Bell’s Theorem Explained: How We Know Quantum Weirdness Is Real

Two particles are created together and fly apart. Measure one, and the result for the other, far away, turns out to be correlated with it in a way no pair of ordinary objects can manage. Is that because the particles carried hidden instructions from the start, or is nature stranger than that? In 1964 John Bell showed that the question can be answered by experiment. Our character Reflexia, the Nonlocal Mirror, on the Reflexia tee, personifies the result. This post goes deeper than our explainer What Is Quantum Measurement?

Where did the question come from?

In 1935 Albert Einstein, Boris Podolsky and Nathan Rosen published “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?” (Einstein, Podolsky & Rosen, 1935^[1]). They argued that if measuring one particle lets you predict the result for its distant partner with certainty, the partner must have had that property all along, so quantum mechanics, which does not assign it in advance, must be incomplete. The extra ingredients Einstein hoped for became known as hidden variables. For the basics of linked particles, see Decoding Quantum Entanglement.

What does Bell’s theorem say?

In 1964 John Bell published “On the Einstein Podolsky Rosen Paradox” (Bell, 1964^[2]). The team behind one of the 2015 experiments summarise his result this way: Bell proved that no theory of nature that obeys locality and realism can reproduce all the predictions of quantum theory, and that in any local realist theory the correlations between distant measurements satisfy an inequality which quantum theory can violate (Hensen et al., 2015^[3]).

Here “realism” means outcomes are fixed in advance by hidden properties, and “locality” means a measurement on one side cannot be affected by the choice of measurement on the other side.

What is the CHSH inequality?

In 1969 John Clauser, Michael Horne, Abner Shimony and Richard Holt generalised Bell’s theorem so that it applies to realizable experiments and proposed what they called a decisive test between quantum mechanics and local hidden-variable theories (Clauser, Horne, Shimony & Holt, 1969^[4]). Their version is the one most experiments use.

Alice chooses between two detector settings, a or a′, and Bob between b or b′. Each measurement gives +1 or −1. E(a, b) is the average of the product of their two results over many pairs, a number between −1 and +1 with no units. Then:

S = E(a, b) − E(a, b′) + E(a′, b) + E(a′, b′), |S| ≤ 2

Any local hidden-variable theory must keep |S| at 2 or less (Stanford Encyclopedia of Philosophy^[5]).

What does quantum mechanics predict? A worked number

For two spin-½ particles in the entangled “singlet” state, quantum mechanics predicts:

E(θ) = −cos θ

where θ is the angle between Alice’s and Bob’s detector directions, in degrees or radians.

Our worked example: choose a = 0°, a′ = 90°, b = 45° and b′ = 135°. The four angles between the settings are 45°, 135°, 45° and 45°. So E(a, b) = −0.707, E(a, b′) = +0.707, E(a′, b) = −0.707 and E(a′, b′) = −0.707. Adding them with the CHSH signs gives S = −0.707 − 0.707 − 0.707 − 0.707 = −2.828, so |S| = 2√2 ≈ 2.828. That is well above 2.

The Stanford Encyclopedia confirms that 2√2 ≈ 2.828 is the maximum violation of the CHSH inequality for any quantum state, as shown by Boris Tsirelson (Cirel’son 1980); it is called the Tsirelson bound. For photons, which are measured by polarisation, the best angles are halved, with maximum violation at 22.5°.

How was Bell’s theorem tested?

• 1972. According to the Nobel Committee’s popular science background, John Clauser and his doctoral student Stuart Freedman built an apparatus that sent pairs of entangled photons to polarisation filters, and showed a clear violation of a Bell inequality, in agreement with quantum mechanics (NobelPrize.org^[6]).
• 1982. Alain Aspect, Jean Dalibard and Gérard Roger used acousto-optical switches near 50 MHz so each analyzer jumped between two orientations in a time short compared with the photons’ flight. Their results agreed with quantum mechanics and violated Bell’s inequalities by 5 standard deviations (Aspect, Dalibard & Roger, 1982^[7]).
• 2015. Three groups reported tests that closed the main loopholes at the same time. Hensen and colleagues entangled electron spins 1.3 km apart, ran 245 trials and found S = 2.42 ± 0.20, with a probability of p = 0.039 that a local-realist model would produce a violation at least as large (Hensen et al., 2015^[3]). Two photon experiments followed in the same year: “Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons” (Giustina et al., 2015^[8]) and “Strong Loophole-Free Test of Local Realism” (Shalm et al., 2015^[9]).

In 2022 Alain Aspect, John Clauser and Anton Zeilinger won the Nobel Prize in Physics “for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science” (Nobel Prize 2022^[10]).

What does it mean, and what is still open?

The experiments rule out every theory that is both local and realist in Bell’s sense. They do not, by themselves, say which assumption nature gives up. That remains an open question of interpretation, and the Stanford Encyclopedia discusses the assumptions behind the theorem in detail. Bell violations also do not let anyone send a signal faster than light: each side’s own results look random, and the correlation only appears when the two lists are compared. Our Quavo & Lia tee and the Entanglement Twins collection are our nods to that shared correlation.

Quick FAQ

What is Bell’s theorem in simple terms?

No theory in which outcomes are set in advance and nothing travels faster than light can match every prediction of quantum mechanics.

What is the CHSH limit?

Local hidden-variable theories give |S| ≤ 2; quantum mechanics can reach 2√2 ≈ 2.828.

Has Bell’s inequality been violated?

Yes, from 1972 onward, and in loophole-free tests in 2015.

Does entanglement allow faster-than-light messages?

No. The correlations only show up when results are compared through ordinary communication.

References

1. A. Einstein, B. Podolsky & N. Rosen (1935), “Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?,” Phys. Rev. 47, 777–780 (DOI 10.1103/PhysRev.47.777; metadata verified via Crossref). https://doi.org/10.1103/PhysRev.47.777
2. J. S. Bell (1964), “On the Einstein Podolsky Rosen paradox,” Physics Physique Fizika 1, 195–200 (DOI 10.1103/PhysicsPhysiqueFizika.1.195; metadata verified via Crossref). https://doi.org/10.1103/PhysicsPhysiqueFizika.1.195
3. B. Hensen et al. (2015), “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526, 682–686 (arXiv:1508.05949). https://arxiv.org/abs/1508.05949
4. J. F. Clauser, M. A. Horne, A. Shimony & R. A. Holt (1969), “Proposed Experiment to Test Local Hidden-Variable Theories,” Phys. Rev. Lett. 23, 880–884 (DOI 10.1103/PhysRevLett.23.880; metadata verified via Crossref). https://doi.org/10.1103/PhysRevLett.23.880
5. W. Myrvold, M. Genovese & A. Shimony, “Bell’s Theorem,” Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/entries/bell-theorem/
6. NobelPrize.org, “How entanglement has become a powerful tool,” popular science background, Nobel Prize in Physics 2022. https://www.nobelprize.org/prizes/physics/2022/popular-information/
7. A. Aspect, J. Dalibard & G. Roger (1982), “Experimental Test of Bell’s Inequalities Using Time-Varying Analyzers,” Phys. Rev. Lett. 49, 1804–1807 (DOI 10.1103/PhysRevLett.49.1804; metadata verified via Crossref). https://doi.org/10.1103/PhysRevLett.49.1804
8. M. Giustina et al. (2015), “Significant-Loophole-Free Test of Bell’s Theorem with Entangled Photons,” Phys. Rev. Lett. 115, 250401 (DOI 10.1103/PhysRevLett.115.250401; metadata verified via Crossref). https://doi.org/10.1103/PhysRevLett.115.250401
9. L. K. Shalm et al. (2015), “Strong Loophole-Free Test of Local Realism,” Phys. Rev. Lett. 115, 250402 (DOI 10.1103/PhysRevLett.115.250402; metadata verified via Crossref). https://doi.org/10.1103/PhysRevLett.115.250402
10. NobelPrize.org, The Nobel Prize in Physics 2022 (Aspect, Clauser, Zeilinger). https://www.nobelprize.org/prizes/physics/2022/summary/

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.