Bell's Theorem: An Overview with Lotsa Links

By David R. Schneider

NOTE: Please feel free to link to this page. I will leave it up permanently for this purpose.

J.S. Bell

On this page:

Photon Polarization
Assumptions leading to Bell's Inequality
Tests of Bell's Inequality

Note: The overview is intended for the those new to this subject. Advanced readers will benefit from the links.

Link Categories:
More About Bell from DrChinese
Bell's Theorem
Tests of Bell's Inequality
The GHZ Theorem and other theorems ruling out Local Realism
Multi-photon and other Exotic Entanglement
Parametric Down Conversion (PDC), the Quantum Eraser, and other physical phenomena
Additional Stanford/Wikipedia/Nobel Site References of Interest
Philosophical and Other Discussion regarding Bell's Theorem

Note: The links include many scholarly (advanced) papers as well as introductory/intermediate references... something for everyone!



Bell's Theorem is the legacy of the late great J.S. Bell. Published in 1965, Bell's Theorem is famous for drawing an important line in the sand between Quantum Mechanics (QM) and the world as we know it intuitively. It is simple and elegant, and at the same time touches upon many of the fundamental philosophical issues that relate to modern physics. If you want to understand the richer meaning of Relativity and Quantum Theory, you will also want to learn about Bell. In it simplest form, Bell's Theorem states:

No physical theory of local Hidden Variables can ever reproduce all of the predictions of Quantum Mechanics.

The significance of this statement is as follows: Quantum Mechanics is the "strange" theory introduced in 1927 by Niels Bohr and Werner Heisenberg to describe the fundamental nature of basic particles: the atomic nucleus, electrons and light (photons, or electromagnetic waves). This theory was a tremendous improvement upon pre-existing theory, and yielded immediate successes. In fact, the same theory exists today as Quantum Mechanics with virtually no change (although it has been extended to explain more phenomena). The 1927 version introduced such novel concepts as: the Heisenberg Uncertainty Principle; Max Born's statistical interpretation of the wave function, including superposition; and Bohr's complementarity (wave-particle duality). In addition, it included important recent advances such as the Schoedinger wave function (1925); the Pauli exclusion principle (1923); Bohr's semi-classical model of the atom (1913); additional contributions from Louis de Broglie and Paul Dirac; and of course the early seminal work of Max Planck (1900) and Albert Einstein (1905).

So what's the problem? Why argue with success? The answer is that the philosophical implications of the new Quantum Theory were troubling to many prominent physicists of the day, including Albert Einstein himself. They felt that Quantum Theory might be a stepping stone, but could not be a final theory. And thus began a long series of attacks on the nascent theory. Einstein and others felt that the holes in QM were obvious, and yet the theory showed no weaknesses in its ability to make useful predictions. Bohr was the defender of QM: if it was wrong, why was its predictions right? So it was argued instead that the theory was not actually "wrong" but rather "incomplete". Yet no significant modifications were forthcoming. Bell's Theorem shows that the dream of a "more complete" theory is just that: a dream. It cannot happen. That a single (and elegantly straightforward) theorem could end such an important debate is what led Stapp to call this theorem "the most profound in science" (1975).

This page is intended to give you an overview of Bell's Theorem: the history, a short explanation, and a summary of tests of Bell's Theorem. In addition, you will find lots of links to subjects closely related to Bell's Theorem.


In 1935, Albert Einstein collaborated with Boris Podolsky and Nathan Rosen to publish a paper which is now simply referred to by the initials of its three authors: EPR (1). In essense, they concluded that Quantum Mechanics was incomplete because there existed so called "Hidden Variables" which must explain at least some of the uncertainty present in QM. Hidden Variables means that there are microscopic properties of fundamental particles that we are unable to observe directly by means of testing, perhaps due to technological limitations that might not exist at some future time. I.e. maybe we simply need a bigger microscope to see the details of what is going on at the very smallest level. But since we can't observe them, they may be "hidden" now - but perhaps if we knew more about them then that might explain the otherwise mysterious behavior of particles.

EPR had a simple but powerful definition of what they called an "element of reality": IF an observable property of a system could be predicted with absolute certainty (100%) without disturbing that system, THEN it must correspond with an element of reality. A simple and succinct definition, to be sure. And one with which few would argue.

EPR provided a proof that says in essence: either there are Hidden Variables, OR particle attributes (such as position, velocity, energy, polarization, etc.) are not real and defined until they are observed. EPR also said that since it is "unreasonable" to believe that these particle attributes require observation to become real, therefore Hidden Variables must exist. Einstein said: "I think that a particle must have a separate reality independent of the measurements. That is: an electron has spin, location and so forth even when it is not being measured. I like to think that the moon is there even if I am not looking at it." (Of course, this statement is not intended to be literal point of debate.)

Bell saw it differently, as many physicists did, and concluded that there were no Local Hidden Variables. I.e. what is "unreasonable" is simply a matter of opinion, and is not definitive proof. QM actually says that an observation does shape reality - and can do so even after the fact! A strange position, to be sure, but not contradicted by the facts. (Note: If you are interested, this part of QM is called the Heisenberg Uncertainty Principle.) Building from this position, Bell then went further: he showed that the Local Hidden Variable scenarios impose a critical requirement which is not obvious. 30 years after EPR, Bell exposed this requirement. It is referred to as Bell's Inequality and can be experimentally tested using entangled particles. The physics is described below, and the results are described.

Photon Polarization

Let's consider the case of the polarization of light. This is the easiest way to relate to Bell's Theorem (although his theorem actually related to electron spin, which is very similar).

A particle of light is called a photon (this is exactly the same as an electromagnetic wave; radio waves are actually photons). Most photons are linearly polarized and this polarization can be observed (measured) using a polarizer filter or a polarizer crystal. Polarizing sunglasses use this polarization characteristic as the basis for their operation. As a general rule, half of all RANDOMLY polarized light will pass through a polarizing filter. The other half will be reflected/absorbed by the filter. This is true regardless of the orientation of the filter. Generally, the filter can be set at any angle from 0 degrees to 360 degrees - i.e. it can be rotated at will in a circle. These effects are seen even if the light source is made to be so weak that a single photon passes though at a time! Once any photon passes through a polarizer lens, its polarization will be aligned exactly with with the lens thereafter (even if it wasn't previously).

Suppose we consider a single particle (photon) of light. We ask a simple question: does it have a definite polarization at the following three angles: 0 degrees (A), 120 degrees (B), and 240 degrees (C)? According to the EPR paper, its polarization at these 3 angles correspond to actual elements of reality IF they can be predicted with certainty without disturbing the system. This can be confirmed via conventional optics: any photon with a known polarization (say 0 degrees) can be checked at a later time by another polarizer at the same angle setting. The predicted result (100% certainty) is that the photon will have the same polarization at that later time, and this has been known to be true for about 200 years (Malus, 1809). By extension, A, B and C must correspond individually to elements of reality, according to the definition of EPR. (In fact, any angle's polarization will meet this test individually.)

Next we come to the $64,000 question: do A, B and C correspond to SIMULTANEOUS elements of reality? According to EPR, an element of reality exists independent of the act of observation. I.E. all elements of reality have definite values at all times, EVEN IF WE DON'T KNOW THEIR VALUES. In fact, EPR says that any other position would be unreasonable. So according to this view, angles A, B and C must have simultaneous definite answers to the questions of their polarizations - even if they cannot be known. So at angle A there might be a hidden variable A; and at angle B we have hidden variable B; and and at angle C we have hidden variable C. If only we had the tools, perhaps we could aspire to someday be able to simultaneously observe the hidden variables A, B and C of a single photon. We want to imagine here that even if the actual values of the photon's "Hidden Variables" seem capricious or random, they must be SOMETHING.


Bell was aware that there were some unusual cases in which pairs of particles could be created that had an unusual property - they remained connected in some way even after they were separated. This connection is called entanglement. Such a pair of particles act as a system until such time as a measurement is performed on one or the other. When one is observed, the other immediately assumes a state compatible with a conservation rule that applied to the pair as a combined system. This happens even if the distance between the pair is too large to be accounted for by a normal signal traveling at the speed of light between the particles (i.e. from one to the other). A faster than light influence could violate relativity - however, it turns out that there is no way to use this "effect" to perform any type of signaling.

Entanglement is the key to testing Bell's Theorem. A pair of entangled particles are allowed to separate, going in opposite directions. The actual spin orientation of the pair is not known. What is known is the relationship between the two. In a typical setup, if the spin of one is "vertical" then the spin of the other is "hortizontal" (and vice versa). This is known as orthogonal (or perpendicular) spins.

Assumptions leading to Bell's Inequality

Returning to the "$64,000" question about the simultaneous existence of A, B and C: Bell's Theorem addresses this mathematically using entangled particles as its base. To see the mathematical treatment, please check out this link (easiest) or this link (more true to the original) or read Bell's original paper itself in PDF form. The math is skipped here, and we instead "cut to the chase" so to speak.

Bell saw three assumptions that he felt any Local Realistic theory would include, regardless of its nature:

1. It should agree with the predictions of Quantum Mechanics (so as to agree with established experiments).
2. It should adhere to the principles of relativity (causes cannot propagate faster than the speed of light) - this is called Locality (sometimes Bell Locality). Specifically, a measurement setting for one member of an entangled particle pair should not affect the results of a measurement on the other member of the pair located at a distance. Otherwise, you would have so-called "spooky action at a distance".
3. There should be simultaneous existence of the elements of reality described above (A, B and C, for example). This is often called "Hidden Variables" or sometimes "Realism".

A person who believes in assumptions 2. and 3. above is called a Local Realist. These two assumptions are very reasonable, and there were a lot of physicists who believed them before Bell. Why not? You simply accepted the predictions of QM and assumed that 2. and 3. were true too. But... Bell showed that the three assumptions above are actually incompatible when combined. Therefore, at least one must be wrong. Bell derived a specific testable prediction related to this called Bell's Inequality. If Bell's Inequality is correct, then the first assumption is wrong. This was the line in the sand for the Local Realist.

Tests of Bell's Inequality

Tests of Bell's Inequality are commonly called tests of the EPR Paradox, or simply Bell tests. If Bell's Inequality is violated (i.e. incorrect), then it must be either the second or the third assumption above which is violated. The experiments of Alain Aspect et al, especially the 1981 tests using time-varying analyzers, have shown that Bell's Inequality IS violated and the predictions of QM have been supported. Therefore, either nature is non-local or there are no Hidden Variables. Either way, the Local Realist position is now discredited. There is no possibility of a future Local Realistic theory which agrees with QM.

The early tests established a pattern which is being improved upon even today. Repeated tests of entangled particles continue to demonstrate that Bell's Inequality is violated with ever increasing statistical accuracy. Recent tests involve numerous variations, improvements, and enhanced efficiencies. They support Quantum Mechanics and reject Local Realistic theories to the tune of over 30 standard deviations - an amazing accomplishment. Research continues on understanding of the true nature of particle entanglement.


Bell's Theorem [return to top Link index]

•  This is a top notch presentation of Bell's Theorem, very comprehensive treatment by one of the pioneers of the field:
Abner Shimony: "Bell's Theorem" , Stanford's Plato (updated as of 2005)

•  Mermin's overview of Bell's Theorem, covers a lot of ground (.PDF, 267k):
N. David Mermin: "Is the moon there when nobody looks? Reality and the quantum theory" , Physics Today (April 1985)

•  This is a great presentation of the EPR Paradox:
Arthur Fine: "The Einstein-Podolsky-Rosen Argument" , Stanford's Plato (updated as of 2004)

•  A recap of the current "state of the debate", by Alain Aspect himself (.PDF):
Aspect: "Bell's inequality test: more ideal than ever" (PDF) , Nature (2001)

•  A recap of Bell's Theorem in Scholarpedia - please note that this is written from a Bohmian
perspective (non-local mechanics) and does not entirely follow standard protocols on the subject:
Goldstein, Norsen, Tausk, Zanghi: "Bell's theorem" , Scholarpedia (2011)

Tests of Bell's Inequality [return to top Link index]

•  The 1998 Innsbruck Experiment (EPR with 1 kilometer of separation) is the gold standard of Bell tests:
Weihs, Jennewein, Simon, Weinfurter and Zeilinger: Violation of Bell's inequality under strict Einstein locality conditions (PDF)

•  Bell test which closes the fair sampling loophole:
Rowe, Kielpinski, Meyer, Sackett, Itano, Monroe, & Wineland: Experimental violation of a Bell's inequality with efficient detection (PDF) , Nature, 1998.

•  Bell tests are now being done at the undergrad level:
Dietrich Dehlinger and M. W. Mitchell: Entangled photons, nonlocality and Bell inequalities in the undergraduate laboratory (PDF) , American Journal of Physics, 2002.

•  Comprehensive review of Bell tests as of 2006 (with 505 references):
Marco Genovese: Research on Hidden Variable Theories: a review of recent progresses , arXiv, 2007.

More About Bell from DrChinese [return to top Link index]

•  The world's shortest (and guaranteed simplest!!) proof of Bell's Theorem:
David Schneider: Bell's Theorem with Easy Math

•  Derivation of Bell's Theorem demonstrating that negative probabilities are a paradoxical consequence:
David Schneider: Bell's Theorem and Negative Probabilities

•  The original papers that say it all:
David Schneider: EPR, Bell & Aspect: The Original References (in PDF format)

The GHZ Theorem, and other theorems ruling out Local Realism [return to top Link index]

•  The important GHZ Theorem (1989) is hard to locate in the original, here is the best I have for you:
"GHZ experiment" , Wikipedia

•  Realization of the GHZ Theorem experimentally, unfortunately is not free:
"Experimental test of quantum nonlocality in three-photon Greenberger–Horne–Zeilinger entanglement" , Pan, Bouwmeester, Daniell, Weinfurter & Zeilinger (2000)

•  Experimental tests of GHZ Theorem using 3 photons, which includes their discussion of the theorem in section 16.2 (.PDF):
J. Pan and A. Zeilinger: Multi-Photon Entanglement and Quantum Nonlocality, from the book Quantum [Un]Speakables. From Bell's Theorem to Quantum Information, R. Bertlmann, A. Zeilinger, editors, (2002).

•  Hardy's Paradox is tested as a way to rule out local realism (.PDF):
Carlson, Olmstead and Beck: Quantum mysteries tested

•  Another contradiction with local realism:
Carston Held: The Kochen-Specker Theorem , Stanford's Plato

•  Experimental realization of the above, ruling out non-contextual theories:
Bartosik, Klepp, Schmitzer, Sponar, Cabello, Rauch, Hasegawa: Experimental test of quantum contextuality in neutron interferometry , (2009)

•  Experimental realization of Hardy's Paradox, also ruling out realistic theories:
Yokota, Yamamoto, Koashi and Imoto: Direct observation of Hardy's paradox by joint weak measurement with an entangled photon pair , New Journal of Physics (2009)

•  Leggett's Theorem shows the limitations on describing a particle's unobserved past:
Bacciagaluppi: Leggett's theorem without inequalities , Foundations of Probability and Physics 5 (2008)

Multi-Photon and Other exotic Entanglement [return to top Link index]

•  Two is not the limit to entangled particles:
Eibl, Gaertner, Bourennane, Kurtsiefer, Zukowski, Weinfurter: Experimental observation of four-photon entanglement from down-conversion , Physical Review Letters (2003)

•  Entangled entanglement:
P. Walther, K. Resch, C. Brukner, A. Zeilinger: Experimental Entangled Entanglement

•  Hyper-entanglement:
Julio Barreiro: Hyper-entangled photons

Parametric Down Conversion (PDC), the Quantum Eraser, and other physical phenomena [return to top Link index]

•  Using an EPR setup to demonstrate the particle nature of light in a relatively inexpensive tabletop setup (repeating an experiment performed by Aspect and others) (.PDF):
Thorn, Neel, Donato, Bergreen, Davies and Beck : Observing the quantum behavior of light in an undergraduate laboratory (PDF) , American Journal of Physics (2002)

•  A discussion of quantum erasers (.PDF):
Herzog, Kwiat, Weinfurter, Zeilinger: Complementarity and the Quantum Eraser , Physical Review Letters (1995)

Additional Stanford/Wikipedia/Nobel Site References of Interest [return to top Link index]

•  To learn more about quantum entanglement:
Jeffry Bub: "Quantum Entanglement and Information" , Stanford's Plato (updated as of 2001)

•  Great historical perspective:
Jan Faye: "The Copenhagen Interpretation" , Stanford's Plato (updated as of 2002)

•  Wikipedia's discussion of Bell's Theorem:
Various: "Bell's Theorem" , Wikipedia

•  Wikipedia's discussion of Bell test experiments:
Various: "Bell Test Experiments" , Wikipedia

•  For the discovery of Planck's constant:
Max Planck: "The Nobel Prize in Physics, 1918" , Nobel site

•  For the law of the photoelectric effect:
Albert Einstein: "The Nobel Prize in Physics, 1921" , Nobel site

•  For the structure of the hydrogen atom and quantized radiation:
Niels Bohr: "The Nobel Prize in Physics, 1922" , Nobel site

•  For his description of the electron as both a particle and a wave:
Louis De Broglie: "The Nobel Prize in Physics, 1929" , Nobel site

•  For the discovery of the Heisenberg Uncertainty Principle and inventing Quantum Theory:
Werner Heisenberg: "The Nobel Prize in Physics, 1932" , Nobel site

•  For their contributions:
Schoedinger and Dirac: "The Nobel Prize in Physics, 1933" , Nobel site

•  For the Pauli Exclusion Principle:
Wolfgang Pauli: "The Nobel Prize in Physics, 1945" , Nobel site

•  For the statistical interpretation of the wave function:
Max Born: "The Nobel Prize in Physics, 1954" , Nobel site

Philosophical and Other Discussion regarding Bell's Theorem and Experimental Tests [return to top Link index]

•  What is "Bell Locality"? This explains, plus definitions for Parameter Independence and Outcome Independence:
Travis Norsen: EPR and Bell Locality , arXiv (2005)

•  Interference is related to uncertainty:
Thomas Marcella: Quantum interference with slits , European Journal of Physics (2002)

(c) 2005-2009 David Schneider/DrChinese

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