# Bell's theorem

Bell's theorem is a "no-go theorem" that draws an important distinction between quantum mechanics (QM) and the world as described by classical mechanics. It proves that quantum physics is incompatible with certain types of local hidden-variable theories. This theorem is named after John Stewart Bell.

## Quotes

* I’ve had experts in quantum field theory – people who’ve spent years calculating path integrals of mind-boggling complexity – ask me to explain the Bell inequality to them, or other simple conceptual things like Grover’s algorithm. I felt as if Andrew Wiles had asked me to explain the Pythagorean Theorem.
** Scott Aaronson, Quantum Computing Since Democritus (2013), Ch. 9 : Quantum
* The purpose of the first part is to convince the reader that the formalism leading to Bell's inequalities is very general and reasonable. What is surprising is that such a reasonable formalism conflicts with quantum mechanics. In fact, situations exhibiting a conflict are very rare, and quantum optics is the domain where the most significant tests of this conflict have been carried out
** Alain Aspect, "Bell's Theorem: The Naive View of an Experimentalist", in Quantum [Un]speakables (2002) edited by Reinhold A. Bertlmann and Anton Zeilinger
* One of these articles, written by N. David Mermin, gave me a tremendous shock. Mermin described the results of experiments that had been carried out as recently as 1982 to test something called Bell's theorem using two-photon 'cascade' emission from excited calcium atoms. Put simply, Bell's theorem says that my idea of naive realism is in conflict with the predictions of quantum theory in a way that can be tested in the laboratory in special experiments on pairs of quantum particles. These experiments had been done: quantum theory had been proved right and naive realism wrong! There in a montage was a pictorial history of the debate about reality and the experiments that had been done to test it (reproduced opposite). This work struck me as desperately important to my understanding of physical reality, something that as a scientist I felt I ought to know about. This discovery also made me feel rather embarrassed. Here I was, proud of my scientific qualifications and with almost 10 years' experience in chemical physics research at various prestigious institutions around the world, and I had been going around with a conception of physical reality that was completely wrong! Why hadn't somebody told me about this before?
** Jim Baggott, The Meaning of Quantum Theory (1992), Preface
* Again, part of that psychohistorical study I would like to see is why it did not impress the Copenhagen people, especially Bohr. But in the end it turns out that these other people were, in a way, right, because what I am notorious for, the so-called Bell's theorem, is just for showing that Einstein's explanation doesn't work. Einstein's explanation works so long as you have perfect correlations, which means measuring the same component of spin on the two sides [spin is a measure of a property similar but not identical to the rotation of a particle on its axis]. But as soon as you are measuring in a nonparallel direction, you get results that cannot be explained by Einstein's idea that the answers existed before the experiment.
** John Stewart Bell, in interview, Omni, May 1988, p. 90
* The theorem tells you that maybe there must be something happening faster than light, although it pains me even to say that much. The theorem certainly implies that Einstein's concept of space and time, neatly divided up into separate regions by light velocity, is not tenable. But then, to say that there's something going faster than light is to say more than I know.
** John Stewart Bell, in interview, Omni, May 1988, p. 90
* The main impact of Bell's theorem is from a philosophical-historical perspective: it reinforces, outside the physics of quantum entanglement, the incompatibility of hidden variable theories with quantum mechanics.
** in F. J. Duarte (2022). "30". Fundamentals of Quantum Entanglement. Bristol: Institute of Physics (UK). ISBN 978-0-7503-5265-9.
* That's all. That's the difficulty. That's why quantum mechanics can't seem to be imitable by a local classical computer. I've entertained myself always by squeezing the difficulty of quantum mechanics into a smaller and smaller place, so as to get more and more worried about this particular item. It seems to be almost ridiculous that you can squeeze it to a numerical question that one thing is bigger than another. But there you are—it is bigger than any logical argument can produce, if you have this kind of logic.
** Richard Feynman, "Simulating Physics with Computers", International Journal of Theoretical Physics, volume 21, 1982, p. 467-488
* Bell’s theorem is the most profound discovery of science.
** Henry P. Stapp, "Bell's Theorem and World Process", Nuovo Cimento, Vol. 29B, No. 2, p. 270 (1975).
* The gist of Bell's theorem is this: no local model of reality can explain the results of a particular experiment.
** Nick Herbert Quantum Reality - Beyond The New Physics Chapter 11, The Einstein-Podolsky-Rosen Paradox, p. 199
* Bell himself managed to devise such a proof which rejects all models of reality possessing the property of "locality". This proof has since become known as Bells theorem. It asserts that no local model of reality can underlie the quantum facts. Bell's theorem says that reality must be non-local.
** Nick Herbert Quantum Reality - Beyond The New Physics Chapter 12, Bell's Interconnectedness Theorem, p. 212
* Physicists continue to debate whether Bell's theorem is airtight or not. However, the real question is not whether Bell can prove beyond doubt that reality is non-local, but whether the world is in fact non-local.
** Nick Herbert Quantum Reality - Beyond The New Physics Chapter 13, The Future Of Quantum Reality, p. 238
* There's an interesting scientific principle that a wrong answer can be much more stimulating to the field than just sort of finding the answer that's in the back of the book. A wrong result gets people excited. Worried. Obviously, you don't really want that to be happening—it's OK for a theorist to come up with a speculative new theory that gets shot down, but experimentalists are supposed to be very careful and their error limits are supposed to be realistic. Unfortunately, with this experiment, whenever you're looking for a stronger correlation, any kind of systematic error you can imagine typically weakens it and moves it toward the hidden-variable range. It was a hard experiment. In those days, at any rate, with the kind of equipment I had, and … well, what can I say? I screwed up.
** Richard Holt (physicist), as quoted by Louisa Gilder, in The Age of Entanglement, Vintage Books, 2008, p. 286: Quote regarding his wrong experimental results that implied that quantum mechanics yielded the wrong prediction regarding Bell's theorem.
* The experimental verification of violations of Bell’s inequality for randomly set measurements at space-like separation is the most astonishing result in the history of physics. Theoretical physics has yet to come to terms with what these results mean for our fundamental account of the world. Experimentalists, from Freedman and Clauser and Aspect forward, deserve their share of the credit for producing the necessary experimental conditions and for steadily closing the experimental loopholes available to the persistent skeptic. But the great achievement was Bell’s. It was he who understood the profound significance of these phenomena, the prediction of which can be derived easily even by a freshman physics student. Unfortunately, many physicists have not properly appreciated what Bell proved: they take the target of his theorem— what the theorem rules out as impossible—to be much narrower and more parochial than it is. Early on, Bell’s result was often reported as ruling out determinism, or hidden variables. Nowadays, it is sometimes reported as ruling out, or at least calling in question, realism. But these are all mistakes. What Bell’s theorem, together with the experimental results, proves to be impossible (subject to a few caveats we will attend to) is not determinism or hidden variables or realism but locality, in a perfectly clear sense. What Bell proved, and what theoretical physics has not yet properly absorbed, is that the physical world itself is non-local.
** Tim Maudlin, "What Bell Did", Journal of Physics A: Mathematical and Theoretical (2014)
* No physical theory of local hidden variables can ever reproduce all of the predictions of quantum mechanics.
** C.B. Parker (1994). McGraw Hill Encyclopaedia of Physics (2nd ed.). McGraw Hill. p. 542. ISBN 0-07-051400-3.
* Bell's theorem, for which he is most famous, was more a triumph of character than of intellect. The difficult thing about it was the realization of what was understood and what was not understood in the discussion of hidden variables. Bell's honesty about his own understanding provided the impetus for his formulation and proof of the theorem.
** Abner Shimony, "A Tribute to John S. Bell," PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association (1990)
* In the same paper, Bell also discussed two rather unwelcome properties of hidden-variables theories. The first was contextuality. This tells us that, except in trivial cases, any hidden-variable theory must be such that the result of measuring a particular observable will depend on which other observable) are measured simultaneously. The second was nonlocality. All me hidden-variable models that Bell examined, including Bohm's, had the unpleasant feature that the behaviour of a particular particle depended on the properties of all others, however far away they were. In the EPR case, the measurement result obtained on one particle would depend on what measurement is performed on the second. As Bell said, this was the resolution of the EPR problem that Einstein would have liked least, and it is in this sense that it may be said that Bell proved Einstein wrong.
** Andrew Whitaker, "John Bell and the most profound discovery of science", Physics World (December 1998)
* At the very least, Bell's Theorem prevents us from interpreting quantum amplitudes as probability in the obvious way. You cannot point at a single configuration, with probability proportional to the squared modulus, and say, "This is what the universe looked like all along."
** Eliezer Yudkowsky, "Bell's Theorem: No EPR "Reality" " (04 May 2008)

## See also

* John Stewart Bell
* EPR paradox
* Quantum entanglement

## External links

* Bell's Theorem by Abner Shimony (2004) in the Stanford Encyclopedia of Philosophy.