Bell theorem

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

The force of the Bell theorem is that a disagreement about a local hidden-variable theory becomes a constraint on frequencies in a Bell test. Quantum entanglement alone does not guarantee a violation for every measurement in quantum mechanics: the quantum state, analyzer settings and treatment of missed events all matter. A violation of the CHSH inequality tests the conjunction of Bell local causality, measurement independence and the stated outcome and sampling assumptions; it remains compatible with quantum no-signalling.

Quantum correlations incompatible with Bell local causality under additional assumptions including measurement independence. This is an operational statement about joint probabilities, not controllable faster-than-light signaling: quantum predictions also obey quantum no-signalling. Entanglement alone does not guarantee violation of a specified Bell inequality in a given measurement arrangement.

A two-setting nonlocality argument in which several zero-probability events forbid a further coincidence in any deterministic local hidden-variable model, but quantum theory assigns that coincidence positive probability. For binary indicators, suppose $C_{+}^R{C}_{-}^R=0$, $D_{+}^I=1\Rightarrow C_{-}^R=1$ and $D_{-}^I=1\Rightarrow C_{+}^R=1$. Any assignment with $D_{+}^I{D}_{-}^I=1$ would violate the first relation. A quantum implementation can nevertheless give such a coincidence. Real imperfect experiments require an inequality that accounts for their nonzero error probabilities. Lucien Hardy gave a two-particle inequality-free argument in 1992.^[ref]

A Bell test compares experimental probabilities or expectation values with a Bell inequality. In a two-setting photon polarization test, each station chooses a polarizer orientation and records an outcome. A coincidence circuit identifies candidate pairs using an agreed timing rule. The choice of which trials enter the statistic is part of the experiment: correlations calculated only from detected pairs can differ from correlations of all emitted pairs. The Polarization CHSH count model separates ideal expectation values from this experimental selection problem.

The Bell test reported by Bas Hensen and collaborators used electron spins separated by $1.3\mathrm{km}$, efficient spin readout and rapid random measurement settings. Those features addressed detection and locality loopholes together. Its result tests Bell local causality with the stated experimental assumptions, rather than ruling out every possible hidden-variable theory.^[ref]

In 1972, Stuart J. Freedman and John F. Clauser excited a calcium beam from an oven with a filtered deuterium arc lamp. A calcium emission cascade supplied paired photons; an optical lens system, optical filters, removable glass-plate polarizers and cooled photomultipliers selected their linear polarizations. They compared coincidence circuit rates at relative angles $22.5^{\circ }$ and $67.5^{\circ }$, normalized with polarizers removed. Alternating configurations controlled drift; delayed counts estimated accidental coincidences. The violation agreed with quantum mechanics, conditional on their explicit assumption that passage through a polarizer did not change a photon's detection probability. This was a static-analyzer test with a detection loophole, rather than the ideal four-output CHSH inequality statistic.^[ref]

In 1982, Alain Aspect, Jean Dalibard and Gérard Roger used a calcium cascade and an acoustic standing wave in water to route each photon toward one of two fixed polarizers. Switching took about $10\mathrm{ns}$, against $40\mathrm{ns}$ for light to cross between stations. Photomultiplier coincidence circuit rates, with delayed-background subtraction and polarizer-removal normalization, violated their Bell inequality. Supplementary polarizers checked each channel against Malus's law. The timing attacked the locality loophole, but the authors emphasized that switching was quasiperiodic, not random; incomplete switching and inefficient collection also limited the test. Its normalized statistic differs from the CHSH inequality convention used below.^[ref]

In 1998, Jian-Wei Pan, Dik Bouwmeester, Harald Weinfurter and Anton Zeilinger created two photon pairs by two passages of a pulsed pump through a crystal using spontaneous parametric down-conversion. Interfering the inner photons at a nonpolarizing beam splitter implemented a partial Bell-basis measurement. Narrow optical filters and spatial filtering removed distinguishing information; changing the outer polarizer bases checked the conditional correlations. Fourfold detections selected the entanglement swapping events. The reported interference supported conditional quantum entanglement, but the paper reserved a statistically significant Bell inequality violation for improved visibility; it was not a demonstration closing the detection loophole.^[ref]

The ideal calculation makes the distinction between conditional and unconditional quantum states explicit. Start with spin singlet states $|{\psi }^{-}{⟩}_{12}|{\psi }^{-}{⟩}_{34}$, where $|{\psi }^{-}⟩=(|HV⟩-|VH⟩)/\sqrt{2}$ and $H,V$ denote orthogonal photon polarizations. Taking the inner product with $⟨{\psi }^{-}{|}_{23}$ gives $-|{\psi }^{-}{⟩}_{14}/2$: expanding the four terms leaves precisely the two antisymmetric outer terms. That Bell-basis measurement outcome has probability $1/4$ and leaves the outer pair in a spin singlet state. Ignoring the inner outcome instead leaves the outer density matrix $I_4/4$, a maximally mixed state. A conditional preparation flag is useful for later Bell tests only with the timing and setting conditions in setting-independent heralding preserves Bell locality.

In 2015, Marissa Giustina, Anton Zeilinger and collaborators combined spontaneous parametric down-conversion with rapid random analyzer choices and efficient transition-edge sensors. Local records included settings and detector signals; their Bell inequality used detected and undetected trials, avoiding fair sampling. Locally fixed time slots prevented coincidence-based trial selection. Calibrated optical and electrical delays checked spacelike separation of each choice from the remote readout. The violation addressed detection and locality together. Measurement independence still requires assumptions about the hidden state's origin; the experiment cannot exclude every possible common cause of source and settings.^[ref]

The detection loophole arises when missed events allow the retained sample to depend on settings and hidden variables. For example, a local hidden-variable theory may retain different underlying pairs at different polarizer angles. Applying the CHSH inequality to that selected sample does not test the original setting-independent ensemble. A valid protocol specifies outcomes and trials, including failures, or justifies an appropriate fair sampling assumption. Subtracting accidental coincidences addresses background contamination; it does not by itself repair this selection problem.

For a specified Bell test statistic, fair sampling assumes that conditioning on successful detections preserves the correlations needed to represent the intended ensemble. Constant detector efficiency in an independently specified loss model can ensure this; constant observed count rates alone do not establish independence from the hidden state. The Polarization CHSH count model explicitly assumes independent losses.

The locality loophole permits a station's outcome to be influenced by the distant setting through an ordinary signal arriving before readout. Spacelike separation excludes such communication at or below the speed of light. Fast changes only help if the relevant choice and detection events have the required timing. Predictable switching also leaves open correlations with earlier settings; closing a communication channel does not prove measurement independence.

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.

A bound on the correlations attainable by measurement in quantum measurements for a specified Bell expression. The CHSH Tsirelson bound is $2\sqrt{2}$; the number depends on which Bell expression is being optimized.

For dichotomic observables with $A_i^2=B_j^2=I$, let $\mathcal{B}=A_0\otimes (B_0+B_1)+A_1\otimes (B_0-B_1)$. Expanding gives ${\mathcal{B}}^2=4I-[A_0,A_1]\otimes [B_0,B_1]$. Each commutator has operator norm at most two, so $\Vert \mathcal{B}{\Vert }^2\le 8$ and $|⟨\mathcal{B}⟩|\le 2\sqrt{2}$. Appropriate spin measurements of a Bell state attain equality. The local hidden-variable bound of the corresponding CHSH inequality is two.

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.

// Target: probability-and-statistics.bigb

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.

In a local hidden-variable description of a Bell experiment, shared hidden data have a distribution independent of the measurement settings. Each outcome depends on the local setting and those shared data, without dependence on the remote setting. Local randomness can be included in the hidden data. Perfect correlations then constrain simultaneous assignments to the possible local measurements. The GHZ theorem shows an all-versus-nothing inconsistency between such assignments and certain three-qubit correlations.

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.

A Bell-test loophole in which hidden information determining the device outputs is correlated with the chosen measurement settings. The usual local CHSH inequality averages a common hidden-variable distribution over independently selected settings. If that distribution can instead depend on the selected pair, the four counterfactual assignments need not obey the same averaging argument. An adversary controlling both the devices and their setting generators can exploit this dependence without violating the local-bound proof's actual hypotheses.

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 a short proof of the local bound, fix a deterministic local hidden-variable model assignment. The combination is $A_0(B_0+B_1)+A_1(B_0-B_1)$. Exactly one bracket vanishes and the other is $\pm 2$, so every assignment contributes $\pm 2$. Averaging with a single setting-independent probability distribution preserves the interval $[-2,2]$. Mixtures also cover stochastic local hidden-variable theories. Rejecting a different sample at each setting would invalidate precisely that common-average step.

The quantum bound follows just as economically. For Hermitian operators with $A_i^2=B_j^2=I$, the CHSH inequality operator $C=A_0\otimes (B_0+B_1)+A_1\otimes (B_0-B_1)$ obeys $C^2=4I-[A_0,A_1]\otimes [B_0,B_1]$. Each commutator has operator norm at most two, giving $\Vert C\Vert \le 2\sqrt{2}$. The Polarization CHSH count model attains this bound without allowing either station to control the distant marginal probability distribution.

For the Bell state $|{\varphi }_{+}⟩$, let both zero-input measurements use the computational basis, and let the one-input bases be rotated by $\theta$ for Alice and $-\theta$ for Bob in the real qubit plane. In the CHSH game, the four conditional winning probabilities are $1,{\cos }^2\theta ,{\cos }^2\theta ,{\sin }^2(2\theta )$. They follow by contraction of the Bell state with vectors from the real orthonormal bases used for the basic measurements. Their average is the displayed expression. On $0\le \theta \le \pi /2$, this beats $3/4$ exactly when $0<\theta <\pi /4$ and reaches $13/16$ at $\theta =\pi /6$, since $t-t^2=1/4-(t-1/2{)}^2$.

Two separated players receive uniformly random input bits $a,b$ and output bits $x,y$ without communicating after receiving their inputs. They win when $x\oplus y=ab$, with exclusive or $\oplus$. For deterministic local response functions, the four desired equalities have inconsistent parity, so at most three can hold. Shared randomness only averages such strategies, proving classical winning probability at most $3/4$. In terms of correlations $E_{ab}=\mathbb{E}[(-1{)}^{x+y}\mid a,b]$, the winning probability is $1/2+(E_{00}+E_{01}+E_{10}-E_{11})/8$, directly connecting the game to the CHSH inequality.

The Hermitian operator $\mathcal{B}=\sqrt{2}(X\otimes X-Z\otimes Z)$ has eigenvalues $2\sqrt{2},0,0,-2\sqrt{2}$ on ${\Psi }^{+},{\Phi }^{+},{\Psi }^{-},{\Phi }^{-}$, respectively. Thus any density matrix satisfying $\mathrm{Tr}(\rho \mathcal{B})=2\sqrt{2}$ must be $|{\Psi }^{+}⟩⟨{\Psi }^{+}|$: an average can equal the largest eigenvalue only when all weight lies in its one-dimensional eigenspace. The purity decouples a subsystem from its purification lemma then excludes correlations with an external environment. This argument assumes the stated measurement operators; CHSH self-testing handles unknown representations with additional qualifications.

Maximal quantum CHSH inequality violation certifies an embedded maximally entangled qubit pair and its ideal measurement action, up to local isometries and unused ancillary systems, under the separated-device quantum assumptions. It does not identify every microscopic component of an arbitrary device. Near-maximal observed correlations require robust error bounds rather than exact identification.^[ref]

Let $H,V$ be orthogonal linear polarizations, and prepare the Bell state $|{\Phi }^{+}⟩=(|HH⟩+|VV⟩)/\sqrt{2}$. The $+1$ analyzer vector is $|a⟩=\cos a|H⟩+\sin a|V⟩$; its orthogonal companion has outcome $-1$. Direct inner products with the Bell state give $P_{++}=P_{--}={\cos }^2(a-b)/2$ and $P_{+-}=P_{-+}={\sin }^2(a-b)/2$. Thus the expectation value of the outcome product is $E(a,b)=\cos 2(a-b)$. These are physical polarizer angles; their corresponding Bloch sphere angles are doubled.

For a stipulated isotropic admixture, use the density matrixThe maximally mixed state contributes zero correlation, so $P_{st}(a,b)=[1+stv\cos 2(a-b)]/4$ for $s,t\in \{-1,1\}$. Each local marginal is $1/2$, independently of the other setting, illustrating quantum no-signalling. In this model $v$ is the interferometric visibility of the coincidence fringe; other noise models need not have one common visibility for every basis.

Choose $a_0=0^{\circ }$, $a_1=45^{\circ }$, $b_0=22.5^{\circ }$ and $b_1=-22.5^{\circ }$. The four correlations in the displayed CHSH inequality are $v/\sqrt{2},v/\sqrt{2},v/\sqrt{2},-v/\sqrt{2}$. Consequently $S=2\sqrt{2}v$, exceeding the local bound exactly when $v>1/\sqrt{2}$. Perfect correlation at a single common setting would not yield this comparison between incompatible quantum measurements.

To connect probabilities with expected counts, stipulate $M$ emitted pairs per setting pair and independent detection efficiencies ${\eta }_A,{\eta }_B$, independent also of settings, outcomes and hidden variables. The expected joint counts are $n_{st}=M{\eta }_A{\eta }_B{P}_{st}$. Their normalized correlation isFor original teaching inputs $M=1000$, ${\eta }_A=0.8$, ${\eta }_B=0.75$ and $v=0.8$, the $(a_0,b_0)$ expected counts are $n_{++}=n_{--}\simeq 234.85$ and $n_{+-}=n_{-+}\simeq 65.15$. They total $600$ and give $E_{\det }\simeq 0.56569$, hence $S\simeq 2.26274$. Fractional expected counts are not measured data. Efficiency cancels here because the loss model was specified to be independent; this algebra supplies no justification for discarding failures in an arbitrary Bell test with a detection loophole.

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.