Bell’s Theorem

Bell’s theorem is an incompatibility result: no model satisfying Bell-local factorization and measurement independence can reproduce every prediction of quantum mechanics. The theorem is not merely the observation that quantum theory contains entangled states. It identifies an experimentally testable boundary between quantum correlations and a precisely defined class of local common-cause explanations.

This page owns that logical conclusion, its assumptions, and its interpretive limits. The CHSH Inequality owns the full inequality derivation.

Required background. Local Hidden Variables defines Bell-local factorization and measurement independence. The CHSH Inequality proves the local bound used below.

Helpful background. Entangled States distinguishes entanglement from classical mixtures of product states.

The theorem as an incompatibility result

Consider two parties. Alice chooses a setting $x$, Bob chooses a setting $y$, and they record outcomes $a$ and $b$. A Bell-local hidden-variable model has the form

$$p(a,b\mid x,y) = \int_\Lambda \mu(d\lambda)\, p_A(a\mid x,\lambda)\, p_B(b\mid y,\lambda),$$

where the distribution $\mu$ is independent of the settings. The variable $\lambda$ may be discrete or continuous, and the local response functions may be deterministic or stochastic.

Bell’s theorem. There are quantum states and local measurements whose joint probabilities admit no representation of this form.

Equivalently, every behavior of the displayed Bell-local form obeys every valid Bell inequality, whereas some Born-rule probabilities violate one. The CHSH scenario supplies the simplest standard witness.

Bell’s 1964 argument used perfect singlet anticorrelation to motivate predetermined responses before deriving an inequality. The later CHSH form does not require perfect correlations and allows stochastic response functions from the outset. These are related formulations of the same broad obstruction, but their premise lists should not be conflated.

Assumptions ledger

The phrase local realism is too compressed to serve as a premise. The operational CHSH version uses the following ledger.

Ingredient	Mathematical role	What it does not assume
Recorded settings and outcomes	Each trial has observed variables $x,y,a,b$ governed by one probability model.	It does not assign quantum operators simultaneous values.
Common-cause variable	A variable $\lambda$ can carry any shared information relevant to the outcomes.	It need not be directly observable or finite dimensional.
Measurement independence	$\mu(d\lambda\mid x,y)=\mu(d\lambda)$.	It does not require the settings to lack all physical causes; it excludes the correlations with $\lambda$ that would bias the Bell comparison.
Bell-local causality	Given a complete $\lambda$, $p(a,b\mid x,y,\lambda)=p_A(a\mid x,\lambda)p_B(b\mid y,\lambda)$.	It does not follow merely from no-signaling of the observed marginals.
Ordinary probability theory	The hidden-variable distribution is nonnegative and normalized, and averages use the same ensemble across settings.	It does not require deterministic response functions.

Bell-local factorization can be split into two conditional-independence conditions. Parameter independence removes the remote setting from each local response once $\lambda$ is given. Outcome independence makes the two outcomes conditionally independent once the settings and $\lambda$ are given. The split is useful diagnostically; Bell’s local-causality condition is the full factorization.

Determinism is not an additional premise of CHSH. Any local stochastic model can absorb independent local random seeds into an enlarged variable $\lambda$ and thereby be represented as a convex mixture of deterministic response tables. Nor is an undefined blanket principle called realism needed. What matters is the displayed probabilistic structure.

In an experiment, further conditions connect recorded data to this theorem: setting choices must be sufficiently independent of the source, the relevant events must have the intended spacetime separation, and selection or loss must be handled without an unjustified fair-sampling step. Those are implementation conditions, not extra lines in the mathematical proof.

From CHSH to Bell

Take $x,y\in\{0,1\}$ and $a,b\in\{-1,+1\}$. Define

$$E_{xy} = \sum_{a,b=\pm1}ab\,p(a,b\mid x,y)$$

and

$$S = E_{00}+E_{01}+E_{10}-E_{11}.$$

The canonical CHSH derivation proves that every Bell-local model satisfies

$$|S|\leq2.$$

The proof of Bell’s theorem is then a short implication, not a second CHSH derivation:

1. If a Bell-local, measurement-independent model reproduced all quantum predictions, it would reproduce the quantum behavior chosen below.
2. Because that model is Bell-local, its behavior would satisfy $|S|\leq2$.
3. The Born rule instead gives $|S|=2\sqrt2$ for that behavior.
4. Therefore no model satisfying the premises reproduces all quantum predictions.

This logic establishes an impossibility theorem even before any experiment is performed. An experiment addresses the separate empirical question of which statistical model describes nature under the implemented conditions.

An explicit quantum witness

Let the two-qubit state be the singlet

$$|\psi^-\rangle = \frac{|01\rangle-|10\rangle}{\sqrt2}.$$

For spin measurements along unit vectors $\mathbf a$ and $\mathbf b$, the Born rule gives

$$E(\mathbf a,\mathbf b) = \langle\psi^-| (\mathbf a\cdot\boldsymbol\sigma) \otimes (\mathbf b\cdot\boldsymbol\sigma) |\psi^-\rangle = -\mathbf a\cdot\mathbf b.$$

Choose

$$A_0=\sigma_z, \qquad A_1=\sigma_x,$$

and

$$B_0=\frac{\sigma_z+\sigma_x}{\sqrt2}, \qquad B_1=\frac{\sigma_z-\sigma_x}{\sqrt2}.$$

The four correlations are

$$E_{00}=E_{01}=E_{10}=-\frac1{\sqrt2}, \qquad E_{11}=+\frac1{\sqrt2}.$$

Consequently,

$$S=-2\sqrt2, \qquad |S|=2\sqrt2>2.$$

The local bound $2$, the quantum Tsirelson bound $2\sqrt2$, and the algebraic or PR-box value $4$ belong to three different correlation sets. The quantum violation therefore does not mean that quantum theory permits every no-signaling behavior.

Theorem, prediction, and experiment

These three layers answer different questions.

Layer	Input	Output	Appropriate conclusion
Mathematical theorem	Bell-local factorization, measurement independence, and ordinary probability	A Bell-inequality constraint such as $\lvert S\rvert\leq2$	Every model with those premises lies inside the local correlation set.
Quantum prediction	A specified state, observables, and the Born rule	A probability distribution, here with $\lvert S\rvert=2\sqrt2$	Quantum theory predicts a behavior outside the local set.
Experiment	A physical source, setting generators, detectors, spacetime arrangement, and statistical protocol	Counts, confidence measures, and systematic checks	The implemented Bell-local null model is disfavored to the reported statistical level.

A serious Bell test specifies its event-ready or trial-selection rule, setting generation, measurement windows, loss treatment, and statistical analysis in advance. Spacelike separation addresses locality-related communication explanations during a trial. High-efficiency detection avoids inferring the unseen sample from a possibly biased detected subset. Setting-generation designs make measurement independence empirically credible over a stated causal range; no finite experiment can establish metaphysical independence from the entire past light cone.

Experiments commonly called loophole-free Bell tests close the major traditional locality and detection loopholes within explicit physical and statistical models. They do not make the theorem assumption free, and they do not turn a finite-sample rejection into deductive certainty.

Separable states have a local model

Entanglement is necessary for Bell-inequality violation. Let

$$\rho_{AB} = \sum_k q_k\, \rho_A^{(k)}\otimes\rho_B^{(k)}$$

be separable, and let $\{M_{a|x}\}$ and $\{N_{b|y}\}$ be arbitrary local POVMs. Then

$$\begin{aligned} p(a,b\mid x,y) &= \operatorname{Tr} \!\left[ \rho_{AB} (M_{a|x}\otimes N_{b|y}) \right] \\ &= \sum_k q_k\, \operatorname{Tr}(\rho_A^{(k)}M_{a|x})\, \operatorname{Tr}(\rho_B^{(k)}N_{b|y}). \end{aligned}$$

This is a Bell-local model with $\lambda=k$. Within quantum theory, a Bell violation therefore certifies entanglement without requiring a detailed device model for the state.

The converse fails: not every entangled mixed state violates a Bell inequality in a given measurement scenario, and some entangled states admit local models for broad classes of measurements. Entanglement and Bell nonlocality are therefore related but distinct resources.

Bell nonlocality is not signaling

For local quantum POVMs,

$$p(a,b\mid x,y) = \operatorname{Tr} \!\left[ \rho_{AB} (M_{a|x}\otimes N_{b|y}) \right].$$

Summing over Bob’s outcome gives

$$\begin{aligned} p(a\mid x,y) &= \sum_b p(a,b\mid x,y) \\ &= \operatorname{Tr} \!\left[ \rho_{AB} (M_{a|x}\otimes I_B) \right], \end{aligned}$$

which is independent of $y$. The analogous statement holds for Bob. Quantum theory can violate a Bell inequality while preserving these no-signaling marginals.

Bell-local behaviors form a proper subset of no-signaling behaviors. Hence no-signaling is necessary for compatibility with operational relativistic causality but is too weak to imply a Bell-local common-cause explanation. The word nonlocality in this setting means failure of Bell-local factorization, not a usable superluminal communication channel.

What Bell’s theorem does and does not establish

Bell’s theorem rules out a specific conjunction: Bell-local causality, measurement independence, ordinary probability, and reproduction of all the relevant quantum predictions cannot all hold together.

It does not by itself establish any of the following:

• that every hidden-variable theory is impossible;
• that nature permits controllable faster-than-light signaling;
• that entanglement and Bell nonlocality are identical notions;
• that measurement outcomes must be predetermined;
• that consciousness, observers, or wave-function collapse explain the violation;
• that one interpretation of quantum theory is uniquely selected;
• that a laboratory result tests the ideal theorem without an explicit model of losses, timing, setting choice, and statistics.

Nonlocal hidden-variable theories can reproduce quantum predictions, and models that relax measurement independence or other premises evade the inequality at the cost of changing the causal explanation. Bell’s theorem tells us where that cost must be paid; it does not pay it on behalf of an interpretation.

Common pitfalls

Saying that Bell disproves all hidden variables. The theorem excludes the Bell-local, measurement-independent class. A hidden-variable theory may evade the conclusion by being nonlocal or by rejecting another explicit premise.

Using “realism” as an unexplained premise. Different authors use the word for determinism, counterfactual definiteness, mind independence, or an ontological state. Write the conditional-probability assumptions instead.

Equating violation with communication. Bell violation concerns the joint distribution. Signaling concerns whether a party can change the other party’s local marginal by changing a setting.

Treating every entangled state as a Bell violator. Separable states are local, but the reverse implication does not hold for mixed states and fixed measurement classes.

Calling an experiment a proof of the theorem. The inequality is proved mathematically. Experiments compare physical data with a statistical null model and report a finite-confidence result.

Exercises

1. Reproduce the singlet violation

Using $E(\mathbf a,\mathbf b)=-\mathbf a\cdot\mathbf b$, calculate the four correlations for the settings in the quantum witness and verify $|S|=2\sqrt2$.

Solution

The dot products are

$$\mathbf z\cdot\frac{\mathbf z+\mathbf x}{\sqrt2} = \mathbf z\cdot\frac{\mathbf z-\mathbf x}{\sqrt2} = \mathbf x\cdot\frac{\mathbf z+\mathbf x}{\sqrt2} = \frac1{\sqrt2},$$

while

$$\mathbf x\cdot\frac{\mathbf z-\mathbf x}{\sqrt2} = -\frac1{\sqrt2}.$$

The singlet adds a minus sign, so $E_{00}=E_{01}=E_{10}=-1/\sqrt2$ and $E_{11}=+1/\sqrt2$. Therefore $S=-4/\sqrt2=-2\sqrt2$.

2. Build the local model for a separable state

For $\rho_{AB}=\sum_kq_k\rho_A^{(k)}\otimes\rho_B^{(k)}$, identify the hidden variable, its distribution, and both response functions for arbitrary local POVMs.

Solution

Take $\lambda=k$ with probability $\mu(k)=q_k$. Define

$$p_A(a\mid x,k) = \operatorname{Tr}(\rho_A^{(k)}M_{a|x}), \qquad p_B(b\mid y,k) = \operatorname{Tr}(\rho_B^{(k)}N_{b|y}).$$

The Born rule factorizes on each product term, and averaging with $q_k$ reproduces the quantum joint probability. The distribution $q_k$ is independent of $x$ and $y$, so this is a measurement-independent Bell-local model.

3. Separate no-signaling from Bell locality

Show that a Bell-local model is no-signaling when $\mu$ is independent of the settings. Identify the step that can fail if measurement independence is abandoned.

Solution

Summing over $b$ gives

$$\begin{aligned} p(a\mid x,y) &= \int\mu(d\lambda)\, p_A(a\mid x,\lambda) \sum_b p_B(b\mid y,\lambda) \\ &= \int\mu(d\lambda)\, p_A(a\mid x,\lambda), \end{aligned}$$

which is independent of $y$. If instead $\mu(d\lambda\mid x,y)$ depends on $y$, that dependence remains after the sum and the displayed argument no longer proves no-signaling.

4. Locate the measurement-independence loophole

Suppose the source variable is allowed to depend on both settings. Show that any target distribution $p_*(a,b\mid x,y)$ can be reproduced by local deterministic responses if $\lambda$ carries the settings and outcomes for the actual trial.

Solution

Let $\lambda=(x',y',a',b')$ and, conditional on the chosen settings, set

$$\mu(\lambda\mid x,y) = \delta_{x',x}\delta_{y',y}\,p_*(a',b'\mid x,y).$$

Choose deterministic local responses $p_A(a\mid x,\lambda)=\delta_{a,a'}$ and $p_B(b\mid y,\lambda)=\delta_{b,b'}$. Averaging reproduces $p_*$. The model is formally factorized at fixed $\lambda$, but its source distribution depends on the settings, so it violates measurement independence. This construction shows why that premise must be stated rather than hidden inside the word local.

References

• J. S. Bell, “On the Einstein Podolsky Rosen paradox,” Physics Physique Fizika 1, 195–200, 1964, doi:10.1103/PhysicsPhysiqueFizika.1.195.
• J. S. Bell, “The theory of local beables,” Epistemological Letters 9, 11–24, 1976; reprinted in Speakable and Unspeakable in Quantum Mechanics, Cambridge University Press, 1987.
• J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, “Proposed experiment to test local hidden-variable theories,” Physical Review Letters 23, 880–884, 1969, doi:10.1103/PhysRevLett.23.880.
• M. Giustina et al., “Significant-loophole-free test of Bell’s theorem with entangled photons,” Physical Review Letters 115, 250401, 2015, doi:10.1103/PhysRevLett.115.250401.
• B. Hensen et al., “Loophole-free Bell inequality violation using electron spins separated by 1.3 kilometres,” Nature 526, 682–686, 2015, doi:10.1038/nature15759.
• L. K. Shalm et al., “Strong loophole-free test of local realism,” Physical Review Letters 115, 250402, 2015, doi:10.1103/PhysRevLett.115.250402.
• R. F. Werner, “Quantum states with Einstein–Podolsky–Rosen correlations admitting a hidden-variable model,” Physical Review A 40, 4277–4281, 1989, doi:10.1103/PhysRevA.40.4277.

Further reading

• N. Brunner, D. Cavalcanti, S. Pironio, V. Scarani, and S. Wehner, “Bell nonlocality,” Reviews of Modern Physics 86, 419–478, 2014, doi:10.1103/RevModPhys.86.419.
• H. M. Wiseman, “The two Bell’s theorems of John Bell,” Journal of Physics A: Mathematical and Theoretical 47, 424001, 2014, doi:10.1088/1751-8113/47/42/424001.