John Stewart BELL theorem

Bell’s Theorem is the collective name for a family of results, all of which involve the derivation, from a condition on probability distributions inspired by considerations of local causality, together with auxiliary assumptions usually thought of as mild side-assumptions, of probabilistic predictions about the results of spatially separated experiments that conflict, for appropriate choices of quantum states and experiments, with quantum mechanical predictions. These probabilistic predictions take the form of inequalities that must be satisfied by correlations derived from any theory satisfying the conditions of the proof, but which are violated, under certain circumstances, by correlations calculated from quantum mechanics. Inequalities of this type are known as Bell inequalities, or sometimes, Bell-type inequalities. Bell’s theorem shows that no theory that satisfies the conditions imposed can reproduce the probabilistic predictions of quantum mechanics under all circumstances.

The principal condition used to derive Bell inequalities is a condition that may be called Bell locality, or factorizability. It is, roughly, the condition that any correlations between distant events be explicable in local terms, as due to states of affairs at the common source of the particles upon which the experiments are performed.
In the mid 1960s Bell brought a major innovation to another field, foundations of quantum mechanics. After exposing flaws in John von Neumann’s proof against the possibility of modifying quantum mechanics via the introduction of additional “hidden” variables, Bell reexamined the famous Einstein-Podolsky-Rosen thought experiment suggested by Albert Einstein and his collaborators in 1935. Bell was able to prove that no theory complying with the assumptions of realism and locality could reproduce all quantum mechanics results—a proof that we now call Bell’s theorem.
Bell's Theorem (Reality must be non-local) is remarkable for several reasons: 1) it is a mathematical proof, not a conjecture or speculation; 2) it is a proof about Reality not Appearances. How often does one find such a window into the nature of reality? 3) it is counter-intuitive: why should everywhere local facts need to be supported by a non-local reality?
The EPR Paradox ; Let us suppose a pair of quantum objects, electrons for example. We perform some interaction onto the pair of electrons, and hence make them into an entangled pair. Without measuring either of the two states, we transport them over hugely large distances, say across the Milky Way. For the sake of experimental simplicity, we assume that this transportation was perfect, and no interaction was done with either of the quantum states we originally had. As a result, the entanglement is preserved. Quantum mechanics says that when we measure either of the state, we would instantaneously know the state of the other object. The authors of the article used this fact to highlight the “incomplete quantum picture”. They argued that this information must be incorrect as it violates a fundamental law of nature: the speed of light!
Bell’s inequality is mainly illustrated through two examples: the polarization of light, and the the spin of an electron. We will be focusing primarily on the electronic spin here.

So let’s imagine 100 singlet pairs of electrons, travelling somehow in two opposite directions free of any interactions. By virtue of their singlet pairing, their spins are already counter-entangled, where if the spin of one electron of a pair is clockwise, then that of the other becomes counter-clockwise and vice versa.

Now, let the 100 electrons with spin ‘up’/clockwise travel immensely far from the other 100 electrons (with spin ‘down’/counter-clockwise), such that two mutually entangled electrons cannot exchange information, locally, in a reasonable amount of time, though they are quantumly entangled. Let’s try to assign a real hidden variable, λ, to the electrons, which defines the bias of the electrons towards getting measured along horizontal or vertical axis. Let the electrons with spin up be with Alice and those with spin down be with Bob (a naming convention).

Alice and Bob, now perform reasonable measurements on the two spin states of the electrons using S-G magnets inclined inversely with each-other. Now, we follow three different cases:

The S-G magnets are inclined completely inversely with each-other; this means that two entangled-electrons would give the same outcome (as their spins are aligned inversely with each-other). Hence 100% of the electrons shall give the same outcome. Hence all 200 (or 100 pairs) electrons give the same outcome.
Now, let’s orient one SG magnet slightly away from the other (δ= θ°, say); hence the electrons with a strong bias towards horizontal shall give different outcomes. Let the number of electrons giving different outcomes in this case be N₁. (Even if we orient the other SG magnet by δ=θ°, keeping this SG magnet as δ=0, the number of electrons, N₂, giving different outcomes would be the same, as the relative angle between the SG magnets matters and not the absolute angle, hence N₁=N₂).
Now, let’s orient both of the SG magnets by δ₁=θ°, but in the opposite directions; this means, by convention, δ₁=θ° for magnet-1, and δ₂= –θ° for magnet-2. Hence, the relative angle between the two SG magnets would be δ = δ₁– δ₂ = 2θ°. Therefore, corresponding to our hidden variable, λ, the number of electrons giving different outcomes, N should NOT be greater than (N₁+N₂ =) 2N₁.
But, this number does not match with the predictions of quantum mechanics. According to the calculations of quantum mechanics, the number N₁ (=N₂), after proper approximations, should be equal to θ²/4. And N=(2θ)²/4=4θ²/4 =4N₁. But this contradicts our previous result that N ≵2N₁. Hence this contradiction prohibits the existence of a real and local hidden variable in quantum mechanics.

This, in a nutshell, is Bell’s Theorem. In mathematical form, this is represented in the form of an inequality, in the original paper, as:


Realism: The idea that nature exists independently of whether somebody is witnessing it or not.

Locality: The principle that an image is directly influenced only by its immediate surroundings, in other words, no information or cause can be transmitted faster than the speed of light.