Experiment: Bell Tests
Bell's theorem [Bell:1964] converted a philosophical dispute about the completeness of quantum mechanics [Einstein:1935] into an inequality between measurable correlation rates, and thereby into an experiment. This chapter is the record of that experiment: the cascade-photon tests of the 1970s [Freedman:1972] [Aspect:1981], the switched-analyzer measurement that first enforced relativistic separation between the choice of setting at one station and the detection at the other [Aspect:1982b], the two experiments that closed the locality and detection loopholes one at a time [Weihs:1998] [Rowe:2001], and the three independent 2015 experiments that closed them simultaneously [Hensen:2015] [Giustina:2015] [Shalm:2015]. The theory that the data test — the derivation of the inequality, the meaning of a violation, and what is and is not entailed — is developed in Entanglement and Bell Tests; this chapter supplies the apparatus, the numbers and the error budgets, and states the residual assumptions without softening them.
When written the chapter will use the structured experiment environment, with apparatus, procedure, observations, interpretation and references as separate fields, and the tabulated CHSH parameters with their uncertainties lodged as a dataset in the evidence store. Three features distinguish a Bell test from every other experiment in this treatise and shape the sections below. Its result is a bound rather than a value, so the figure of merit is a \(p\)-value under an explicitly stated null hypothesis, not an agreement with a predicted number. Its validity rests on the spacetime layout of the apparatus, so the timing diagram is part of the physics and not part of the engineering. And one assumption — that the setting choices are not themselves correlated with the hidden variables — cannot be removed by any experiment, only pushed back in time, which is what the cosmic Bell tests do [Handsteiner:2017] [Rauch:2018]. The 2022 Nobel Prize recognized this line of work [KVA:2022]; the reviews are [Brunner:2014] [Larsson:2014].
Experiment: Bell Tests: all derivations of this chapter are pending.
Historical context and the prediction under test
From EPR to a testable inequality
[Reserved: the Einstein–Podolsky–Rosen argument that quantum mechanics is either incomplete or non-local [Einstein:1935]; Bohm's reformulation with two spin-half particles in a singlet, which replaced continuous variables by a two-outcome measurement and made the problem experimental in principle [Bohm:1951]; Bell's derivation of an inequality obeyed by every local hidden-variable model and violated by quantum mechanics [Bell:1964], with his own later commentary on what the assumptions mean [Bell:1987]; the reduction of the whole dispute to a counting experiment.]
When the two analysers of a Bell apparatus are set to the same direction, the two recorded outcomes are found to be opposite on every detected pair, up to the imperfections of source and detectors, while each outcome taken alone is a fair coin. This is the observation on which the Einstein–Podolsky–Rosen argument rests [Einstein:1935]: the result at one station can be predicted with certainty from the result at the other, at any separation, without the second particle having been touched. It is also the observation a Bell test must reproduce as a calibration before its inequality means anything, and it is measured as a matter of routine in every experiment of this chapter.
Derivation. Write the singlet \(\ket{\Psi^{-}}=\bigl(\ket{\uparrow\downarrow} -\ket{\downarrow\uparrow}\bigr)/\sqrt{2}\) in the eigenbasis of the spin component along the common analyser direction \(\hat{\vect{a}}\). Both surviving terms have the two spins opposite and no term with them parallel appears at all, so a joint measurement of that component returns \((+,-)\) or \((-,+)\) and never \((+,+)\) or \((-,-)\): the outcomes are opposite on every pair, and \(E(\hat{\vect{a}},\hat{\vect{a}})=-1\). The two terms carry equal amplitude, so each occurs with probability \(\tfrac{1}{2}\) and the marginal recorded at either station is uniform, carrying no information about which direction was chosen. Finally, the singlet is the unique two-spin state of zero total angular momentum and is therefore invariant under simultaneous rotations of both spins, so the same expansion holds in the eigenbasis of any direction: the anticorrelation is not a property of a preferred axis.
Note what this does not establish. Nothing in the argument is inconsistent with local hidden variables — a pair of particles carrying opposite predetermined values reproduces every statement above exactly. That is why the settings in Section 89.1.2 must be unequal, and why the dispute could not be decided until Bell found an inequality that unequal settings violate.
∎CHSH: the inequality that is measured
[Reserved: the Clauser–Horne–Shimony–Holt inequality \(\abs{S}\leq2\), built from four correlation measurements at two settings per side and requiring no perfect anticorrelation [Clauser:1969]; the Clauser–Horne form, which uses only singles and coincidence rates and so needs no auxiliary no-enhancement assumption [Clauser:1974b]; Eberhard's inequality, optimized for low detection efficiency and used in the 2015 photonic experiments [Eberhard:1993]; the quantum maximum \(S=2\sqrt{2}\approx2.828\) and Tsirelson's proof that no quantum state exceeds it [Cirelson:1980], which fixes the scale against which every measured \(S\) is read.]
The three loopholes
[Reserved: the detection or fair-sampling loophole, exhibited constructively by Pearle, who built a local model reproducing the quantum correlations when undetected pairs are discarded [Pearle:1970], with the efficiency thresholds — about \(83\,\mathrm{\%}\) for CHSH with a maximally entangled state, about \(67\,\mathrm{\%}\) for the Eberhard inequality with a partially entangled one [Eberhard:1993]; the locality loophole, closed only when the setting choice at one station is spacelike separated from the detection at the other; the freedom-of-choice loophole, and the honest statement that superdeterminism is not testable and that the experiments therefore constrain, and never eliminate, it [Larsson:2014].]
The first tests
[Reserved: Kocher and Commins's measurement of the polarization correlation of the calcium cascade photons, made for other reasons and later recognized as the first relevant data [Kocher:1967]; Freedman and Clauser's dedicated test [Freedman:1972], which measured \(\delta=0.050\pm0.008\) against the local bound \(\delta\leq0\) — the first reported violation; the conflicting null result of Holt and Pipkin and its resolution, together with Clauser's own re-measurement that traced the anomaly to the source [Clauser:1976] and Fry and Thompson's fast confirmation with an isotopically selected mercury cascade [Fry:1976]. The episode is kept because a disagreement resolved by finding a systematic is better evidence than unanimity.]
Apparatus
Atomic cascade sources
[Reserved: the two-photon radiative cascade in calcium \(4p^{2}\,^{1}S_{0}\to4s4p\,^{1}P_{1}\to4s^{2}\,^{1}S_{0}\), whose angular-momentum selection rules correlate the polarizations of the two emitted photons; the resonant two-photon laser excitation that raised the rate by orders of magnitude in Aspect's apparatus [Aspect:1981]; the intrinsic limitation of the geometry — the correlation is degraded by the finite solid angle, since the photons are not emitted back to back — and why this made the detection loophole unclosable with cascade sources.]
Parametric down-conversion
[Reserved: spontaneous parametric down-conversion in a noncentrosymmetric crystal, in which one pump photon splits into a correlated pair; the type-II geometry with its two intersecting cones, which produces polarization-entangled pairs directly and brightly [Kwiat:1995]; phase matching, spectral filtering, and the collection optics that determine the heralding efficiency — the single parameter on which the detection loophole turns; the source used in Section 89.6.1 and, with far higher collection efficiency, in the 2015 experiments.]
Detectors
[Reserved: photomultipliers with quantum efficiencies of a few percent in the cascade era, which is why every experiment before 2001 assumed fair sampling; avalanche photodiodes; transition-edge sensors operating at millikelvin temperatures, which reach detection efficiencies above \(90\,\mathrm{\%}\) in the near infrared [Lita:2008] and are what made the photonic loophole-free experiments possible; state readout by resonance fluorescence in trapped ions and nitrogen–vacancy centres, which is essentially deterministic and closes the detection loophole by construction [Rowe:2001] [Hensen:2015].]
Entangled matter systems
[Reserved: entanglement between two trapped ions in the same trap [Rowe:2001]; entanglement swapping through a Bell-state measurement on photons emitted by remote emitters, the event-ready scheme proposed by Żukowski and collaborators [Zukowski:1993] and used to entangle nitrogen–vacancy centres \(1.3\,\mathrm{km}\) apart [Hensen:2015]; entangled atoms in separate laboratories [Rosenfeld:2017]; superconducting circuits entangled through a cryogenic waveguide over \(30\,\mathrm{m}\) [Storz:2023], where the whole spacelike-separation budget is metres rather than kilometres because the readout is nanoseconds rather than microseconds.]
Setting generators
[Reserved: the progression of setting choices — fixed orientations in the earliest tests; the acousto-optic switches of Aspect, Dalibard and Roger, which redirected the light periodically rather than randomly [Aspect:1982b]; physical random-number generators driven by a beam splitter [Weihs:1998]; photons from Milky Way stars, whose emission moved the last possible common cause back centuries [Handsteiner:2017]; high-redshift quasars, moving it back billions of years [Rauch:2018]; and the choices of about one hundred thousand human participants [BIGBellTest:2018]. Each is a statement about when the setting was fixed, which is the only thing that matters.]
Procedure
[Reserved: the spacetime layout, drawn as a diagram with the two stations' light cones — setting choice, analyzer setting completed, detection, and the requirement that each station's setting event be spacelike separated from the other's detection event; the coincidence window and the pitfalls of defining coincidences by post-selection on arrival times; the run structure, with the number of trials fixed in advance; the pre-registered analysis and the martingale-based hypothesis test that avoids assuming independent and identically distributed trials; how the \(p\)-value is computed and why it, and not a number of standard deviations, is the honest statistic [Giustina:2015] [Shalm:2015].]
Observations
[Reserved: the quantities to be tabulated for each experiment, in SI with uncertainties per [JCGM:2008] — source type, station separation, detection and heralding efficiencies, trial count, the measured CHSH parameter and its uncertainty, and the \(p\)-value under local realism. The series to be given: Freedman and Clauser's \(\delta=0.050\pm0.008\) [Freedman:1972]; Aspect's \(S=2.697\pm0.015\) with two-channel polarizers [Aspect:1982a] and \(S=2.404\pm0.080\) with time-varying analyzers [Aspect:1982b]; Weihs's \(S=2.73\pm0.02\) under strict Einstein locality [Weihs:1998]; Rowe's \(S=2.25\pm0.03\) with unit detection efficiency [Rowe:2001]; the 2015 results, \(S=2.42\pm0.20\) with \(p=0.039\) over \(1.3\,\mathrm{km}\) [Hensen:2015], \(p\leq3.74\times10^{-31}\) [Giustina:2015] and \(p\leq2.3\times10^{-7}\) [Shalm:2015]. Every entry is to be read against the quantum bound \(2\sqrt{2}\) of Section 89.1.2.]
Coincidence counting on entangled pairs, with two alternative analyser settings at each station, yields values of
exceeding \(2\) in modulus — which no local hidden-variable model can produce — and consistent with the quantum prediction, whose maximum is \(2\sqrt{2}\). The result has been obtained with atomic cascade photons [Freedman:1972] [Aspect:1982a], with parametric down-conversion [Weihs:1998] [Giustina:2015] [Shalm:2015], with trapped ions [Rowe:2001], with nitrogen–vacancy centres [Hensen:2015] and with superconducting circuits [Storz:2023]; at station separations from micrometres to kilometres; and with settings fixed by hand, by periodic switches, by physical random-number generators, by starlight, by quasar light and by human choices. The excess over \(2\) is therefore not one number from one apparatus but a stable feature of Nature reproduced on every platform on which it has been sought.
Derivation. Both numbers being compared are short to obtain. For the local bound, suppose the outcomes are functions \(A(a,\lambda),A(a',\lambda),B(b,\lambda),B(b',\lambda)\in\{-1,+1\}\) of the setting on their own side and of a shared variable \(\lambda\) with density \(p(\lambda)\), so that \(E(a,b)=\int\dd\lambda\,p(\lambda)A(a,\lambda)B(b,\lambda)\). At fixed \(\lambda\), group the four terms as \(S(\lambda)=A[B+B']+A'[B-B']\). Since \(B,B'\in\{-1,+1\}\), one bracket vanishes and the other has modulus \(2\), so \(\abs{S(\lambda)}=2\) for every \(\lambda\), and averaging gives \(\abs{S}\leq2\). No quantum mechanics is used, which is what gives the bound its force; the same argument is set out at greater length in Entanglement and Bell Tests.
For the quantum value, take the singlet, whose correlation between analyser directions at angle \(\theta\) is \(-\cos\theta\), and choose coplanar directions \(a=0\), \(a'=\pi/2\), \(b=\pi/4\), \(b'=-\pi/4\). Then \(E(a,b)=E(a,b')=E(a',b)=-1/\sqrt{2}\) while \(E(a',b')=-\cos(3\pi/4)=+1/\sqrt{2}\), so Equation (89.1) gives \(S=-4/\sqrt{2}=-2\sqrt{2}\). The two statements are the whole experiment: the apparatus is asked to produce a number, and the number it produces must fall on one side or the other of \(2\). Because the predicted excess is a factor \(\sqrt{2}\) rather than a marginal effect, the question was answerable with the detectors of the 1970s once fair sampling was granted, and the fifty years since have gone into removing that grant rather than into improving the statistics.
∎Interpretation
What the violations exclude
[Reserved: the exact statement — no theory in which each outcome is determined by variables carried by the particles and unaffected by the distant setting can reproduce the observed rates; that this is a constraint on explanations, not the observation of a signal, since the marginal distributions are uniform and no information is transmitted; the no-signalling property checked directly in the data as a control; and the standing warning that “non-locality” names the failure of an assumption, not a mechanism [Brunner:2014].]
In every Bell experiment the singles rate recorded at one station is compatible, within its statistical uncertainty, with being independent of the setting chosen at the other. The comparison is made directly on the recorded data and reported as a control, so this is an experimental result and not only a theorem. Consequently a violation of Equation (89.1) is not the observation of a signal: nothing detectable at one station changes when the distant setting changes, and the correlation becomes visible only once the two records are brought together, which needs a classical channel and therefore takes at least the light travel time between the stations. The word “non-locality” in this subject names the failure of an assumption, not the discovery of an influence.
Derivation. Everything one station can record is computed from its reduced state \(\rho_{A}=\tr_{B}\rho\). The most general thing the distant station can do to its own factor is a trace-preserving map \(\mathcal{E}_{B}\), of which “set the analyser to \(b\) and measure, without telling anyone the result” is one instance. Since \(\tr_{B}\bigl[(\mathrm{id}_{A}\otimes\mathcal{E}_{B})\rho\bigr] =\tr_{B}\rho\) whenever \(\mathcal{E}_{B}\) preserves the trace, the reduced state \(\rho_{A}\) is unchanged, and with it
for every local observable \(X\). Only linearity and trace preservation were used, so the conclusion holds for any state and any separation. The reason the check is nonetheless performed on the data is that the apparatus might signal even though the physics does not — through cross-talk between the two arms, a shared clock, or a setting generator that drifts in step with a detector. A measured dependence of the marginals on the distant setting would therefore indicate a fault, and finding none is a statement about the experiment.
∎What remains open
[Reserved: the alternatives that survive — explicitly nonlocal hidden-variable theories, retrocausal accounts, and superdeterministic models in which the setting choices are correlated with the state at preparation; the honest observation that the cosmic Bell tests constrain the last of these only by pushing the required conspiracy back in time [Handsteiner:2017] [Rauch:2018], and that no experiment can close it. The interpretive consequences belong to Interpretations (Evidence-Anchored); what this chapter contributes there is the datum that any interpretation must accommodate.]
Why a violation is not fragile
[Reserved: the quantitative role of decoherence (Open Quantum Systems and Decoherence) — entanglement degrades with photon loss, mode mismatch and dephasing, so a measured \(S\) below the Tsirelson bound is an apparatus statement rather than a physics statement; the visibility threshold \(1/\sqrt{2}\) below which CHSH cannot be violated at all; why every reported violation is therefore also a bound on the imperfection of the source and detectors, and how the two are separated in the error budget.]
The measured \(S\) falls below \(2\sqrt{2}\) in every real apparatus, and it falls monotonically with the interference visibility of the source. No violation at all is seen once that visibility drops below \(1/\sqrt{2}\), about \(71\,\mathrm{\%}\). Photon loss, mode mismatch, dark counts, dephasing and drift all act through this one parameter, and a reported \(S\) is therefore two statements at once: one about Nature and one about the apparatus. Separating them is what the error budget of a Bell test is for, and it is the reason a value of \(S\) short of \(2\sqrt{2}\) is a statement about the equipment rather than a discrepancy with the theory.
Derivation. Model the imperfection by mixing the intended singlet with unpolarized noise,
where \(V\) is the visibility read off the correlation fringe. The maximally mixed part contributes nothing to any correlator, since \(\tr\bigl[\identity\,(\vect{\sigma}\cdot\hat{\vect{a}})\otimes (\vect{\sigma}\cdot\hat{\vect{b}})\bigr] =\tr(\vect{\sigma}\cdot\hat{\vect{a}})\, \tr(\vect{\sigma}\cdot\hat{\vect{b}})=0\), so every correlator is simply scaled: \(E(\hat{\vect{a}},\hat{\vect{b}})=-V\cos\theta\). Substituting into Equation (89.1) at the optimal angles multiplies the ideal value by \(V\),
so \(\abs{S}>2\) demands \(V>1/\sqrt{2}\approx0.707\). Below that visibility no choice of settings violates the inequality, however many pairs are counted — the deficiency cannot be bought off with statistics — and for \(V\leq\tfrac{1}{2}\) there is in addition an explicit local hidden-variable model reproducing every projective measurement on Equation (89.3), so the state is then provably useless for the purpose. Above the threshold the number of pairs required grows only as the inverse square of the excess of \(S\) over \(2\), which is why the loophole-free experiments could afford to trade brightness for efficiency.
∎The 2022 Nobel citation
[Reserved: the award to Clauser, Aspect and Zeilinger for experiments with entangled photons establishing the violation of Bell inequalities and pioneering quantum information science [KVA:2022]; the scientific background document as a compact history of the fifty years covered by this chapter; the observation, worth stating, that the prize was given for closing loopholes rather than for a discovery of a new effect — the discipline of eliminating alternatives is itself the achievement.]
Modern repetitions and precision
Closing the locality loophole
[Reserved: the Innsbruck experiment, with independent quantum random-number generators, fast electro-optic modulators and a \(400\,\mathrm{m}\) fibre separation, in which every setting choice was spacelike separated from the distant detection [Weihs:1998]; the timing budget that makes the claim, given as numbers; the extension to \(144\,\mathrm{km}\) between the Canary Islands, which also addressed the freedom-of-choice loophole [Scheidl:2010].]
Closing the detection loophole
[Reserved: the trapped-ion experiment in which every prepared pair was detected, so that no fair-sampling assumption was needed [Rowe:2001] — at the cost of a separation of a few micrometres, which leaves the locality loophole wide open; why the two loopholes pulled the design in opposite directions for fourteen years, and what had to improve in detector efficiency [Lita:2008] and in entanglement distribution before both could be closed at once.]
The loophole-free experiments
[Reserved: the three independent 2015 experiments — Delft, with nitrogen–vacancy centres \(1.3\,\mathrm{km}\) apart and event-ready entanglement swapping, 245 trials and \(p=0.039\) [Hensen:2015]; Vienna, with a high-efficiency photonic source and the Eberhard inequality [Giustina:2015]; and Boulder, with transition-edge sensors and a fully pre-registered analysis [Shalm:2015]; the differences in their residual assumptions, and why three experiments with different systematics are worth more than any one of them repeated.]
In 2015 three independent experiments [Hensen:2015] [Giustina:2015] [Shalm:2015] observed violations of Equation (89.1) while enforcing together the two conditions that had previously been met only one at a time: each station's setting was chosen and its measurement completed in a region spacelike separated from the distant station's detection event, and essentially every prepared pair yielded a recorded outcome, so that no assumption of fair sampling was needed. The three used different physical systems — nitrogen–vacancy centres on a kilometre-scale baseline with event-ready entanglement swapping at Delft, and two high-efficiency photonic sources with pre-registered analyses at Vienna and at Boulder — and therefore different systematics, so that no single misunderstood effect can account for all three. Their combined conclusion is a statement about a class of theories, and it is the strongest statement the data support: no local hidden-variable theory reproduces the observed rates. The one assumption that remains, that the setting choices are not themselves correlated with the pair at preparation, cannot be removed by any experiment and can only be pushed back in time (Section 89.6.4).
The detection-efficiency threshold: the least heralding efficiency at which the observed coincidence and singles rates cannot be reproduced by any local model free to decide which pairs go undetected, worked out both for the CHSH inequality with a maximally entangled state and for the Eberhard inequality with a partially entangled one, the latter threshold being the lower and being the reason the photonic experiments deliberately used a non-maximally entangled state
Pushing back the setting choice
[Reserved: cosmic Bell tests using the colour of photons from Milky Way stars [Handsteiner:2017] and from quasars at redshifts above 3 [Rauch:2018] to choose the settings; the BIG Bell Test, in which about one hundred thousand people supplied the setting bits for thirteen experiments on five continents in a single day [BIGBellTest:2018]; what each achieves precisely — an upper bound on the time available for a common cause — and what none of them can achieve.]
Other platforms and longer baselines
[Reserved: atoms in separate buildings [Rosenfeld:2017]; superconducting circuits in a cryogenic link [Storz:2023], the first loophole-free test with massive solid-state systems; satellite-distributed entanglement over \(1200\,\mathrm{km}\), which extends the correlations to space without closing the loopholes [Yin:2017]; multipartite tests, where the Greenberger–Horne–Zeilinger argument turns the statistical contradiction into an all-or-nothing one [Greenberger:1989] and its three-photon realization confirms the prediction [Pan:2000].]
From test to instrument
[Reserved: the inversion of the logic, in which a measured violation certifies rather than tests — Ekert's entanglement-based key distribution [Ekert:1991]; device-independent randomness generation, certified by the impossibility of superluminal signalling [Bierhorst:2018]; the first device-independent quantum key distribution between two trapped-ion nodes [Nadlinger:2022]; and the point that a violation used as an instrument must be loophole-free for exactly the reasons catalogued in Section 89.1.3.]