Bell Nonlocality: The Structure of Quantum Correlations

The entanglement and Bell's theorem page establishes the headline fact — the CHSH inequality $|S|\le 2$ holds for every local hidden-variable theory, and quantum mechanics reaches $2\sqrt{2}$ — and the philosophy remark dissects which assumption to blame. This topic sits between them: it develops the mathematical structure that the single CHSH inequality is a shadow of. The right object is not one inequality but a convex geometry of correlations, with three nested bodies — local, quantum, and no-signalling — whose boundaries encode exactly how far nature departs from classical common causes and why it stops where it does.

Everything is phrased device-independently, in terms of the observable statistics $P(ab\mid xy)$ alone: the probability that Alice and Bob obtain outcomes $a,b$ given that they freely chose measurement settings $x,y$. No Hilbert space is assumed until §4.

1. Behaviours and the three convex sets

Fix a bipartite Bell scenario with $m$ settings per party and $d$ outcomes each (the workhorse is $(m,d)=(2,2)$). A behaviour is the full table of conditional probabilities

$$\mathbf{P}=\{P(ab\mid xy){\}}_{a,b,x,y},P(ab\mid xy)\ge 0,\sum _{a,b}P(ab\mid xy)=1.$$

Three properties carve out three convex sets, nested as $\mathcal{L}\subseteq \mathcal{Q}\subseteq \mathcal{N}\mathcal{S}$:

• Local ($\mathcal{L}$). There is a hidden variable $\lambda$ with density $\rho (\lambda )$ and local response distributions such that $$P(ab\mid xy)=\int d\lambda \rho (\lambda )P_A(a\mid x,\lambda )P_B(b\mid y,\lambda ).$$ This is the factorizability condition — a common cause screening off the wings.
• Quantum ($\mathcal{Q}$). There is a state $\hat{\rho }$ on ${\mathcal{H}}_A\otimes {\mathcal{H}}_B$ and local POVMs $\{{\hat{E}}_x^a\},\{{\hat{F}}_y^b\}$ with $P(ab\mid xy)=\mathrm{Tr}[\hat{\rho }({\hat{E}}_x^a\otimes {\hat{F}}_y^b)].$
• No-signalling ($\mathcal{N}\mathcal{S}$). Each party's marginal is independent of the distant setting: $$\sum _{b}P(ab\mid xy)=\sum _{b}P(ab\mid x{y}^{\prime })=:P_A(a\mid x)\forall y,y^{\prime },$$ and symmetrically for Bob. This is the only constraint relativity imposes directly (§10 of the foundations page).

The two outer inclusions are the whole story: $\mathcal{L}⊊\mathcal{Q}$ is Bell's theorem, and $\mathcal{Q}⊊\mathcal{N}\mathcal{S}$ is Tsirelson's bound (§4). Both inclusions are strict.

2. The local polytope and Bell inequalities as its facets

The local set has a decisive geometric feature: $\mathcal{L}$ is a polytope. Any stochastic local model is a convex mixture of deterministic ones (absorb the local coins into $\lambda$ — Fine's theorem), and there are only finitely many deterministic strategies: each party fixes an outcome for each of its $m$ settings, giving $d^m$ local assignments per party and $d^{2m}$ joint deterministic vertices ${\mathbf{P}}_{\lambda }$. Hence

$$\mathcal{L}=\mathrm{conv}\{{\mathbf{P}}_{\lambda }{\}}_{\lambda }\text{(finitely many vertices)}.$$

By the Minkowski–Weyl theorem a bounded polytope is equally the intersection of finitely many half-spaces. Those half-spaces are exactly the tight Bell inequalities:

$$\boxed{\text{facets of }\mathcal{L}⟺\text{tight (non-redundant) Bell inequalities}.}$$

Deciding membership in $\mathcal{L}$ is therefore, in principle, a linear program; finding all facets is the (computationally hard) facet enumeration problem. For the simplest scenario $(m,d)=(2,2)$ the answer is clean: modulo the trivial positivity facets, the only nontrivial facets are the eight symmetry-related copies of CHSH,

$$|E(x,y)\pm E(x,y^{\prime })\pm E(x^{\prime },y)\mp E(x^{\prime },y^{\prime })|\le 2,$$

obtained by relabelling settings and outcomes. This is why CHSH is not one inequality among many but the inequality of the elementary scenario.

Fine's theorem (1982), restated geometrically. A $(2,2)$ behaviour is local iff a single joint distribution $P(A_1{A}_2{B}_1{B}_2)$ over all four counterfactual outcomes exists reproducing every pair marginal iff all eight CHSH inequalities hold. Locality, the existence of a global joint distribution, and CHSH-satisfaction are one and the same condition.

3. The no-signalling polytope and the PR box

The no-signalling conditions of §1 are also linear equalities, so $\mathcal{N}\mathcal{S}$ is a polytope too, strictly larger than $\mathcal{L}$. Its most important extremal point is the Popescu–Rohrlich (PR) box (1994): for binary settings/outcomes $x,y,a,b\in \{0,1\}$,

$$P_{\mathrm{PR}}(ab\mid xy)=\{\begin{array}{ll}\frac{1}{2}& a\oplus b=x\wedge y,\\ 0& \text{otherwise.}\end{array}$$

It has uniform marginals (hence no-signalling) yet reaches the algebraic maximum $S=4$. The PR box shows that no-signalling alone does not pin down quantum theory: there is logical room for correlations stronger than any state can produce. The foundational question "why $2\sqrt{2}$ and not $4$?" is the question of what extra principle — beyond relativity — nature obeys (information causality, macroscopic locality, local orthogonality; see §7).

$$\underset{\text{local }\mathcal{L}}{\underbrace{S\le 2}}⊊\underset{\text{quantum }\mathcal{Q}}{\underbrace{S\le 2\sqrt{2}}}⊊\underset{\text{no-signalling }\mathcal{N}\mathcal{S}}{\underbrace{S\le 4}}.$$

Unlike $\mathcal{L}$ and $\mathcal{N}\mathcal{S}$, the quantum set $\mathcal{Q}$ is not a polytope: its CHSH-facing boundary is the curved Tsirelson surface, so no finite list of linear Bell inequalities describes it.

4. Tsirelson's bound and the quantum set

Promote the $\pm 1$ outcomes to observables ${\hat{A}}_x={\hat{A}}_x^{†}$, ${\hat{B}}_y={\hat{B}}_y^{†}$ with ${\hat{A}}_x^2={\hat{B}}_y^2=\hat{1}$ and $[{\hat{A}}_x,{\hat{B}}_y]=0$ (spacelike commutation). The CHSH operator $\hat{S}={\hat{A}}_0{\hat{B}}_0-{\hat{A}}_0{\hat{B}}_1+{\hat{A}}_1{\hat{B}}_0+{\hat{A}}_1{\hat{B}}_1$ obeys the sum-of-squares identity

$${\hat{S}}^2=4\hat{1}+[{\hat{A}}_0,{\hat{A}}_1][{\hat{B}}_1,{\hat{B}}_0].$$

Since each commutator of two $\pm 1$ observables has norm $\le 2$, $\Vert {\hat{S}}^2\Vert \le 8$, giving Tsirelson's bound

$$\boxed{|⟨\hat{S}⟩|\le \Vert \hat{S}\Vert \le 2\sqrt{2}}$$

saturated by the singlet with settings at successive $45^{\circ }$ (derived in full in §5 of the foundations page).

Characterizing $\mathcal{Q}$. Deciding membership in the quantum set is far subtler than for the polytopes. The NPA hierarchy (Navascués–Pironio–Acín, 2007) gives a convergent sequence of semidefinite outer approximations $\mathcal{Q}\subseteq \cdots \subseteq {\mathcal{Q}}_2\subseteq {\mathcal{Q}}_1$: one asks whether a certain moment matrix of operator products can be positive semidefinite, an SDP feasibility test at each level. The hierarchy converges to the commuting-operator quantum set ${\mathcal{Q}}^c$.

Tsirelson's problem. Does the tensor-product definition (${\mathcal{H}}_A\otimes {\mathcal{H}}_B$) give the same set as the commuting-operator one? For finite dimensions yes; in general $\overline{{\mathcal{Q}}^{\text{tensor}}}={\mathcal{Q}}^c$ was refuted by the result ${\mathrm{MIP}}^{\ast }=\mathrm{RE}$ (Ji–Natarajan–Vidick–Wright–Yuen, 2020): the two sets differ, a striking bridge from Bell nonlocality to the theory of operator algebras (Connes' embedding problem) and computational complexity.

5. All-versus-nothing: GHZ and Mermin

CHSH is a statistical contradiction. With three or more parties the clash becomes a logical one, refuted by a single run rather than a margin. Take the Greenberger–Horne–Zeilinger state

$$|\mathrm{GHZ}⟩=\frac{1}{\sqrt{2}}(|000⟩+|111⟩),$$

and consider the four commuting observables built from Pauli operators $X,Y$: $XXX,XYY,YXY,YYX$. The GHZ state is a simultaneous eigenstate with

$$XXX|\mathrm{GHZ}⟩=+|\mathrm{GHZ}⟩,XYY=YXY=YYX=-|\mathrm{GHZ}⟩.$$

A local hidden-variable theory must pre-assign values $x_i,y_i\in \{\pm 1\}$ to each $X_i,Y_i$. Multiplying the four predicted products and using $y_i^2=1$ forces $(+1)(-1)(-1)(-1)=(x_1{x}_2{x}_3)(x_1{y}_2{y}_3)(y_1{x}_2{y}_3)(y_1{y}_2{x}_3)=(x_1{x}_2{x}_3{)}^2(y_1{y}_2{y}_3{)}^2\cdot (\dots )$, which collapses to $-1=+1$: no value assignment exists. The Mermin inequalities extend this to $n$ parties with a violation growing exponentially in $n$, and underlie the Mermin–Peres magic square (§6). GHZ is the sharpest possible face of Bell's theorem: quantum mechanics and local realism disagree with certainty on a single ideal measurement.

6. Contextuality: the ambient obstruction

Bell nonlocality is a spatial instance of a wider impossibility. The Kochen–Specker theorem (1967) states that in any Hilbert space of dimension $\ge 3$ there is no non-contextual value assignment: no map $v$ sending each projector to $\{0,1\}$ that respects the orthogonality/sum rules of quantum mechanics independently of which compatible set the projector is measured alongside. A finite set of vectors (the classic constructions use 18 in dimension 4) already admits no consistent $\{0,1\}$-colouring.

The Peres–Mermin square exhibits this state-independently with nine two-qubit observables arranged in a $3\times 3$ grid, each row and column a commuting triple whose product is $+\hat{1}$ except one column whose product is $-\hat{1}$; no assignment of $\pm 1$ can reproduce all six product constraints. Locality is then exposed as spatial non-contextuality (compatibility enforced by spacelike separation rather than by commutation), and Bell's theorem as contextuality's most physically dramatic special case. The modern graph- and sheaf-theoretic frameworks (Cabello–Severini–Winter; Abramsky–Brandenburger) unify inequalities, KS colourings, and the quantum bounds under one combinatorial roof.

7. Device independence: turning the boundary into a resource

The convex geometry is not merely descriptive — its boundary is operationally certifying. Because a behaviour outside $\mathcal{L}$ cannot have been produced by any predetermined local mechanism, observing it licenses conclusions about a black box:

• Self-testing. Achieving the maximal CHSH value $2\sqrt{2}$ certifies, uniquely up to a local isometry, that the boxes shared a singlet and measured anticommuting observables — the entire physical setup is pinned down by the statistics alone (Mayers–Yao; Tsirelson's rigidity). Extremal points of $\mathcal{Q}$ are the self-testable ones.
• Device-independent randomness. A CHSH value $S>2$ lower-bounds the min-entropy of the outcomes against a quantum adversary holding the purifying system — the violation guarantees genuine unpredictability, enabling randomness expansion and amplification certified without trusting the hardware.
• Device-independent key distribution (DIQKD). The same certification secures a cryptographic key whose secrecy rests on the Bell violation, not on modelling the devices — the technological cash value of the no-signalling structure and the no-cloning theorem.

These applications are why the shape of $\mathcal{Q}$ matters: the closer a behaviour sits to the Tsirelson boundary, the more certified randomness and security it yields.

8. No-cloning and the hiddenness of hidden variables

The no-cloning theorem is usually filed beside no-signalling as a curiosity, but in the geometry above it plays a structural role, and its relation to hidden variables is worth stating precisely — the slogans routinely mislead.

No-cloning does not forbid hidden variables. It constrains unitary operations on the quantum state $|\psi ⟩$; it says nothing about ontology. A theory that supplements $|\psi ⟩$ with extra variables $\lambda$ is entirely consistent with it — Bohmian mechanics, a deterministic, $\lambda$-carrying, realist theory, satisfies no-cloning exactly. The theorem rules hidden variables neither in nor out.

The real link runs through no-signalling. The chain is $$\text{cloning}⟹\text{readout of the reduced state }{\hat{\rho }}_A⟹\text{learn Alice’s distant setting}⟹\text{signalling}.$$ Given many copies of one wing of an entangled pair, tomography would reveal ${\hat{\rho }}_A={\mathrm{Tr}}_B|\psi ⟩⟨\psi |$ — but by Parameter Independence (§1) that marginal is exactly what cannot depend on the distant setting. Cloning would defeat that protection and turn the nonlocal correlations into a usable signal. This is not hypothetical: no-cloning was proved precisely to kill such a scheme (Herbert's FLASH, 1981, which cloned Bob's photon to read Alice's basis). No-cloning is thus the barrier that keeps Shimony's "passion at a distance" from becoming "action at a distance."

Inside a hidden-variable theory, no-cloning is the inaccessibility of $\lambda$. In a nonlocal HV theory such as Bohm, the hidden variable $\lambda$ (the particle positions) is instantaneously affected by the distant setting. What stops that from being a telegraph is that the distribution of $\lambda$ is locked to the Born rule — Bohmian quantum equilibrium $\rho =|\psi {|}^2$, with its associated "absolute uncertainty" (Dürr–Goldstein–Zanghì): one can never know $\lambda$ more sharply than $|\psi {|}^2$ permits. Perfect cloning would let one sample $\lambda$ finely enough to detect the nonlocal influence, breaking equilibrium and enabling signalling. Hence

$$\underset{\text{orthodox formalism}}{\underbrace{\text{no-cloning}}}⟷\underset{\text{hidden-variable formalism}}{\underbrace{\text{no read access to }\lambda \text{ beyond }|\psi {|}^2}},$$

two faces of the same epistemic wall.

Equivalent framing. No-cloning is essentially equivalent to the impossibility of determining an unknown $|\psi ⟩$ from a single copy. Any $\lambda$ that fixed outcomes for all measurement bases would become fully measurable if it could be cloned, collapsing the epistemic quantum state into the ontic $\lambda$; forbidding cloning is exactly what guarantees a $\psi$-supplementing $\lambda$ must stay hidden. So no-cloning is the mechanism that renders whatever nonlocal, predetermined structure underlies the correlations harmless to relativity — the nonlocality remains passion, never signal.

9. Remarks and open problems

• Why $2\sqrt{2}$? Several information-theoretic principles — information causality (Pawłowski et al., 2009), macroscopic locality, local orthogonality — recover the Tsirelson bound from physically motivated axioms, but no single principle is yet known to single out $\mathcal{Q}$ exactly. This is the reconstruction programme flagged in the philosophy remark §5.
• Characterizing $\mathcal{Q}$. No finite Bell-inequality description exists (the boundary is curved and, by ${\mathrm{MIP}}^{\ast }=\mathrm{RE}$, its commuting-operator version is not even computably approximable in general).
• Nonlocality is a strict subset of entanglement. Some entangled mixed states admit a local model for all projective measurements (Werner states below a visibility threshold): entanglement is necessary but not sufficient for a Bell violation.
• Interpretational reading. Which premise the violation of $\mathcal{L}⊊\mathcal{Q}$ indicts — locality, definiteness, single-outcomes, or measurement independence — is the subject of the philosophy of Bell's theorem; the deterministic, avowedly nonlocal completion that lives inside this geometry is Bohmian mechanics.

See also

• Entanglement, EPR, and Bell's Theorem — the core CHSH derivation, the singlet violation, and no-signalling/no-cloning.
• The Philosophy of Bell's Theorem — which assumption to blame; loopholes, interpretations, and the metaphysical ledger.
• Bohmian Mechanics — a nonlocal hidden-variable theory consistent with the whole geometry above.
• The Measurement Problem and Interpretations — the interpretive options sorted by premise.
• Spin — the two-level observables that realize the CHSH measurements.