CHSH inequality and the rejection of realism

I will start now a series on Bell's theorem and its importance to quantum foundations. Today I will talk about the Clauser, Horne, Shimony, and Holt inequality and its implication. 

Quantum mechanics is a probabilistic theory which does not make predictions of the outcome of individual experiments, but makes statistical predictions instead. This opens the door to consider "subquantic" or "hidden variable" theories which would be able to restore full determinism. However, even with the statistical nature of quantum mechanics predictions there is something more which can be investigated: correlations. What distinguishes quantum from classical mechanics is how the observables of a composite system are related to the observables of the individual systems. In the quantum case there is an additional term related to the generators of the Lie algebras of each individual systems and this in turn prevents the neat factorization of the Hermitean observables of the composite system. It is this lack of factorization which prevents in general the factorization of the quantum states. In literature this goes under the (bad) name nonlocality.

Now suppose we have two spatially separated laboratories which receive from a common source pairs of photons. The "left lab" L chooses to measure the polarization of the photons on two directions \(\alpha\) an \(\gamma\), while the "right lab" R chooses to measure the polarization of the photons on two directions \(\beta\) an \(\delta\), Let's call the outcome of the experiments: \(a, b, c, d\) for the directions of measurement \(\alpha, \beta\, \gamma, \delta\), respectively. The values \(a, b, c, d\) can take are +1 or -1.

Now let us compute the following expression:

\(C=(a+c)b-(a-c)d\)

Suppose \((a+c) = 0\), then \((a-c)=\pm  2\) and so \(C=\pm 2\)

Similarly if \((a-c) = 0\), then \((a+c)=\pm  2\) and again \(C=\pm 2\)

Either way \(C=\pm 2\)

Now suppose we have many runs of the experiment and for each run \(i\)we get:

\(a_i b_i + b_i c_i + c_i d_i - d_i a_i = \pm 2\)

from which we deduce on average that:

\(|\langle ab\rangle + \langle bc\rangle + \langle cd\rangle - \langle da\rangle|\leq 2\)

This is the CHSH famous inequality. Now under appropriate circumstances nature violates this inequality:

in an experiment with photons the average correlation between measurements on two distinct directions \(\alpha, \beta\) is: \(\cos 2(\alpha - \beta)\) and the inequality to be obeyed is:

\(| \cos 2(\alpha - \beta) + \cos 2(\beta - \gamma) + \cos 2(\gamma - \delta) - \cos 2(\delta - \alpha)| \leq 2\)

but if the angle differences are at 22.5 degrees we get that \(2\sqrt{2} \leq 2\) so what is going on here?

A natural first objection is that not all 4 measurements can be simultaneously be performed and so we are reasoning counterfactually. But in N runs of the experiment we get 2N experimental results and there is a finite number of ways we can fill in the missing 2N data and in each counterfactual way of filling in the unmeasured data the CHSH inequality is still obeyed.

A second potential objection is that there is no free will and there is a conspiracy going on which prevents an unbiased choice of the 4 directions. There is no counteragument for this objection except that I know I have free will. If free will does not exists then mankind has much deeper troubles than explaining quantum mechanics: try to explain morality and justify the existence of the judicial system. 

The introduction of bias can affect correlations and if the detection rate depends on the angle, then for appropriate dependencies one can obtain the quantum correlations. This is the so-called detection loophole, However, if such a dependency exists, it can be tested in additional experiments and the introduction of angle dependency only for Bell test experiments is indefensible. Loophole free Bell experiments while important to push the boundary of experimental technology have no scientific importance and they count only towards experimentalist's bragging rights.

Another way to obtain correlations above 2 is by appealing to contextuality: for example the value of \(a\) when measured by lab L when lab R measures \(b\) may not be the same when lab R measures \(d\). While quantum mechanics is contextual, in this case such an argument means that the the choice lab R makes influences the result of measurement at lab L which is spatially separated!!!

Last, if the values of \(a, b, c, d\) do not exist prior to measurement, this decouples again the value of \(a\) when lab R measures \(b\) from the value of \(a\) when lab R measures \(d\). 

Assuming free will is true, we have only two choices at out disposal to be able to obtain correlations above 2: 

• measurement in a spatially separated lab affects the outcome on the remote lab
• the outcome of measurement does not exist before measurement.

The first choice is taken by dBB theory because the quantum potential changes instantaneously and the second option is advocated by the Copenhagen camp. (I am excluding the MWI proposal because in it there is no valid derivation of Born rule. I am also excluding collapse models because they are a departure from quantum mechanics and experiments will soon be able to reject them).

Now here is the catch: the two labs need not be spatially separated and one experiment in lab L can unambiguously happen before the experiment in lab R. When the R lab measurement takes place it cannot affect the outcome in the L lab because that is in the past and already happened! 

But can the first measurement affect the second one? In dBB this is possible as long as the first particle and its quantum potential is still around to "guide" the second particle. However, if after the first measurement the first particle is annihilated by its antiparticle then its quantum potential vanishes. The behavior of quantum potential after annihilation  is a reason why a relativistic second quantization dBB theory is not possible: either the quantum potential sticks around and messes up subsequent measurements, or vanishes and then the correlations cannot occur in the case above. (dBB supporters pin their hopes on a "future to be discovered" relativistic dBB quantum field theory which never materialized and cannot exists for several reasons.)

So from the two choices above only one remains valid:

the outcome of measurement does not exist before measurement

Realism is rejected by Bell's theorem. However in literature Bell's result is presented instead as a rejection of locality. But this is an abuse of language: locality=state factorization. Nature and quantum mechanics are incompatible with a state factorization. State factorization is just factorization, not locality. Rejection of realism is the only viable option left.