# Bell's theorem

**Bell's theorem** is a family of results in the foundations of quantum mechanics, beginning with John S. Bell's 1964 paper "On the Einstein Podolsky Rosen paradox", showing that no physical theory satisfying a locality condition can reproduce all the statistical predictions of quantum mechanics for entangled systems.<sup>[1](https://journals.aps.org/ppf/abstract/10.1103/PhysicsPhysiqueFizika.1.195)</sup><sup> • </sup><sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup> The theorem converts a philosophical dispute, begun by the 1935 Einstein-Podolsky-Rosen argument about whether the quantum state description is complete, into an experimentally testable inequality.<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup> [Experiments](https://www.edgepedia.org/experiments) have repeatedly found violations of Bell inequalities, and the 2022 [Nobel Prize in Physics](https://www.edgepedia.org/nobel-prize-in-physics) was awarded to [Alain Aspect](https://www.edgepedia.org/alain-aspect), John F. Clauser, and Anton Zeilinger "for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science".<sup>[3](https://www.nobelprize.org/uploads/2022/10/press-physicsprize2022-2.pdf)</sup>

| Key fact | Detail |
|---|---|
| Original publication | J. S. Bell, "On the Einstein Podolsky Rosen paradox", *Physics Physique Fizika* **1**, 195–200 (November 1964)<sup>[1](https://journals.aps.org/ppf/abstract/10.1103/PhysicsPhysiqueFizika.1.195)</sup> |
| Core claim | No theory satisfying Bell's locality condition (factorizability) can reproduce all quantum probabilities for entangled systems<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup> |
| CHSH inequality | Any locally explicable theory gives S ≤ 2; quantum mechanics reaches 2√2 ≈ 2.828, the Tsirelson bound<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup> |
| First loophole-free tests | 2015: electron spins 1.3 km apart (Hensen et al., S = 2.42 ± 0.20 over 245 trials) and entangled-photon tests (Giustina et al.; Shalm et al.)<sup>[4](https://pubmed.ncbi.nlm.nih.gov/26503041/)</sup><sup> • </sup><sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup> |
| Recognition | Nobel Prize in Physics 2022 to Aspect, Clauser, and Zeilinger<sup>[3](https://www.nobelprize.org/uploads/2022/10/press-physicsprize2022-2.pdf)</sup> |
| Practical use | Device-independent quantum key distribution and certified random-number generation rest on Bell-inequality violations<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup> |

## Background

Bell's paper was a response to the Einstein-Podolsky-Rosen (EPR) argument of 1935, which examined correlations between two well-separated particles and concluded that, to avoid an appeal to nonlocality, the correlations could only be explained by "elements of physical reality" in each particle, effectively an argument for a hidden-variables interpretation in which the quantum state is incomplete.<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup> Bell cited Einstein's own formulation: "the real factual situation of the system S2 is independent of what is done with the system S1, which is spatially separated from the former".<sup>[1](https://journals.aps.org/ppf/abstract/10.1103/PhysicsPhysiqueFizika.1.195)</sup>

The theorem also grew out of Bell's examination of the hidden-variables program, the attempt to supplement the quantum state with further variables that restore determinism. A common misconception treats Bell's theorem as a no-go theorem for hidden-variables theories in general; there can be no such theorem, since the de Broglie-Bohm theory is a functioning hidden-variables theory. What the theorem rules out are hidden-variables theories that are local in Bell's sense.<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup>

## The assumptions and the inequality

In the standard framework, each pair of particles is characterized by a "complete state" λ containing the entirety of the pair's properties at generation, sampled from a distribution ρ. The distinctive condition is factorizability, often called Bell locality: for any settings a, b, and state λ, the joint probability of outcomes factors into a product of probabilities depending only on the local setting and λ. Bell regarded this condition not as the definition of "local causality" but as a consequence of it.<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup>

A further assumption, made explicit after an exchange with Shimony, Horne, and Clauser in 1976, is the Measurement Independence Assumption: the distribution ρ of complete states is independent of how the measurement settings are chosen. This assumption has also been called the Free Will assumption, the Freedom of Choice Assumption, and the No-conspiracies Assumption.<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup>

For experiments with two outcomes labeled ±1, define the correlation E(a, b) as the expectation of the product of outcomes. The CHSH combination, introduced by Clauser, Horne, Shimony, and Holt in 1969, is

\[ S(a, a', b, b') = \lvert E(a, b) + E(a, b') \rvert + \lvert E(a', b) - E(a', b') \rvert. \]

If factorizability holds, then S ≤ 2. [Quantum mechanics](https://www.edgepedia.org/quantum-mechanics) predicts, and experiments confirm, values exceeding this bound. For spin-1/2 particles in the singlet state, choosing the four settings coplanar with equal angular spacing φ gives a maximum violation at φ = π/4, where

\[ S = 2\sqrt{2} \approx 2.828. \]

This value is the maximum violation of the [CHSH inequality](https://www.edgepedia.org/chsh-inequality) achievable by any quantum state, a limit shown by Tsirelson in 1980 and called the Tsirelson bound.<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup>

Bell's original 1964 derivation relied on perfect anticorrelation of aligned measurements on spin-1/2 particles in the singlet state, from which outcome determinism follows via an EPR-type argument. Later work by Bell himself (1971) and by Clauser and Horne (1974) removed the need for perfect anticorrelation and outcome determinism.<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup>

## Experimental tests and loopholes

Early tests used photons from atomic cascades and positronium annihilation. The first experiment using two-channel polarization analyzers, realizing the theoretical scheme directly, was performed in the early 1980s by Aspect, Grangier, and Roger with cascade photons from laser-excited calcium atoms; its result deviated from the Bell inequality limit by 40 standard deviations. A follow-up by Aspect, Dalibard, and Roger in 1982 addressed the communication loophole by switching analyzer orientations rapidly enough that the choices were spacelike separated events.<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup>

Two loopholes limited the early tests. The communication loophole allows a subluminal or light-speed influence between the two wings; it is addressed by spacelike separation of setting choices and detections. The detection loophole arises when only a biased subset of pairs is detected, allowing local models to fit the data; it is addressed by high-efficiency detectors. The 1998 experiment of Weihs, Jennewein, Simon, Weinfurter, and Zeilinger used stations 400 m apart with independent rapid random generators, with total electronic and optical delays under 100 ns against the 1.3 μs light-travel time between stations.<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup>

**Loophole-free tests.** In 2015 three experiments appeared. Hensen, Bernien, and colleagues entangled electron spins in diamond nitrogen-vacancy centers in two laboratories 1.3 km apart, performing 245 trials of the CHSH test and finding S = 2.42 ± 0.20, a violation free of the detection and communication loopholes, though with limited statistical significance of about 2 standard deviations.<sup>[4](https://pubmed.ncbi.nlm.nih.gov/26503041/)</sup><sup> • </sup><sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup> The same issue of [Physical Review Letters](https://www.edgepedia.org/physical-review-letters) carried two photon experiments using high-efficiency transition-edge sensors: Shalm et al. reported a p-value of 2.3 × 10⁻⁷, and Giustina et al. reported 3.4 × 10⁻³¹, corresponding to an 11.5 standard deviation effect.<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup> In 2023, Storz and colleagues at [ETH Zurich](https://www.edgepedia.org/eth-zurich) achieved a loophole-free violation with superconducting circuits connected by a 30-meter cryogenic link, evaluating more than 1 million trials and finding an average S of 2.0747 ± 0.0033 with a P value below 10⁻¹⁰⁸.<sup>[5](https://www.nature.com/articles/s41586-023-05885-0)</sup> Loophole-free tests have also been realized with neutral atoms (Rosenfeld et al., 2017).<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup>

Separation has grown from laboratory scales to 10.9 km between fiber-linked stations in Geneva (Tittel et al., 1998), 144 km (Scheidl et al., 2010), and 1200 km in satellite experiments (Yin et al., 2017).<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup>

## What the violation rules out, and what it does not

Taken together with the loophole-free experiments, the theorem shows that quantum-mechanical systems cannot satisfy all of realism, locality, and measurement independence; at least one must be given up.<sup>[6](https://doi.org/10.1007/s10701-026-00919-9)</sup> The options form a recognized taxonomy. Everettian or many-worlds theories reject unique outcomes. Superdeterminism, a term Bell used in an interview, supposes a common cause in the past that determines both settings and outcomes, rejecting measurement independence; retrocausal approaches instead allow settings to influence the source state backwards in time. Rejecting local causality itself is the route taken by the de Broglie-Bohm theory, in which Bell noted "an explicit causal mechanism exists whereby the disposition of one piece of apparatus affects the results obtained with a distant piece".<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup>

**No faster-than-light signaling.** Violation of Bell inequalities does not enable controllable faster-than-light communication. The Nobel popular science background states plainly that "a signal cannot travel faster than light", while quantum correlations show no need for a signal to connect the parts of an extended system.<sup>[7](https://www.nobelprize.org/prizes/physics/2022/popular-information/Using/)</sup> The no-signaling property is what makes the correlations usable in practice rather than a channel for messages.<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup>

## Significance for quantum information

Bell's theorem reveals that entanglement-based correlations differ in kind from locally explicable classical correlations, and it provided part of the impetus for quantum information theory.<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup> It also plays a direct technical role. [Device-independent quantum key distribution](https://www.edgepedia.org/device-independent-quantum-key-distribution) derives provable security from Bell-inequality-violating correlations at spacelike separation, assuming only that the underlying physics permits no superluminal signaling; a result of Colbeck and Renner shows an eavesdropper cannot subvert such a scheme without disrupting the correlations.<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup> Loophole-free Bell tests also underpin random-number generators whose output is unpredictable to any adversary limited only by general physical principles such as special relativity.<sup>[2](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)</sup>

## References

1. [J. S. Bell (1964). On the Einstein Podolsky Rosen paradox. *Physics Physique Fizika* 1, 195–200.](https://journals.aps.org/ppf/abstract/10.1103/PhysicsPhysiqueFizika.1.195)
2. [Bell's Theorem, Stanford Encyclopedia of Philosophy (Fall 2024 Edition)](https://plato.stanford.edu/archives/fall2024/entries/bell-theorem/)
3. [Press release: The Nobel Prize in Physics 2022, Royal Swedish Academy of Sciences](https://www.nobelprize.org/uploads/2022/10/press-physicsprize2022-2.pdf)
4. [Hensen et al. (2015). Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres. *Nature* 526, 682–686.](https://pubmed.ncbi.nlm.nih.gov/26503041/)
5. [Storz et al. (2023). Loophole-free Bell inequality violation with superconducting circuits. *Nature* 617, 265–270.](https://www.nature.com/articles/s41586-023-05885-0)
6. [A Heptalemma for Quantum Mechanics, *Foundations of Physics* (2026)](https://doi.org/10.1007/s10701-026-00919-9)
7. [The Nobel Prize in Physics 2022, Popular science background](https://www.nobelprize.org/prizes/physics/2022/popular-information/Using/)

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*Topic: Encyclopedia › Physical world and mathematics › Physics › Quantum physics › Quantum mechanics › Quantum phenomena and measurement › Entanglement and nonlocal correlations › Bell's theorem and Bell inequalities*

*Initially written Sep 17, 2026 · Reviewed: Sep 17, 2026; Sep 30, 2026; Oct 15, 2026 · Edited: Sep 30, 2026; Oct 15, 2026 · Last review: Oct 15, 2026*

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