Bell inequalities separate three ideas that are often blurred together: ordinary classical correlation, quantum correlation and unrestricted correlation consistent only with no faster-than-light signalling. The CHSH inequality gives the cleanest mathematical comparison.
Quantum entanglement can produce correlations stronger than any local hidden-variable model permits, yet still weaker than the strongest correlations allowed by the no-signalling principle alone. The numerical ladder is memorable:
- local hidden-variable CHSH bound: 2;
- quantum Tsirelson bound: 2√2;
- algebraic/no-signalling maximum: 4.
This guide derives each level, shows exactly where the assumptions enter, and explains why a Bell violation does not permit communication faster than light.
Local realism sets a bound. Quantum mechanics violates it. Operator geometry sets a stronger quantum bound. No-signalling permits still stronger correlations.
1. The Bell scenario
Two separated observers, conventionally Alice and Bob, receive parts of a joint physical system. Alice chooses one of two measurement settings x∈{0,1}; Bob independently chooses y∈{0,1}. Each produces a binary outcome, written a,b∈{−1,+1}.
The experiment is described by conditional probabilities p(a,b|x,y).
The central question is whether these probabilities can be explained by shared classical information λ established before the measurement choices, together with local response rules.
2. Local hidden-variable factorisation
A local hidden-variable model has the form
p(a,b|x,y)=∫dλ q(λ)p(a|x,λ)p(b|y,λ).
The shared variable λ can correlate Alice and Bob, but Alice’s response does not depend on Bob’s setting y, and Bob’s response does not depend on Alice’s setting x.
Stochastic local response functions can be expressed as mixtures of deterministic ones, so the extremal strategies assign predetermined values A₀,A₁,B₀,B₁∈{−1,+1}.
3. Correlators
For each pair of settings define the correlation
E_xy=Σ_{a,b}ab p(a,b|x,y).
If outcomes always agree, E=+1. If they always disagree, E=−1. If agreement and disagreement are balanced, E=0.
The CHSH expression is
S=E₀₀+E₀₁+E₁₀−E₁₁.
4. Derive the local bound
For one deterministic hidden-variable assignment,
S_λ=A₀B₀+A₀B₁+A₁B₀−A₁B₁.
Factor it:
S_λ=A₀(B₀+B₁)+A₁(B₀−B₁).
Because B₀ and B₁ are each ±1, exactly one of the brackets B₀+B₁ and B₀−B₁ is zero, while the other is ±2. Therefore S_λ=±2.
A probabilistic local model is a convex mixture of deterministic strategies, so averaging cannot move S outside [−2,2]. Thus
|S|≤2
for every local hidden-variable model satisfying the stated assumptions.
5. Why shared randomness does not help
Alice and Bob can pre-share unlimited classical randomness. That changes which deterministic strategy is used in each trial, but each strategy still satisfies |S|=2, and mixtures remain inside the convex hull.
This convexity is the geometric foundation of Bell inequalities: local correlations occupy a polytope whose facets include CHSH inequalities.
6. Quantum correlators
Quantum mechanics assigns a bipartite state ρ and local observables A_x and B_y with eigenvalues ±1. Then
E_xy=Tr[ρ(A_x⊗B_y)].
The CHSH operator is
𝓑=A₀⊗(B₀+B₁)+A₁⊗(B₀−B₁).
The observed S is the expectation Tr(ρ𝓑).
7. Bell-state correlation formula
For the singlet state
|Ψ^-⟩=(|01⟩−|10⟩)/√2,
spin measurements along unit Bloch vectors a and b have correlation
E(a,b)=−a·b.
Thus correlation depends only on the angle between the measurement directions.
8. Worked settings that reach 2√2
Choose Alice’s directions
a₀=za₁=x
and Bob’s directions
b₀=(z+x)/√2b₁=(z−x)/√2.
For the singlet, each relevant dot product has magnitude 1/√2 with signs arranged so that the CHSH terms reinforce. One obtains
|S|=2√2≈2.828427.
This violates the local bound 2.
9. Tsirelson’s bound
Quantum theory does not permit the algebraic maximum 4. For observables satisfying A_x²=B_y²=I, square the CHSH operator. A standard rearrangement gives
𝓑²=4I−[A₀,A₁]⊗[B₀,B₁]
up to the sign convention used for CHSH.
Each dichotomic observable has norm one, so ||[A₀,A₁]||≤2 and ||[B₀,B₁]||≤2. Therefore
||𝓑²||≤8
and hence
||𝓑||≤2√2.
Every quantum expectation therefore satisfies |S|≤2√2. This is Tsirelson’s bound.
10. Why noncommuting measurements matter
If Alice’s observables commute, then [A₀,A₁]=0 and the squared-operator argument collapses back toward the local scale. The same applies if Bob’s pair commute.
CHSH violation therefore requires incompatible local measurement choices as well as entanglement.
Entanglement alone is not enough: a poor choice of measurement axes can produce S≤2 even for a maximally entangled state.
11. The CHSH game
An equivalent viewpoint is a cooperative game. Alice and Bob receive random bits x and y and must output bits a and b without communicating after receiving the inputs. They win when
a⊕b=x·y.
For uniformly random inputs, the winning probability and CHSH value are related by
P_win=1/2+S/8
for the appropriate sign convention.
Thus the local limit S=2 gives P=3/4. The quantum maximum gives
P_quantum=1/2+(2√2)/8=(2+√2)/4=cos²(π/8)≈0.853553.
12. No-signalling conditions
A correlation is no-signalling if Alice’s marginal probability does not depend on Bob’s setting and vice versa:
Σ_b p(a,b|x,y) is independent of y,
and
Σ_a p(a,b|x,y) is independent of x.
These equations prevent either party from using the correlation alone to transmit a controllable message instantaneously.
13. PR-box correlations
A Popescu–Rohrlich box is an idealised no-signalling correlation satisfying
a⊕b=x·y
with certainty, while each local output is individually random.
It wins the CHSH game with probability one and reaches S=4.
Because the local marginals remain uniform, the box is no-signalling. Yet quantum mechanics cannot realise it because S=4 exceeds Tsirelson’s bound.
No-signalling alone is weaker than quantum mechanics: it permits correlations that nature’s quantum formalism forbids.
14. Why Bell violation does not transmit a message
In an entangled Bell experiment, Alice’s choice changes the joint correlation structure but not Bob’s local marginal distribution.
Bob cannot determine Alice’s setting by examining his outcomes alone. To reveal the Bell correlation, their records must later be compared through ordinary classical communication.
Entanglement therefore creates nonclassical correlations without becoming a faster-than-light signalling channel.
15. Bell nonlocality is not the same as entanglement
Every Bell-nonlocal state is entangled, but not every entangled state violates every Bell inequality with every measurement choice.
Some mixed entangled states admit local models for broad classes of measurements. Hidden nonlocality can sometimes be revealed after filtering or with different scenarios.
Entanglement, steering and Bell nonlocality form distinct levels of nonclassical correlation.
16. Device-independent interpretation
A Bell violation constrains possible explanations using only observed input-output statistics plus assumptions such as measurement independence and no unwanted communication.
This supports device-independent protocols in which security or randomness can be certified without trusting a detailed internal model of the measurement devices.
The certification is conditional on closing the relevant loopholes and on the statistical analysis of finite data.
17. Statistical significance in a Bell experiment
Finite experimental data produce an estimate S_hat with uncertainty. Observing 2.01 does not automatically establish convincing violation if the uncertainty is 0.05.
Modern analyses use hypothesis tests or martingale-style methods designed for sequential trials and possible memory effects rather than assuming every trial is perfectly independent and Gaussian.
The mathematical inequality is exact; experimental evidence is statistical.
18. Detection and locality loopholes
If many trials are discarded because detectors fail to register an outcome, a biased detected subset can mimic violation unless detection efficiency is high enough or the protocol handles losses rigorously.
If measurement choices and outcomes are not space-like separated, ordinary subluminal communication could in principle explain correlations.
Loophole-free Bell experiments are designed to close the major detection and locality loopholes simultaneously.
19. Measurement independence
Bell derivations assume the hidden variable is not conspiratorially correlated with the later measurement choices in a way that invalidates the factorisation.
Experiments therefore use fast random setting choices and causal-separation designs to support the intended independence assumptions.
No finite experiment proves metaphysical “free will”; it tests a physical causal model under stated assumptions.
20. Geometry of correlation sets
The local set is convex and polyhedral. The quantum set is also convex but has curved boundaries. The no-signalling set is a larger polytope.
CHSH describes one direction through these sets:
- local region ends at 2;
- quantum region extends to 2√2;
- no-signalling region extends to 4.
This geometric viewpoint leads naturally to semidefinite-programming hierarchies used to bound quantum correlations.
21. Common misconception: Bell disproves causality
Bell violations rule out local hidden-variable explanations satisfying the theorem’s assumptions. Quantum predictions remain no-signalling and compatible with relativistic causal restrictions on controllable information transfer.
22. Common misconception: entanglement gives S=2√2 automatically
The state and measurements must be chosen appropriately. Misaligned observables can reduce S below the local bound even for a maximally entangled state.
23. Common misconception: Tsirelson’s bound follows from no-signalling
No-signalling permits S=4. The stronger 2√2 limit comes from the Hilbert-space and operator structure of quantum theory.
24. Worked synthesis problem
An experiment estimates E00=0.70, E01=0.71, E10=0.69 and E11=-0.72.
Step 1: Calculate CHSH.
S=0.70+0.71+0.69-(-0.72)=2.82.
Step 2: Compare local bound. 2.82>2, so the central estimate violates CHSH.
Step 3: Compare quantum bound. 2√2≈2.828427, so 2.82 is compatible with quantum mechanics and close to its maximum.
Step 4: Do not overclaim. A real conclusion still needs uncertainties, settings independence, detection treatment and finite-sample statistics.
Step 5: No-signalling check. Separately verify that Alice’s and Bob’s marginal outcome frequencies do not depend significantly on the distant setting.
25. Practice set
- Define the CHSH correlator S.
- Why is the deterministic local value always ±2?
- Why does shared classical randomness not exceed the local bound?
- Write a quantum correlation E_xy in operator form.
- State Tsirelson’s CHSH bound.
- What feature of the CHSH operator proof makes incompatible measurements important?
- What is the maximum classical CHSH-game winning probability?
- What is the quantum optimum?
- What is a no-signalling condition?
- What CHSH value can a PR box reach?
- Why does Bell violation not permit faster-than-light messaging?
- Name two experimental loopholes that must be controlled.
Answers
S=E00+E01+E10-E11.- Because one of B₀+B₁ and B₀−B₁ is zero and the other is ±2.
- Convex mixtures of bounded deterministic strategies remain within the same bound.
E_xy=Tr[ρ(A_x⊗B_y)].|S|≤2√2.- The excess over the local scale is controlled by commutators of the local observables.
- 3/4.
cos²(π/8)≈0.853553.- A local marginal distribution is independent of the distant measurement choice.
- 4.
- Local marginals remain independent of the distant choice; joint records must later be compared classically.
- Examples: detection, locality and measurement-independence loopholes.
Sources and further study
[1] John F. Clauser, Michael A. Horne, Abner Shimony and Richard A. Holt, Proposed Experiment to Test Local Hidden-Variable Theories, Physical Review Letters 23, 880 (1969). The original CHSH formulation.
[2] Boris S. Tsirelson, Quantum generalizations of Bell’s inequality, Letters in Mathematical Physics 4, 93–100 (1980). The quantum bound now associated with Tsirelson’s name.
[3] Sandu Popescu and Daniel Rohrlich, Quantum nonlocality as an axiom, Foundations of Physics 24, 379–385 (1994). A canonical discussion of stronger-than-quantum no-signalling correlations.
[4] Michael Epping, Hermann Kampermann and Dagmar Bruß, A quantum mechanical bound for CHSH-type Bell inequalities. A modern derivation and singular-value viewpoint on Tsirelson-type bounds.
Continue through Quantum Mathematics
Guide 30: Quantum Teleportation, Superdense Coding and Entanglement Swapping turns entanglement into communication primitives. Guide 31: Entanglement Transformations, LOCC, Majorisation and Distillation studies which entangled states can be converted into which others. Guide 32: Quantum Shannon Theory, Holevo Bounds, Channel Capacities and Coding Theorems develops information-rate limits.