons that arrived within 19 ns of a photon registered at the (+) detector, then we find, as required, if the first detector is at A(+), then no photons come through at A(-). That was nearly the case.

Given that we count only measured photons, the law of probability requires A(+) =1; A(-) =0. The same will happen at B, and at C. (Some count all photons, so their probabilities are the ones that follow, divided by two.) There would be the occasional equipment failure, but for the present let’s assume all went ideally. This occurs because if we apply the Malus law to polarized photons, if the two filters are at right angles, and if working ideally, and if the two photons have the same polarization, and you only count photons at the second detector that are entangled with the first, there are zero photons going through the second filter. What is so special about the Malus law? It is a statement of the law of conservation of energy for polarized light, or the conservation of probability at 1 per event.

Now, let us put this into Bell’s Inequality, from three independent measurements, because the minus determinations are all zero: {A(+).B(-) + B(+).C(-)} = 0 + 0, while A(+)C(-) = 0. We now have 0 + 0 = 0, in accord with Bell’s inequality.

What Aspect did, however, was to argue that we can do joint tests and measure A(+) and B(-) on a set of entangled pairs. The proposition is, if we leave the first polarizing detector at A(+), but rotate the second we can score B(-) at the same time. Let the difference in clockwise rotations of the detectors be θ, thus in this example θ = 22.5 degrees. Following some turgid manipulations with the state vector formalism, or by simply applying the Malus law, if A(+) = 1, then B(-) = sin^{2} θ, and if we do the same for the others, we find,

{A(+).B(-) + B(+).C(-)} = 0.146 +0.146 while A(+)C(-) = 0.5 Oops! Since 0.292 is not greater or equal to 0.5, Bell’s inequality appears to be violated. At this point, I believe we should carefully re-examine what the various terms mean. In one of Bell’s examples (washing socks!) the socks undergoing the tests at A, B and C were completely separate subsets of all socks, and if we label these as *a, b *and *c* respectively, we can write {*a*} = ~{*b, c*}; {*b*}= ~{*a, c*}; {*c*} = ~ {*a, b*} where the brackets {} indicate sets. What Bell did with the sock washing was to take the result A(+) from the subset {*a*} and B(-) from the subset {*b*} and so on. But that is not what happened in the Aspect experiment, because as seen above, when we do that we have the result, 0 + 0 = 0. So, does this variation have an effect? In my opinion, clearly yes.

My first criticism of this is that the photons that give the B(-) determination are not those entangled with the B(+) determination. By manipulating things this way, B(+) + B(-) > 1. Previously, we decided that 1 represented the fact that an entangled pair was detected only once during a B(+) + B(-) determination, because the minus indicates “photons not detected”, but it has grown by rotating the B(-) filter. If we recall the derivation, we used the fact that B(+) + B(-) =1. Our experiment has not only violated Bell’s Inequality, but it violates our derivation of it.

Let us return to the initial position, The first detector vertical, the second horizontal, and we interpret that as A(-) = 0. That means that no photons entangled with those assigned as A(+) are recorded, and all photons actually recorded there are the other half of the photons, i.e. ¬A(+), or 0*A(+). Now, rotate the second detector by 90^{o}. Now it records all the photons that are entangled with the selection chosen by A(+). It is nothing more than an extension of the first detector, or part of the first detector translated in space, but chosen to detect the second photon. Its value is equivalent to that of A(+), or 0*A(-). Because the second photon to be detected is chosen as only those entangled with those detected at A(+), surely what is detected is still in the subset A, and what Aspect labelled as B(-) should more correctly be labelled 0.146*A(+), and what was actually counted includes 0.854*A(-), in accord with the Malus law. What the first detector does is to select a subset of half the available photons, which means A(+) is not a variable, because its value is set by how you account for the selection. The second detector applies the Malus law to that selection.

Now, if that is not bad enough, then consider that the B(+).C(-) determination is an exact replica of the A(+).B(-) determination, but has been rotated by 22.5 degrees. Now, you cannot prove 2[A(+) B(-)]≧ A(+)C(-), so how can you justify simply rotating the experiment? The rotational symmetry of space says that simply rotating the experiment does not change anything. This fact is, from Nöther’s theorem, the source of the law of conservation of angular momentum, and conservation laws to arise from such symmetries. Thus the law of conservation of energy depends on the fact that if I do an experiment today I should get the same result if I do it tomorrow. The law of conservation of momentum depends on the fact if I move the experiment to the other end of the lab, or to another town, the same result arises. Moving the experiment somewhere else does not change anything physically. The law of conservation of angular momentum depends on the fact that if I orient the experiment some other way, I still get the same result. Therefore just rotating the experiment does not generate new variables. So, we have the rather peculiar fact that it is because of the rotational symmetry of space that we get the law of conservation of angular momentum, and that is why we assert that the photons are entangled. We then reject that symmetry in order to generate the required number of variables to get what we want.

Suppose there were no rotational symmetry? This happens with experiments involving compass needles, where the Earth’s magnetic field orients the needle. Now, further assume energy is conserved and the Malus law applies. If a thought experiment is carried out on a polarized source and we correctly measure the number of emitted photons, now we have the required number of variables, but we find, surprise, Bell’s Inequality is followed. Try it and see.

My argument is quite simple: Bell’s Inequality is not violated in rotating polarizer experiments, but logic is. People wanted a weird result and they got it, by hook or by crook.