Bell’s Theorem: The End of Hidden Variables?
In the year 1964, John Stewart Bell, a physicist, published an article titled ‘On the Einstein-Podolsky-Rosen Experiment’, citing an apparent paradox discovered by three greats of Institute of Advanced Study, Princeton, namely Albert Einstein, Nathan Rosen, and Boris Podolsky.
The paradox, known as the ‘EPR Paradox’, rejected the uncertainties of quantum mechanics. So, what was this Paradox and how did Bell’s theorem solve this problem?
The EPR Paradox
In the year 1935, a paper was published in the journal Physical Review, titled, ‘Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?’. This paper, co-authored by Albert Einstein, Boris Podolsky and Nathan Rosen, famously introduced a paradox, now known as the EPR Paradox.
The paper argued that the existing quantum mechanical theory and mathematical framework of the time was incomplete as it contradicted reality. In the article’s own language,
either (1) the description of reality given by the wave function in quantum mechanics is not complete or (2) two quantum operators cannot have simultaneous reality.
The EPR Paradox illustrates the “incompleteness of quantum theory” through a thought experiment with entanglement (though the original paper does not explicitly talk about the same). It goes as follows:
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!
Special relativity proves that the speed of light is the ultimate speed of the Universe. No object/information in the universe can travel faster than 299, 792,458 m/s. As a result, if two objects are exchanging information in space-time, they must not break this speed barrier. But, if two objects, separated by the width of the Milky Way are interacting with each-other instantaneously, “they were violating a fundamental law of nature”, as believed by the authors of this famous paper.
As a side note, the original paper mathematically builds upon this incomplete description of reality by taking an account of two mutually non-commutative quantum operators, say A and B (non-commutative meaning that AB≠BA), showing their probabilistic incompatibility with each-other and henceforth concluding on the incomplete description of reality.
Some people might find this thought experiment somewhat trivial. Another way to think about it is if you have two spinning tops, identical in all aspects, without seeing either, take one of them out of this world, open the box, see the color of the top, say red, and immediately know that the other top is red in color. This is exactly what is called a hidden variable approach, that the quantum state is already determined previously, all what is left, is to measure the quantum state. But, as we’ll soon see, this is not how nature works!
Actually, a better analogy would be as follows: picture two tops (analogous to quantum states) spinning side-by-side; let them collide with each-other suddenly, elastically (analogous to quantum interactions, such that now the momenta of the two tops is mutually entangled). As soon as they collide, we immediately transport one of them to the far reaches of the galaxy, without interacting with the motion of either of the tops. What entanglement means is that if we see either of the tops coming to rest along, say, North-South axis, (head pointing North, tail pointing South) then this will be a proof of the fact that the other top will point towards South-North axis, i.e. head pointing South and tail pointing North.
The hidden-variable approach was advocated by the authors of the paper previously discussed. They argued against the probabilistic effects of quantum mechanics, and entanglement, especially over very large distances. This approach was formalized into a complete theory by David Bohm, and hence is also called the Bohmian interpretation, or the Pilot-Wave Theory (more information on bohmianmechanics.org).
A possible solution to the hidden-variable discussion was provided by John Stewart Bell in 1964, known as Bell’s Inequality:
Bell’s Inequality
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.
A really good video about the Bell Test on realism and locality is this one, by Minute Physics (and 3Blue1Brown), that I would recommend watching for an introduction to Bell’s Theorem, where they have discussed the other example of Bell’s inequality, focusing on polarization of light.
Though we have disproven the existence of a real, local hidden variable in quantum mechanics, many physicists now are ready to sacrifice the condition of locality to support a hidden-variable theory, including the likes of John Stewart Bell (as quoted in an interview from the 1980s). Though many experiments have confirmed that a real, local variable does not exist, there are many loopholes in these experiments, as a result of which the non-existence of any hidden variable is yet to be proven or tested. The Bohmian Mechanics provide a complete interpretation based on the supposed existence of a non-local hidden variable (find more information at: Bohmian Mechanics).
Bell’s inequality leads to yet another powerful hypothesis, superdeterminism, formulated by Bell himself, according to which everything, literally everything is pre-determined, supposed to happen at that very incident at which it did, irrespective of any external actions.
So, is this the end of hidden variables? You decide!
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