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Alain Aspect, Jean Dalibard and Gérard Roger tested Bell inequalities using pairs of photons and rapidly changing polarization analyzers. An acousto-optical switch in each arm routed light toward one of two polarizers; switching near $50\mathrm{MHz}$ changed the effective setting on a timescale shorter than the photon transit time between the analyzers. The measured correlations violated the tested Bell inequality. These switches were driven at incommensurate frequencies, so their role should be distinguished from the independently randomized settings of later experiments.^[ref]

A physicist researching quantum optics and the foundations of quantum mechanics.^[ref]

Artur Ekert proposed deriving a shared cryptographic key from measurements on entangled particle pairs while testing other measurement settings through a Bell inequality. The different settings therefore serve two purposes: correlated results supply candidate key bits, and the observed correlations constrain possible descriptions of the source and an eavesdropper. The proposal connected a foundational test of quantum mechanics with the operational task of quantum key distribution, under the assumptions of its measurement and security model.^[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]

An experimental physicist who coauthored the 2015 Delft electron-spin Bell test. The experiment entangled electron spins at sites separated by 1.3 kilometres. Efficient readout addressed the detection loophole, while fast random measurement choices and the separation addressed the locality loophole. The observed violation of a Bell inequality tested local realism without relying on fair sampling of undetected events.^[ref]

Give Alice the first qubit $A$ and Bob the second $B$. Compare the Bell state $|{\Phi }^{+}⟩$ with a preparation that tosses a fair classical bit and gives both parties its value. Their density operators, written in the computational basis ordered $00,01,10,11$, areThey have identical diagonal quantum state populations. The off-diagonal entries of the Bell state retain quantum coherence between $00$ and $11$; the classical mixture has none in this basis.

If both parties perform a Pauli measurement of $Z$, the Born rule gives $P(00)=P(11)=1/2$ and $P(01)=P(10)=0$ for both preparations. Perfect agreement here therefore cannot establish entanglement: an ordinary shared random bit already explains it.

Now both measure $X$, whose normalized eigenstates are $|+⟩=(|0⟩+|1⟩)/\sqrt{2}$ and $|-⟩=(|0⟩-|1⟩)/\sqrt{2}$. Changing basis givesThe Bell state again has matching outcomes, each with probability $1/2$. For either product state $|00⟩$ or $|11⟩$, each of the four $X$ outcome pairs has probability $1/4$, so their classical mixture does too. Equivalently, the Pauli correlator $⟨X\otimes X⟩$ is one for the Bell state and zero for this mixture, while $⟨Z\otimes Z⟩$ is one for both.

Taking a partial trace gives ${\rho }_A={\rho }_B=I_2/2$ for both preparations. Thus every measurement confined to Alice's system has the same statistics in the two cases, and so does every measurement confined to Bob's. They reveal the difference by comparing records through classical communication. Conditional on Alice's result, Bob can predict a matching $X$ result for the Bell state; without that result, his outcomes remain evenly split. An uncommunicated local quantum channel, including a measurement with its outcome discarded, leaves his reduced density matrix unchanged. This is quantum no-signalling, even when conditional states change. ^[ref]

The Von Neumann entropy makes the joint-versus-local distinction numerical. With base-two logarithms, the Bell state has eigenvalues $1,0,0,0$, giving $S(AB)=0$, whereas the mixture has $1/2,1/2,0,0$, giving $S(AB)=1$. Both have $S(A)=S(B)=1$. Their quantum mutual information $I(A:B)=S(A)+S(B)-S(AB)$ is consequently two and one bits, respectively. The value two is not two readable classical bits in either local qubit: the matching $Z$ records have ordinary mutual information one bit in both cases. Random records can come from a globally pure state. ^[ref]

These two matching-axis measurements distinguish the specified preparations; they alone are not a test of a Bell inequality. A local hidden-variable theory could assign shared random answers separately to these two axes. Bell theorem concerns incompatible correlations across a suitable larger collection of measurement settings. Bell's 1964 paper established the obstruction for local hidden-variable descriptions of the predicted correlations. ^[ref]

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 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.

Boris Podolsky coauthored the 1935 argument of Albert Einstein, Boris Podolsky and Nathan Rosen about the completeness of the wave function. Their example correlates two distant particles so that a measurement on one can predict either the other's position or its momentum. Combining their certainty-based criterion for an element of reality with the assumption that the remote measurement does not disturb the second particle leads to a tension with a complete wave function description. The argument states assumptions about reality and disturbance; it is distinct from the later experimentally testable Bell inequalities.^[ref]

Entanglement-based key distribution that tests correlations with a Bell inequality. Equal measurement axes generate correlated key data; other settings test the quantum channel. Authentication, statistical parameter estimation, information reconciliation and privacy amplification are needed to turn observed correlations into a secure shared key under specified assumptions.^[ref]

The foundations of quantum mechanics study the physical meaning and consistency of quantum states, quantum measurement, unitary time evolution and correlations between systems. The aim is to connect precise mathematical rules with claims about preparation, observation and physical reality.

The measurement problem can be seen in a simple model. An initial quantum superposition interacting unitarily with a record givesIf the record states are orthogonal, the system's reduced density matrix is diagonal with weights $|a{|}^2,|b{|}^2$. This explains suppression of observable interference, while the relation between the joint state and an individual recorded outcome remains a foundational question. Quantum decoherence, the many-worlds interpretation and de Broglie-Bohm theory address different parts of this problem.

John Stewart Bell made a separate question experimentally sharp. In his singlet-pair model, local hidden-variable response functions cannot reproduce all the quantum spin correlations. A Bell test compares measured correlations across analyzer settings with the appropriate Bell inequality. ^[ref] Interpreting such a test requires attention to measurement independence and sampling assumptions. Quantum nonlocality concerns these joint probabilities and is compatible with quantum no-signalling.

John Clauser and Stuart Freedman tested local hidden-variable theories with photons from a calcium atomic cascade. They measured joint detections behind differently oriented polarizers and compared a combination of those rates with a Bell inequality. The observed violation agreed with quantum mechanics, turning assumptions about distant outcomes into a quantitative laboratory test. The experiment used additional assumptions about detection and the sampled photon pairs; it did not close every later-identified experimental loophole.^[ref]

A physicist who performed experimental tests of Bell inequalities.^[ref]

Lucien Hardy gave an argument against local hidden-variable theories using logical exclusions among two-particle measurement outcomes, rather than a statistical Bell inequality. The structure combines outcomes forbidden by quantum mechanics with another outcome that remains possible: assigning all the relevant local results in advance cannot satisfy these requirements together. His 1992 paper also examined an additional assumption about Lorentz-invariant elements of reality, keeping the role of that assumption explicit in its comparison with quantum mechanics.^[ref]

Homepage: perimeterinstitute.ca/people/lucien-hardy

Alain Aspect, John F. Clauser and Anton Zeilinger shared the award equally for experiments with entangled photons, violations of Bell inequalities and pioneering quantum information science.^[ref]

For quantum key distribution with Bell pairs, a source distributes one photon to Alice and its partner to Bob. One implementation is the Ekert protocol, using the spin singlet state $|{\psi }^{-}⟩=(|01⟩-|10⟩)/\sqrt{2}$. Each party independently chooses a polarization measurement setting after distribution. Equal settings give opposite outcomes; Bob reverses his outcome-to-bit assignment so these rounds give matching random bits. Other setting pairs test a Bell inequality, with ideal spin singlet state correlation $E(\mathbf{a},\mathbf{b})=-\mathbf{a}\cdot \mathbf{b}$ for the corresponding Bloch sphere axes. The optical polarizer angles correspond to twice those angles on the appropriate Bloch sphere great circle.

Over an authenticated public classical channel, Alice and Bob announce settings and detection times, select coincident detection rounds, and perform key sifting. They reveal a test sample to estimate the quantum bit error rate and the chosen Bell inequality signal, discarding all revealed bits. Excess disturbance causes an abort. Otherwise information reconciliation makes the retained strings agree and privacy amplification shortens them to remove an adversary's residual information, including information disclosed during reconciliation. The tested quantum entanglement supplies correlations unavailable to an independently prepared classical mixture; a security claim also requires stated detector and sampling assumptions.^[ref]

Each photon polarization analyzer uses a setting-dependent waveplate followed by a polarizing beam splitter. A single-photon detector on each output converts the routed photon into an electrical click, recording the outcome and its time. For example, a single-photon avalanche diode is a reverse-biased photodiode operated above breakdown: absorption creates a charge carrier, impact ionization triggers an avalanche, and the resulting pulse is registered. A quenching circuit lowers the bias to stop the avalanche and then resets the device. The detector dead time during recovery limits the count rate; dark counts produce false clicks, and imperfect detection loses rounds. The polarization information comes from the preceding analyzer, while the detector registers which output fired. Coincidence timing associates Alice's and Bob's records with the same emitted pair. ^[ref]

Encode $|0⟩=|H⟩$ and $|1⟩=|V⟩$ in two orthogonal linear polarizations. Then $|\pm ⟩=(|H⟩\pm |V⟩)/\sqrt{2}$ are linear polarizations at opposite forty-five-degree angles, while $(|H⟩\pm i|V⟩)/\sqrt{2}$ are the two circular polarizations. Their right/left names depend on the viewing convention. A half-wave plate with its axis at $22.5^{\circ }$ to H implements H up to a common phase, since its Jones matrix has entries $\cos 2\theta ,\sin 2\theta$ and $-\cos 2\theta$.

In principle, a nonlinear interaction producing $e^{i\pi n_{1V}{n}_{2V}}$ supplies a deterministic Controlled-Z gate: only the joint VV component acquires a minus sign. A strongly coupled matter mediator can also provide the required conditional interaction. At optical single-photon energies the direct nonlinear interaction is weak; loss, unwanted noise and maintaining indistinguishable spatial and temporal modes make a high-fidelity deterministic gate difficult. Passive linear optics by itself does not supply this deterministic entangling interaction; measurement-based optical gates introduce additional resources and success conditions.

For the printed circuit, the first three H gates create $|+{⟩}^{\otimes 3}$. The two controlled-Z operations from the first qubit to the others giveThe final H gates on the last two qubits therefore produce the GHZ stateDefine $|s_X⟩=(|0⟩+s|1⟩)/\sqrt{2}$ and $|s_Y⟩=(|0⟩+is|1⟩)/\sqrt{2}$ for $s=\pm 1$. In any of the XYY, YXY and YYX bases, direct overlaps givewith the same coefficient formula when the X position is permuted. Hence the three explicit expansions can be written compactly asIn each sum the four sign triples are $(+,+,-),(+,-,+),(-,+,+),(-,-,-)$. They have equal probabilities $1/4$. Thus two Y outcomes determine the X outcome with certainty: their product times the X outcome is $-1$.

In the XXX basis,The four allowed triples are $(+,+,+),(+,-,-),(-,+,-),(-,-,+)$, so the XXX outcome product is $+1$.

The GHZ theorem follows without a statistical inequality. If local predetermined values $X_i,Y_i\in \{\pm 1\}$ reproduced the three mixed-basis correlations, multiplying them would givebecause every Y value is squared. Quantum mechanics instead predicts the XXX product $+1$. This is an all-versus-nothing contradiction with local realism using ideal perfect correlations. Standard two-qubit Bell-state tests compare statistical correlations through a Bell inequality; the GHZ construction makes the inconsistency visible already at the level of these deterministic parity constraints.

Taking the partial trace over either photon of the pure pair state givesThe two Schmidt coefficients are both $1/\sqrt{2}$, so the entanglement entropy is one bit for every $\phi$. A local phase gate on Alice maps $|{\Psi }^0⟩$ to $|{\Psi }^{\phi }⟩$, proving local-unitary equivalence. In strict four-vector Bell-basis terminology only $\phi =0,\pi$ give the standard $|{\Psi }^{+}⟩,|{\Psi }^{-}⟩$; every phase nevertheless gives a maximally entangled Bell-type state.

The eigenpolarizations of the requested observable areA polarizing beam splitter with its transmitted polarization axis at angle $\alpha /2$ to horizontal separates these two orthogonal states. Assign outcome $+1$ to its first detector and $-1$ to the other. Equivalently, use a fixed horizontal/vertical splitter preceded by a half-wave plate whose fast axis is at $\alpha /4$; the diagram shows this equivalent implementation. Bob uses $\beta /2$ for his splitter axis, or $\beta /4$ for the preceding half-wave plate.

The pair correlations are $⟨Z\otimes Z⟩=-1$, $⟨X\otimes X⟩=\cos \phi$, and $⟨Z\otimes X⟩=⟨X\otimes Z⟩=0$. Expanding the two observables therefore givesThe half-angle between the Bloch observable parameter and the physical linear-polarization analyser angle is essential.

In a local hidden-variable description, assign predetermined values $X_j,Y_j\in \{+1,-1\}$ to the two possible measurements at photon $j$. The perfect correlations of b requireMultiplying these identities cancels each squared X value and gives $\boxed{Y_1{Y}_2{Y}_3=-1}$. Thus local realism predicts only the YYY outcomes $++-,$ $+-+,$ $-++,$ and $---$. Combining this parity rule with the measured unbiased single-Y marginals forces all four frequencies to be $1/4$: four weights subject to normalization and the three zero-mean conditions are uniquely uniform.

Quantum mechanics instead has $k=3$ in the amplitude formula of b, giving ${\mathcal{A}}_3=(1+abc)/4$. ThereforeQuantum mechanics predicts each positive-product outcome with probability $1/4$ and every negative-product outcome with probability zero. The local and quantum allowed sets are disjoint.

This is the GHZ theorem: in the ideal case the contradiction follows from deterministic product constraints, an all-versus-nothing argument. A usual two-party Bell-state test instead violates a statistical Bell inequality assembled from several settings. Both require appropriate locality and measurement-choice assumptions and careful treatment of experimental imperfections; the distinction is the deterministic form of the ideal three-party argument.

Each source has density matrixAssume the two sources and the detector's random choices are independent. In the $p^2$ branch both resources are ideal. If either resource is $I_4/4$, the remote pair remains $I_4/4$ after any specified Bell-basis measurement outcome. An ideal detector and the correction of part (b) therefore yieldWhen the detector instead reports an independent uniformly random label, the corresponding four Pauli gates average the conditional pair to $I_4/4$. ThusHere $q$ is the probability of the reliable-detector branch, not its total chance of being correct: that chance is $q+(1-q)/4$.

For the usual two-setting CHSH inequality, the spin singlet state has correlation $-\mathbf{a}\cdot \mathbf{b}$ and the white-noise part has zero correlation, so $E(\mathbf{a},\mathbf{b})=-v\mathbf{a}\cdot \mathbf{b}$. With Alice's axes $z,x$ and Bob's axes $(z+x)/\sqrt{2},(z-x)/\sqrt{2}$, the absolute CHSH inequality combination is $2\sqrt{2}v$. These settings are optimal for an isotropic correlation tensor, givingThis is the threshold for the standard CHSH inequality with optimized projective polarization measurements. The unqualified phrase “the Bell inequality” needs this choice of test; entanglement alone is a weaker requirement, and no claim about every possible Bell inequality is needed for this calculation.

Consider two spacelike-separated measurement regions $R_A$ and $R_B$, with locally chosen settings $a,b\in \{0,1\}$ and recorded outcomes $A,B\in \{-1,1\}$. A relevant past region supplies a sufficient specification $\lambda$ of the common preparation and other hidden physical information needed to screen the outcomes. This is not necessarily just a quantum state written at the source, nor a conditioning on arbitrary records which already contain the future choices. The spacetime shielding and sufficiency requirements are substantive causal assumptions.

Use Bell local causality: once the relevant past is sufficiently specified, learning facts in the remote measurement region changes no local outcome probability. In particular, requirewhere conditional probabilities are defined. The probability chain rule givesThis factorization combines parameter independence with outcome independence. Neither probability is required to be zero or one: the argument includes stochastic local hidden-variable theories.

Also assume measurement independence, $\rho (\lambda \mid a,b)=\rho (\lambda )$, so all four setting pairs average over one normalized nonnegative past-variable distribution. This is a statistical condition on the setting procedure and relevant hidden information, not a proof of a metaphysical doctrine of free will. Finally the recorded outcomes must represent the same ensemble of trials. One can include nondetections in a specified bounded-outcome rule; if instead one discards events depending on setting or outcome, a further sampling assumption is required. Setting-dependent postselection is not covered silently by the argument.

Let ${\alpha }_a(\lambda )={\sum }_{A}AP(A\mid a,\lambda )$ and ${\beta }_b(\lambda )={\sum }_{B}BP(B\mid b,\lambda )$. They lie in $[-1,1]$, and the factorization givesFor each $\lambda$,Averaging with the same $\rho$ proves the CHSH inequalityNo separate assumption of simultaneous predetermined values for incompatible quantum observables was needed. The common probability model and its screening relations did the work.

A Reichenbach common cause principle can motivate the same calculation: a complete classical common cause in the past screens the two outcomes, while each setting affects only its own wing. Together with setting-independent sampling this gives the displayed factorization. It is not enough to provide a different screening variable for each pair of settings: those constructions need not form one compatible common-cause model for the four correlations.

A stochastic Einstein locality formulation instead constrains objective chances by facts in the appropriate causal past. To use it as a sufficient premise here, specify both local dependence on settings and a joint screening condition for the two outcomes. Merely requiring local marginal chances not to depend on a remote setting gives parameter independence, which is weaker than factorization. Thus the precise stochastic locality version matters; one should not infer a Bell inequality from every condition loosely described as no superluminal influence.

spontaneous parametric down-conversion uses a second-order nonlinear polarization to convert pump energy into correlated signal and idler photons. With a classical undepleted pump, a single-mode interaction-picture model isThe creation term acts on the signal/idler vacuum, so no classical signal seed is required. The pump supplies the emitted energy; vacuum fluctuations are not an extra energy source. In the small-gain limit the SPDC pair state iswith normalization corrections understood at second order. More generally it is a two-mode squeezed vacuum with correlated pair numbers. A classical calculation with both generated fields exactly zero would miss the spontaneous start.

Efficient generation requires ${\omega }_p={\omega }_s+{\omega }_i$ and approximate wave-vector matching ${\mathbf{k}}_p={\mathbf{k}}_s+{\mathbf{k}}_i$. birefringence or a reciprocal grating vector in quasi-phase matching can compensate dispersion. In a continuum-mode treatment the pair amplitude includesThe pump spectrum constrains the sum frequency, while phase matching constrains the allowed frequency combinations and emission directions. type-I phase matching produces daughters in the same polarization class; type-II phase matching produces orthogonal ordinary/extraordinary classes. Correlation of pair numbers or orthogonal polarizations does not alone guarantee a polarization-entangled state in two chosen output paths.

One application is a heralded single-photon source: detect one partner and condition measurements on the other. Low gain suppresses multiple-pair contamination, whose probability begins at fourth order in the gain. Efficient collection and low detector noise matter. Spectral entanglement with an unresolved herald can leave the partner mixed; engineering or filtering the joint spectrum improves purity at a possible cost in brightness. Heralding does not make a photon appear in every pump pulse.

A second application is entangled-pair generation for quantum communication and tests of Bell inequalities. If the assignments $H_s{V}_i$ and $V_s{H}_i$ are indistinguishable except for polarization, their amplitudes can form $(|H_s{V}_i⟩+e^{i\phi }|V_s{H}_i⟩)/\sqrt{2}$. Walk-off, group delay or frequency labels can reveal which assignment occurred and destroy polarization interference. Their compensation is part of preparing the state, not merely an improvement in count rate.

A primary example pumped a 3-mm beta-barium-borate crystal with a $351.1\mathrm{nm}$ argon-ion laser. Collecting the two intersections of its type-II emission cones selected paired output directions. Half-wave rotation and compensating birefringent crystals reduced temporal walk-off, and polarization analysers with avalanche photodiodes measured coincidence fringes. Varying analyser settings tested the correlations; accidentals, mode distinguishability and detector/collection assumptions limited interpretation. The reported visibility exceeded 97 percent and supported a Bell violation under the stated auxiliary assumptions, rather than a loophole-free claim.^[ref]

Other uses include preparing interfering photons for quantum-information experiments, correlation-based imaging, and pair-based detector calibration. All rely on specified spatial and spectral modes and coincidence statistics. A pump-blocked background run, delayed-coincidence estimate, gain dependence and polarization/temporal-overlap scans distinguish correlated pairs from fluorescence, detector dark counts and accidental coincidences. The useful resource is the controlled joint quantum state, not simply two bright output beams.