Bell's Theorem

Prerequisites: entanglement. Full path (18 pages)

Best explanations

Intuitive The Quantum Theory and Reality by d'Espagnat
Concrete A simple proof of Bell's inequality

Intuitive
Concrete
Abstract
Why is it interesting?
History
Criticism

Intuitive

Two particles are prepared together and sent to two distant labs. In each lab an experimenter picks one of several possible measurements and writes down the result. When the two notebooks are compared later, the results turn out to be correlated. A natural explanation is that each particle carried a set of instructions from the source, fixed in advance, telling it how to respond to every possible measurement. The correlations would then be like those between two gloves of a pair: finding a left glove in one box tells you instantly that the other box has a right glove, without any influence travelling between the boxes.

John Bell showed in 1964 that this kind of explanation has a measurable limit. If the outcomes are fixed by shared instructions, and if the choice of measurement in one lab cannot influence the result in the other lab, then certain combinations of correlations can never exceed a fixed bound. Quantum mechanics predicts that entangled particles exceed this bound for suitable choices of measurements. So the question of whether "hidden instructions" could underlie quantum mechanics became a question for the laboratory.

Experiments since the 1970s have found violations of Bell's inequality in agreement with quantum mechanics, and experiments in 2015 closed the main loopholes simultaneously. As a result, at least one of the assumptions behind the bound must fail in nature: outcomes determined by local hidden variables, no influence faster than light between the labs, or measurement settings chosen independently of the particles' properties.

A particularly nice interactive explanation can be found here, and also here.
The best explanation for Bell's theorem in layman terms, can be found in The Quantum Theory and Reality by d'Espagnat
Another nice explanation can be found in this reddit comment.
Another creative explanation in laymen terms can be found in The mystery of the quantum cakes by P. G. Kwiat and L. Hardy

Concrete

The most commonly tested form of Bell's theorem is the CHSH inequality (Clauser, Horne, Shimony and Holt). Alice chooses between two measurement settings $a$ and $a'$ , Bob between $b$ and $b'$ , and every measurement gives a result $\pm 1$ .

Local hidden variables. Assume the source emits a pair with some hidden variable $\lambda$ , distributed with probability density $\rho(\lambda)$ that does not depend on the settings. Alice's result is a function $A(a,\lambda) = \pm 1$ that depends only on her own setting, and similarly Bob's result is $B(b,\lambda) = \pm 1$ . The correlation function is $E(a,b) = \int d\lambda\, \rho(\lambda)\, A(a,\lambda) B(b,\lambda) .$ Define $S = E(a,b) - E(a,b') + E(a',b) + E(a',b') .$ For each fixed $\lambda$ we have $A(a)\big[B(b) - B(b')\big] + A(a')\big[B(b) + B(b')\big] = \pm 2 ,$ because $B(b)$ and $B(b')$ are $\pm 1$ , so one bracket vanishes and the other is $\pm 2$ . Averaging over $\lambda$ gives the CHSH inequality $|S| \leq 2 .$ The same bound holds if the local results are random rather than determined, as long as their probabilities depend only on $\lambda$ and the local setting.

Quantum prediction. Take two spin-1/2 particles in the singlet state $|\psi\rangle = \frac{1}{\sqrt 2}\left( |\uparrow\downarrow\rangle - |\downarrow\uparrow\rangle \right)$ and let Alice measure spin along a unit vector $\vec a$ , with result $\pm 1$ in units of $\hbar/2$ , and Bob along $\vec b$ . Quantum mechanics gives $E(\vec a, \vec b) = \langle \psi | (\vec\sigma\cdot\vec a) \otimes (\vec\sigma\cdot\vec b) | \psi\rangle = - \vec a \cdot \vec b = -\cos\theta_{ab} .$ Choose all four directions in one plane, at angles $a = 0^\circ$ , $a' = 90^\circ$ , $b = 45^\circ$ , $b' = 135^\circ$ . Then $E(a,b) = E(a',b) = E(a',b') = -\tfrac{1}{\sqrt2}, \qquad E(a,b') = +\tfrac{1}{\sqrt2} ,$ so $S = -2\sqrt2$ and $|S| = 2\sqrt 2 \approx 2.83 > 2$ . No local hidden variable model can reproduce these correlations. The value $2\sqrt2$ is the largest that quantum mechanics allows for CHSH (Tsirelson's bound).

In experiments the spins are usually replaced by photon polarisations, where the corresponding correlation is $\pm\cos 2\theta$ because polarisation directions differ by $90^\circ$ for orthogonal states. The measured $S$ is compared with 2. To exclude local explanations, the setting choices must be made at spacelike separation from the other lab's measurement (locality loophole), and enough pairs must be detected so that the selected subsample cannot fake a violation (detection loophole).

A simple proof of Bell's inequality by Lorenzo Maccone
Understanding Bell’s theorem part 1: the simple version | More Quantum
Bell's Theorem with Easy Math by David R. Schneider
See also David Mermin's paper, "Bringing home the atomic world: Quantum mysteries for anybody" Am. J. Phys. 49:940-943(10), Oct. 1981, free PDF.
Daniel M. Greenberger, Michael A. Horne, Abner Shimony, Anton Zeilinger "Bell's theorem without inequalities", American Journal of Physics Vol. 58 (12), December 1990
For another nice discussion, see What could not be in Feynman lectures: Bell inequalities by V. Cerny

Abstract

In the modern formulation one describes an experiment by the conditional probabilities $p(a,b|x,y)$ of outcomes $a,b$ given settings $x,y$ (a "behaviour"). A behaviour is local if it can be written as $p(a,b|x,y) = \int d\lambda\, \rho(\lambda)\, p_A(a|x,\lambda)\, p_B(b|y,\lambda) ,$ with $\rho$ independent of $x,y$ . Two assumptions enter: measurement independence ( $\rho(\lambda|x,y) = \rho(\lambda)$ ) and Bell's local causality (the factorisation of the conditional probabilities given $\lambda$ ). For finitely many settings and outcomes, the local behaviours form a convex polytope whose vertices are the deterministic strategies. Its facets are the Bell inequalities, and CHSH is the only nontrivial facet (up to relabelling) for two settings and two outcomes per party.

Quantum behaviours have the form $p(a,b|x,y) = \operatorname{Tr}\left[\rho_{AB}\, (E^x_a \otimes F^y_b)\right] ,$ with $\{E^x_a\}$ and $\{F^y_b\}$ POVMs on the two Hilbert spaces. The quantum set is convex but not a polytope. It strictly contains the local polytope and is strictly contained in the no-signalling polytope, defined only by the condition that each party's marginal does not depend on the other's setting. The no-signalling Popescu-Rohrlich box reaches $S = 4$ .

The quantum bound follows from an operator identity. Let $A_0, A_1$ and $B_0, B_1$ be Hermitian operators with $A_i^2 = B_j^2 = 1$ , acting on the two factors. The CHSH operator $\mathcal{B} = A_0\otimes(B_0+B_1) + A_1\otimes(B_0 - B_1)$ satisfies $\mathcal{B}^2 = 4\cdot \mathbb{1} - [A_0, A_1]\otimes[B_0,B_1] .$ Since $\|[A_0,A_1]\| \leq 2$ and $\|[B_0,B_1]\| \leq 2$ , we get $\|\mathcal B^2\| \leq 8$ and $|\langle \mathcal B\rangle| \leq 2\sqrt2$ . If the observables of either party commute, the bound reduces to the local value 2, which shows that incompatible measurements on both sides are necessary for a violation. Every pure entangled state violates some Bell inequality (Gisin's theorem), while some mixed entangled states (Werner states) admit a local model for projective measurements.

Bell nonlocality is related to, but distinct from, contextuality (Kochen-Specker), and it underlies device-independent protocols where security or randomness is certified from the observed violation alone, without trusting the internal workings of the devices.

A nice overview is given in this Nature article.
A modern reformulation of Bell's theorem is No extension of quantum theory can have improved predictive power by Roger Colbeck & Renato Renner (Nature)

Why is it interesting?

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).

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

History

John Bell and the most profound discovery of science by Andrew Whitaker

Criticism

Bell’s theorem is wrong at Project Tartarus

Jaynes vs. Bell

"The measurement at A at time t does not change the real physical situation at B; but it changes our state of knowledge about that situation, and therefore it changes the predictions we are able to make about B at some time t 0 . Since this is a matter of logic rather than physical causation, there is no action at a distance and no di culty with relativity [also, it does not matter whether t 0 is before, equal to, or after t".Clearing up the Mysteries, by Jaynes

A critical discussion of Jayne's point of view can be found here.