Bell theorem

Bell's theorem shows that no theory assigning outcomes through local hidden variables can reproduce every quantum correlation.

A Bell inequality is a constraint on observable correlations satisfied by the specified class of local hidden-variable models. In finite-setting scenarios such constraints bound linear combinations of probabilities or expectation values. The CHSH inequality is one example, following from Bell factorization, measurement independence and bounded outcomes. A violation rules out the conjunction of the assumptions defining that model class; it does not alone identify a unique false premise.

For outcomes in $\{0,\dots ,d-1\}$, let $[x{]}_d$ be their nonnegative modular representative. A deterministic local hidden-variable model satisfiesThe unreduced chain telescopes to $-1$, so its reduced nonnegative sum is at least $d-1$ for each assignment. For binary outcomes and two settings, it is a form of the CHSH inequality: $I_{2,2}=2-(E_{11}+E_{21}+E_{22}-E_{12})/2$.

On $|{\Phi }^{+}⟩$, measure ${\sigma }_z\cos \theta +{\sigma }_x\sin \theta$, assigning bit $a$ to eigenvalue $(-1{)}^a$. Choose Alice's angles $2(i-1)\delta$ and Bob's angles $(2i-1)\delta$, with $\delta =\pi /(2N)$. The Bell state correlation is $\cos (\theta -\varphi )$. Each adjacent disagreement term and the last agreement term therefore has probability ${\sin }^2(\pi /(4N))$. For $N\ge 2$, their total violates the chained modular Bell inequality bound one, and it tends to zero as $N\to \infty$. At $N=1$ the value is exactly one.

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A local hidden-variable theory assigns each party's outcome from a shared hidden state and that party's own setting, independently of the spacelike-separated setting chosen by the other party.

If a successful preparation flag is fixed before freely selected Bell inequality settings, conditioning a local hidden-variable theory on that flag only replaces its hidden-variable distribution by the conditional distribution. The later response functions remain local and the distribution is independent of the later settings, so the CHSH inequality still holds. Setting-dependent rejection during the Bell measurements does not satisfy this argument.

Outcome independence requires $P(A,B\mid a,b,\lambda )=P(A\mid a,b,\lambda )P(B\mid a,b,\lambda )$. It says that, once settings and the proposed complete state are fixed, learning the distant outcome does not change the local outcome probability. Combining it with parameter independence gives Bell factorization. Entangled quantum states generally fail outcome independence even though their observable marginal distributions obey quantum no-signalling.

For spacelike-separated measurement regions, a sufficient specification $\lambda$ of their relevant past should screen each outcome from additional information in the other wing. In a two-setting experiment a Bell-local model has $P(A,B\mid a,b,\lambda )=P(A\mid a,\lambda )P(B\mid b,\lambda )$. Together with measurement independence and bounded outcomes, this implies the CHSH inequality. Bell local causality is stronger than quantum no-signalling and is not an assumption that every outcome was predetermined.

A setting-independent hidden variable fixes responses for every measurement choice, and each response depends only on the local setting. Stochastic local hidden-variable theories can be represented as mixtures of these assignments by including local random seeds in the hidden variable. Deterministic counterfactual assignments are the starting point of the chained modular Bell inequality.

Outcome determinism says that a complete hidden state and the measurement settings fix each measurement outcome with probability one.

Parameter independence says that, conditional on the complete hidden state, the probability distribution of one party's outcome is independent of the other party's measurement setting.

Measurement independence says that the distribution of the hidden state is independent of the measurement settings: $\rho (\lambda |a,b)=\rho (\lambda )$. Its failure is the measurement-dependence loophole in a Bell test.

For local outcomes $A_0,A_1,B_0,B_1\in \{-1,1\}$, the Clauser–Horne–Shimony–Holt inequality isAn arbitrary no-signalling correlation can reach the algebraic maximum $4$, while quantum correlations are bounded by $2\sqrt{2}$.

For spin-projective measurements on a two-qubit state, let ${\lambda }_1\ge {\lambda }_2\ge {\lambda }_3$ be the eigenvalues of $T^{T}T$, where $T$ is its Pauli correlation tensor. Bob's sum and difference of unit axes are orthogonal and have lengths $2\cos \varphi$ and $2\sin \varphi$. Maximizing Alice's two axes aligns them with the corresponding images under $T$. Maximizing $\varphi$ then gives $2\sqrt{\Vert T\mathbf{u}{\Vert }^2+\Vert T\mathbf{v}{\Vert }^2}$ for orthonormal $\mathbf{u},\mathbf{v}$. The maximum of this quadratic-form sum is ${\lambda }_1+{\lambda }_2$, by diagonalizing $T^{T}T$ and selecting its top two eigenvectors. Thus the displayed expression is both an upper bound and attainable. The same maximum applies to the form with the sum of two absolute values, since local outcome signs can select their positive branches.

Every entangled pure state of two qubits violates a CHSH inequality for a suitable choice of local measurement axes. If its two Schmidt coefficients are $c_0,c_1>0$, the optimal construction already gives a value $2\sqrt{1+(2c_0{c}_1{)}^2}>2$.

For the Schmidt-basis Pauli correlation tensor, choose Alice's axes ${\mathbf{e}}_z,{\mathbf{e}}_x$ and Bob's $({\mathbf{e}}_z\pm s{\mathbf{e}}_x)/\sqrt{1+s^2}$. The CHSH inequality expression is $2\sqrt{1+s^2}$. It exceeds two exactly when both Schmidt coefficients are nonzero, reaching $2\sqrt{2}$ for maximal quantum entanglement. A product state gives $s=0$ and saturates rather than violates the inequality.

A no-signalling box is a conditional probability distribution $P(x,y|a,b)$ whose marginal distribution for either party is independent of the other party's input. It can therefore exhibit correlations stronger than quantum correlations without transmitting information superluminally.

A Popescu–Rohrlich box is a bipartite no-signalling box attaining the algebraic maximum of the CHSH inequality.

Take a separated Boolean decomposition $f(x,y)={⨁}_{\ell }{u}_{\ell }(x)v_{\ell }(y)$. Feed $(u_{\ell },v_{\ell })$ to one independent PR box, obtaining outputs with $a_{\ell }\oplus b_{\ell }=u_{\ell }{v}_{\ell }$. Alice forms $A={⨁}_{\ell }{a}_{\ell }$, Bob forms $B={⨁}_{\ell }{b}_{\ell }$, and their parities satisfy $A\oplus B=f$. Sending the single bit $A$ lets Bob compute the answer with certainty. The truth-table decomposition needs at most $2^{\min (m,n)}$ boxes for input lengths $m,n$. Each box's local output is uniform, so Bob's uncommunicated output string carries no information about Alice's input; the final parity bit is essential for a general function depending on that input.