Bell Inequalities and Bell Tests

Take the quantum-mechanical prediction for a pair of polarization-entangled photons — the singlet correlation $E(a,b)=-\cos(2(a-b))$ that field quantization and the two-photon state hand down — and ask a sharper question than Einstein, Podolsky and Rosen dared in 1935. EPR argued that because you can predict a distant photon's polarization with certainty, that polarization must be an "element of reality" fixed in advance, and so quantum mechanics must be an incomplete description of a deeper, locally real world. For thirty years this looked like philosophy: two stories, the same predictions, no way to choose. John Bell's 1964 insight was that the two stories do not make the same predictions. Any theory in which the outcomes are set by shared local variables $\lambda$ carried from the source — and in which a setting on one side cannot touch the outcome on the other — obeys a quantitative ceiling on how strongly two measurement streams can be correlated. Entangled photons climb straight through that ceiling. The workhorse is the CHSH inequality: from four correlation functions $E(a,b)$ , built with two analyzer settings per side, any local hidden-variable model must satisfy $|S|\le 2$ . Quantum mechanics with the singlet reaches $S=2\sqrt2\approx2.828$ — Tsirelson's bound — and the gap between $2$ and $2\sqrt2$ is exactly what a Bell test weighs. In optics the pairs come from atomic cascades or, dominantly, from spontaneous parametric down-conversion; each photon meets a polarizer at a chosen angle and coincidence counts furnish the correlations. A measured $S>2$ , with the locality and detection loopholes shut, means no local realistic account of nature — however baroque its hidden variables — reproduces the data.

Key takeaways

A local hidden-variable theory fixes each outcome in advance through a shared variable $\lambda$ and forbids one side's setting from touching the other's outcome; every such theory obeys the CHSH bound $|S|\le 2$ .
For the entangled singlet $|\Psi^-\rangle$ quantum mechanics predicts $E(a,b)=-\cos(2(a-b))$ , which violates CHSH up to Tsirelson's bound $|S|_{\max}=2\sqrt2\approx2.828$ .
Polarizers at the magic angles $0^\circ,45^\circ,22.5^\circ,67.5^\circ$ make all four terms reinforce, giving the flagship prediction $S=2\sqrt2$ ; experiments (Aspect 1982, Weihs 1998, loophole-free 2015) confirm the quantum value.
A conclusive test must close the locality loophole (spacelike-separated, fast random settings) and the detection loophole (efficiency above $\eta_{\mathrm{crit}}\approx82.8\%$ ) — achieved together only in 2015.
Bell violation certifies genuine entanglement and nonlocal correlations yet never breaks no-signalling, which is why it underpins device-independent cryptography and certified randomness.
Key idea: for any local hidden-variable theory the combination $S=E(a,b)-E(a,b')+E(a',b)+E(a',b')$ satisfies $|S|\le2$ ; quantum entanglement violates this up to $2\sqrt2$ , so measuring $S>2$ with spacelike-separated, efficient detectors falsifies local realism.

The core insight

For any local hidden-variable theory the combination $S=E(a,b)-E(a,b')+E(a',b)+E(a',b')$ satisfies $|S|\le2$ ; quantum entanglement violates this up to $2\sqrt2$ , and measuring $S>2$ with spacelike-separated, efficient detectors falsifies local realism.

Where this lives today

The modern engine of optical Bell tests is spontaneous parametric down-conversion: a pump photon splitting inside a nonlinear crystal (β-barium borate, or periodically poled lithium niobate in waveguides) into a polarization-entangled pair, collected into single-mode fibre and analyzed with a rotatable waveplate, a polarizing beam splitter, and single-photon detectors. What made the loophole-free tests of 2015 possible was not a new source but new detectors — transition-edge sensors and superconducting-nanowire detectors with system efficiencies above $90\%$ , finally clearing the $82.8\%$ CHSH floor (or, with Eberhard's non-maximal states, the gentler $\sim67\%$ target). Giustina et al. in Vienna and Shalm et al. at NIST reported loophole-free photonic violations within weeks of the Delft nitrogen-vacancy experiment, and the 2022 Nobel Prize to Aspect, Clauser and Zeilinger sealed the programme. The payoff is no longer a test of principle but a resource. Because a genuine violation cannot be forged by any local classical process, the measured $S$ itself certifies both entanglement and intrinsic randomness without trusting the boxes' internals — the foundation of device-independent quantum key distribution and certified random-number generators. Loophole-free links have since been stretched across kilometres of city fibre and used to close the setting-choice loophole with cosmic and even quasar light, and CHSH at the four magic angles remains the standard bench test every entangled-photon source is graded against, its visibility read straight off $V=S/2\sqrt2$ .

The physical picture

Picture two players, Alice and Bob, each handed one particle of a pair from a common source, each free to press one of two buttons. A local-realist referee insists that every particle leaves the source carrying a complete instruction sheet $\lambda$ — a prewritten answer for either button — and that Alice's answer depends only on her own button and her own sheet, never on Bob's distant choice. This is the most reasonable-sounding assumption in physics: outcomes fixed in advance, nothing travelling faster than light to coordinate the two wings. Bell's theorem is the discovery that this innocuous picture has teeth. However the source authors its sheets — deterministic, random, correlated any way you like — the four agreement rates built from the two-by-two settings are bounded: $|S|\le2$ . Entangled photons produce $S=2\sqrt2$ , so the instruction sheets simply do not exist. The geometry behind that ceiling is one picture. Quantum correlations trace a smooth cosine, $E=-\cos(2\Delta\theta)$ , while the strongest local model can do no better than a straight line between the same endpoints. Near the working angles the cosine sags below the line — more strongly anti-correlated than any local theory is allowed to be — and that excess curvature is precisely the $2\sqrt2$ -versus- $2$ violation.

Figure

Quantum versus local correlations. The singlet gives a smooth cosine E equals minus cos of twice the relative analyzer angle, while the best local hidden-variable model is bounded by the straight dashed line. At the working angle Δθ equals 22.5° the quantum value E equals minus 0.707 sits below the local ceiling of minus 0.5 — that gap is the Bell violation

The four magic angles $0^\circ,22.5^\circ,45^\circ,67.5^\circ$ are chosen so this local-beating excess appears in every term of $S$ at once and never cancels: three correlations equal $-\cos45^\circ=-1/\sqrt2$ , and the fourth — carried into $S$ with a minus sign — flips to $+\cos135^\circ=-1/\sqrt2$ as well, so all four add coherently to $4/\sqrt2=2\sqrt2$ . The nonlocality this reveals is about correlations, never communication: each wing alone sees perfectly random $50/50$ outcomes whatever the far analyzer is set to, and only when the two data streams are brought together afterward does the impossible-for-a-local-theory structure surface.

Figure

The Bell-test scoreboard. Entangled photons from a down-conversion source reach Alice and Bob, whose polarizers sit at the magic angles; coincidence counts give S. On the scale, local realism is capped at S equals 2, quantum mechanics reaches the Tsirelson value 2√2 ≈ 2.828, and only a hypothetical super-quantum box would reach the algebraic maximum 4. Aspect's 1982 result S equals 2.697 sits deep in quantum-only territory

The core equations

The object under test is a maximally entangled two-photon state. In the polarization basis the singlet is the antisymmetric combination, the one whose correlations are rotationally invariant — the same whatever pair of matched axes Alice and Bob choose.

|\Psi^-\rangle=\tfrac{1}{\sqrt2}\big(|H\rangle_A|V\rangle_B-|V\rangle_A|H\rangle_B\big) \qquad (1)

where: $|H\rangle,|V\rangle$ — horizontal and vertical single-photon polarization states; $A,B$ — the two spatially separated photons, sent to Alice and Bob. The minus sign and equal weights make this a spin-singlet analogue: measure both photons along any common axis and they always disagree. Each run gives a $\pm1$ result on each side. The correlation function collapses the four joint probabilities into one number running from $-1$ (always disagree) through $0$ to $+1$ (always agree).

E(a,b)=P_{++}+P_{--}-P_{+-}-P_{-+} \qquad (2)

where: $a,b$ — analyzer angles at Alice and Bob; $P_{\pm\pm}$ — joint probabilities of the four outcome combinations. Agreements count positive and disagreements negative, so $E$ is simply the average product of the two $\pm1$ results. Projecting the singlet onto analyzers at $a$ and $b$ gives the central quantum prediction. The factor of two is the signature of photon polarization: a $90^\circ$ rotation (not $180^\circ$ ) already carries $H$ into $V$ , so the physical angle enters doubled.

E(a,b)=-\cos\big(2(a-b)\big) \qquad (3)

where: $a-b$ — relative analyzer angle; the factor $2$ is the spin-1 vector character of the photon. At $a=b$ the value is $-1$ (perfect anti-correlation) and swings to $+1$ when the analyzers are crossed at $45^\circ$ . Bell's move was to combine four such correlations, at two settings per side, into one testable quantity — arranged so a local theory cannot make all four terms large and same-signed at once.

S=E(a,b)-E(a,b')+E(a',b)+E(a',b') \qquad (4)

where: $a,a'$ — Alice's two settings; $b,b'$ — Bob's two settings. The lone minus sign is essential: it forces a local model to trade one term against another, capping the total. The two ceilings the whole subject turns on follow from (3) and (4): the Bell/CHSH bound obeyed by every local hidden-variable theory, and the maximum quantum mechanics itself allows.

|S|\le 2\quad(\text{LHV}),\qquad |S|_{\max}=2\sqrt2\quad(\text{QM}) \qquad (5)

where: $2$ — the local-realist ceiling (derived below); $2\sqrt2\approx2.828$ — Tsirelson's bound, the largest $S$ any quantum state and dichotomic measurements can reach. Nature lives in the forbidden strip between them. To hit the quantum maximum the settings must be chosen well. The magic angles space the analyzers by a uniform $22.5^\circ$ , so every relative angle in $S$ is $\pm22.5^\circ$ or $\pm67.5^\circ$ .

a=0^\circ,\; a'=45^\circ,\; b=22.5^\circ,\; b'=67.5^\circ\;\Rightarrow\;S=2\sqrt2 \qquad (6)

where: the four angles maximize $|S|$ for the singlet. Only relative angles matter — a common rotation of all four leaves $S$ unchanged, a direct consequence of the singlet's rotational invariance. In the laboratory $E$ is a ratio of counts, built from the four coincidence tallies accumulated over many pairs at a fixed setting pair.

E_{\mathrm{exp}}(a,b)=\frac{N_{++}+N_{--}-N_{+-}-N_{-+}}{N_{++}+N_{--}+N_{+-}+N_{-+}} \qquad (7)

where: $N_{\pm\pm}$ — coincidence counts for the four detector combinations. Dividing by the total normalizes away the pair rate, so $E_{\mathrm{exp}}$ estimates $E$ with a statistical error shrinking as $1/\sqrt{N}$ . No inference is clean unless enough photons are caught. The detection loophole sets a hard floor on the per-photon efficiency below which a local model can counterfeit violation by hiding in the undetected events.

\eta_{\mathrm{crit}}=\frac{2}{1+\sqrt2}=2(\sqrt2-1)\approx0.828 \qquad (8)

where: $\eta$ — overall detection efficiency per photon (source to click); $0.828$ — the minimum for a maximally entangled state under CHSH. Below it, no-click events form a sub-ensemble a hidden variable can exploit; above it, even the worst-case assignment to misses cannot rescue a local model. Finally, the condition that lets Bell violation coexist with relativity: nothing on Alice's side leaks her setting into Bob's marginal statistics.

P(a\,|\,x,y)=P(a\,|\,x) \qquad (9)

where: $P(a\,|\,x,y)$ — Alice's outcome distribution given both settings; $x,y$ — Alice's and Bob's settings. Her marginals are blind to $y$ , so no message rides the correlations — the nonlocality is real but unusable for signalling.

Derivation AThe CHSH bound from locality and realism

The goal is to show that $|S|\le2$ follows from just two assumptions — realism, that each outcome is fixed in advance by a hidden variable $\lambda$ , and locality, that Alice's outcome depends on her own setting but not on Bob's distant choice. We write the correlations as averages over $\lambda$ , combine the four CHSH terms in one integral, and bound the integrand pointwise. The power of the argument is that it never invokes quantum mechanics: it is a theorem about any conceivable local realistic world, so a laboratory violation refutes the whole class at once, not one model within it.

Step 1. Encode both assumptions in one line. Let each side return $A(a,\lambda),B(b,\lambda)=\pm1$ with $\lambda$ drawn from a normalized $\rho(\lambda)\ge0$ . Then $E(a,b)=\int\mathrm{d}\lambda\,\rho(\lambda)\,A(a,\lambda)B(b,\lambda)$ . Why: the factorized product is locality made concrete — Alice's factor carries no $b$ , Bob's no $a$ — while the fixed $\pm1$ values are realism.

Step 2. Assemble the four terms, suppressing $\lambda$ : $S=\int\mathrm{d}\lambda\,\rho(\lambda)\big[A(a)\big(B(b)-B(b')\big)+A(a')\big(B(b)+B(b')\big)\big]$ . Why: pairing $b,b'$ with $A(a)$ in a difference and $A(a')$ in a sum isolates a quantity that can only ever be $\pm2$ .

Step 3. Bound the bracket pointwise. Since $B(b),B(b')=\pm1$ , exactly one of $|B(b)-B(b')|$ and $|B(b)+B(b')|$ equals $2$ and the other $0$ ; with $A=\pm1$ the whole bracket is $\pm2$ for every $\lambda$ . Why: however cleverly the source correlates the four hidden answers, two of them are forced to cancel.

Step 4. Average the bound: $|S|\le\int\mathrm{d}\lambda\,\rho(\lambda)\cdot2=2$ . Why: a quantity of magnitude $2$ at every $\lambda$ averages to at most $2$ , so the CHSH ceiling holds for every local realistic theory. QED.

Go deeper · Derivation B · 1 minThe quantum value at the magic angles and Tsirelson's bound

Derivation BThe quantum value at the magic angles and Tsirelson's bound

The goal is to compute $S$ for the singlet at the optimal settings, show it equals $2\sqrt2$ , and then establish that $2\sqrt2$ is not just this state's value but the ceiling for all of quantum mechanics. We evaluate the four correlations from $E=-\cos(2(a-b))$ , watch the CHSH minus sign convert one term so all four reinforce, and close with Tsirelson's operator argument. The number $2\sqrt2\approx2.828$ is the single most important figure in the subject: it is exactly how far past the classical wall of $2$ nature will go, and no further.

Step 1. Insert the magic angles and read off the relative angles: $a-b=-22.5^\circ$ , $a-b'=-67.5^\circ$ , $a'-b=22.5^\circ$ , $a'-b'=-22.5^\circ$ . Why: the correlation depends only on the relative angle, so these four differences are all that enter — three of them $\pm22.5^\circ$ and the odd one out $-67.5^\circ$ .

Step 2. Evaluate with $E=-\cos(2(a-b))$ : $E(a,b)=-\cos45^\circ=-1/\sqrt2$ , $E(a,b')=-\cos135^\circ=+1/\sqrt2$ , $E(a',b)=-\cos45^\circ=-1/\sqrt2$ , $E(a',b')=-\cos45^\circ=-1/\sqrt2$ . Why: doubling sends $22.5^\circ\to45^\circ$ and $67.5^\circ\to135^\circ$ , and $\cos135^\circ=-\cos45^\circ$ is the sign flip that makes the middle term cooperate.

Step 3. Combine: the subtraction hits the one positive term, $S=(-1/\sqrt2)-(+1/\sqrt2)+(-1/\sqrt2)+(-1/\sqrt2)=-4/\sqrt2$ , so $|S|=2\sqrt2\approx2.828$ . Why: the CHSH minus sign turns the lone $+1/\sqrt2$ into a fourth $-1/\sqrt2$ , so all four stack instead of cancelling — the whole reason the magic angles are magic.

Step 4. Establish the ceiling. For dichotomic observables with eigenvalues $\pm1$ , forming $\hat S$ and squaring gives $\hat S^2=4\hat{\mathbb 1}-[\hat A_a,\hat A_{a'}][\hat B_b,\hat B_{b'}]$ , of norm at most $4+2\cdot2=8$ , so $\|\hat S\|\le2\sqrt2$ for every state and measurement. Why: this is Tsirelson's 1980 theorem — the singlet saturates it, so quantum mechanics stops exactly at $2\sqrt2$ and never reaches the algebraic maximum of $4$ .

Go deeper · Derivation C · 2 minThe detection-efficiency threshold

Derivation CThe detection-efficiency threshold

The goal is to see why a real experiment can violate CHSH only if the detectors catch enough photons, and to pin the maximally entangled threshold at $\eta_{\mathrm{crit}}=2/(1+\sqrt2)\approx0.828$ . The danger is post-selection: if the analysis keeps only the runs where both photons were detected, a hidden variable that decides which runs those are can bias the surviving sub-ensemble to fake $S>2$ . We show how fair sampling smuggles that loophole in, and how assigning outcomes to the misses closes it once efficiency is high enough.

Step 1. Name the loophole. Suppose each detector fires with probability $\eta$ and otherwise gives no click, and the analysis conditions on coincidences only. Then a local model can let $\lambda$ dictate both the outcomes and whether a click occurs, so the detected sub-ensemble need not resemble the full one. Why: conditioning on detection is exactly the freedom a local realist needs — the visible data can violate CHSH while the complete data do not.

Step 2. Remove the freedom. Instead of discarding no-clicks, assign each a definite outcome (say $+1$ ) and compute $S$ over all prepared pairs. There is now no hidden sub-ensemble; the loophole is shut, at the cost that misses dilute the correlation. Why: a genuinely refuting statistic must be computed on the whole population, not a self-selected slice — fair sampling is an assumption, and closing the loophole means dropping it.

Step 3. Find the break-even efficiency. Balancing the dilution from misses against the singlet's quantum excess gives, for the maximally entangled state under CHSH, $\eta>2/(1+\sqrt2)\approx0.8284$ . Why: below this fraction the diluted $S$ falls under $2$ and a local model survives; above it the violation is real however the misses are assigned.

Step 4. Read the consequence. Ordinary avalanche detectors sit far below $83\%$ end-to-end, which is why every optical Bell test before 2015 relied on fair sampling; only transition-edge and superconducting-nanowire detectors ( $\eta>90\%$ ) made the loophole-free optical tests of 2015 possible. Why: Eberhard showed in 1993 that a non-maximal state with the CH inequality relaxes the floor toward $2/3\approx66.7\%$ , and that slack is what the 2015 photon experiments used.

Limiting cases

The measured $S$ interpolates between regimes, and each ceiling has a distinct physical origin.

Case	Result	Meaning
Local hidden variables (classical)	$\vert S\vert\le2$	Bell/CHSH bound; any factorizable local model, deterministic or stochastic, obeys it
Quantum mechanics (Tsirelson)	$\vert S\vert\le2\sqrt2\approx2.828$	Maximum for the singlet at the magic angles; saturates Tsirelson's operator bound
General no-signalling (PR box)	$\vert S\vert\le4$	Algebraic maximum; a hypothetical Popescu–Rohrlich box reaches it, but quantum mechanics does not
Noisy state, visibility $V$	$S=2\sqrt2\,V$ , violation for $V>1/\sqrt2$	Werner state; correlations must survive above $70.7\%$ visibility to exceed $2$
Product (separable) state	$\vert S\vert\le2$	No entanglement means the state is already local-realistic; it cannot violate

Common trapBell violation does not permit faster-than-light signalling

The nonlocality of a Bell test is real, but it is a property of correlations, not a channel. Alice's marginal outcome distribution obeys $P(a\,|\,x,y)=P(a\,|\,x)$ : whatever Bob sets, Alice sees the same random $50/50$ stream, so nothing about his choice is legible on her side alone. The impossible-for-a-local-theory structure lives only in the joint statistics, which can be assembled only after the two data records are brought together over an ordinary subluminal channel. A second, subtler trap: violation is not certified by entanglement or visibility alone. With detection efficiency below $\approx83\%$ a purely local model can reproduce $S>2$ on the post-selected coincidences, so an experiment that has high visibility but low efficiency, or that assumes fair sampling, has not actually closed the door on local realism — efficiency, not just correlation strength, is decisive.

Worked examples

1 · The flagship number: S at the magic angles

A source emits the polarization singlet $|\Psi^-\rangle$ . Alice sets her polarizer to $a=0^\circ$ or $a'=45^\circ$ , Bob to $b=22.5^\circ$ or $b'=67.5^\circ$ . Compute the four correlations and the CHSH value $S$ , and state by how much it beats the classical bound.

Step 1. Use $E(a,b)=-\cos(2(a-b))$ and form each relative angle: $a-b=-22.5^\circ$ , $a-b'=-67.5^\circ$ , $a'-b=+22.5^\circ$ , $a'-b'=-22.5^\circ$ ; doubling gives $\mp45^\circ$ and $-135^\circ$ . Cosine is even, so the sign of the relative angle does not matter.

Step 2. Evaluate: $E(a,b)=-\cos45^\circ=-0.7071$ , $E(a,b')=-\cos135^\circ=+0.7071$ , $E(a',b)=-0.7071$ , $E(a',b')=-0.7071$ . Three terms are equally anti-correlated; the crossed pair $(a,b')$ is the positive outlier.

Step 3. Combine with the minus sign on the second term: $S=(-0.7071)-(+0.7071)+(-0.7071)+(-0.7071)=-2.8284$ , so $|S|=2\sqrt2=\mathbf{2.828}$ . The subtraction turns the lone positive term into a fourth negative contribution, and all four add.

Step 4. Compare to the ceilings. The local bound is $2$ , so the prediction overshoots by $2\sqrt2-2=0.828$ (a $41\%$ excess); it reaches but does not pass Tsirelson's $2\sqrt2$ and stays well short of the algebraic $4$ . The singlet at the magic angles gives $S=2\sqrt2\approx2.828$ , violating the local bound of $2$ by $0.828$ .

2 · From coincidence counts to a violation

Over a run at settings $(a,b)=(0^\circ,22.5^\circ)$ an experiment records $N_{++}=75$ , $N_{--}=70$ , $N_{+-}=340$ , $N_{-+}=360$ coincidences. Find the measured correlation, infer the state's visibility, and predict the full CHSH value assuming a uniform Werner-type noise.

Step 1. Apply the counts formula (7). Total $N=75+70+340+360=845$ ; signed sum $=(75+70)-(340+360)=-555$ . So $E_{\mathrm{exp}}=-555/845=\mathbf{-0.657}$ . The preponderance of disagreements ( $+-,-+$ ) is the anti-correlation the singlet demands.

Step 2. Compare to the ideal $E=-\cos45^\circ=-0.7071$ at $\Delta\theta=22.5^\circ$ . The measured magnitude is smaller; their ratio is the visibility $V=0.657/0.7071=\mathbf{0.929}$ . Real down-conversion sources routinely reach $V\approx0.92$ – $0.99$ .

Step 3. Predict $S$ . For a Werner state the correlations scale uniformly, $E=-V\cos(2\Delta\theta)$ , so $S=2\sqrt2\,V=2.8284\times0.929=\mathbf{2.63}$ — the value the full four-setting run should return under uniform noise.

Step 4. Test the bound. Since $2.63>2$ the state violates CHSH (loopholes aside). The threshold is $V_{\mathrm{crit}}=1/\sqrt2=0.707$ , so any $V$ above $70.7\%$ suffices and the measured $92.9\%$ clears it comfortably. $E_{\mathrm{exp}}=-0.657$ , $V=0.929$ , $S=2.63>2$ — a clear violation with margin to spare.

Physics of the portrait

Bell inequalities are where a question everyone thought was philosophy became an experiment — and the portrait is the man who found it had an answer.

Origins · 1 minJohn Stewart Bell and how this was discovered

John Stewart Bell

1928–1990 · foundations of quantum mechanics and quantum optics

John Stewart Bell was born in Belfast in 1928, worked as a laboratory technician before studying physics at Queen's University, and spent his career at CERN as a particle and accelerator theorist — Bell inequalities were, in his own words, a hobby pursued on the side. In 1964, on leave in the United States, he published two papers that reshaped the foundations of physics. The first showed that the standard "impossibility" proofs against hidden variables (chiefly von Neumann's) rested on an unjustified assumption; the second, "On the Einstein Podolsky Rosen paradox," derived the inequality that bears his name and turned the EPR debate into a laboratory question. Bell's own sympathies lay with Einstein — he found nonlocality genuinely troubling and hoped a local completion might yet be found — which makes his theorem a model of honesty: he built the very tool that would rule out the picture he preferred. He also clarified the measurement problem, championed the de Broglie–Bohm pilot-wave theory as proof that a coherent realist quantum mechanics was possible, and coined the term "beable" for what a theory takes as real. Clauser, Horne, Shimony and Holt recast his inequality into experiment-ready CHSH form in 1969, and Aspect's group tested it decisively in the early 1980s. Bell died suddenly in 1990, the year he had been nominated for a Nobel Prize; the 2022 Prize to Aspect, Clauser and Zeilinger honoured the vindication of his theorem.

History

1935

— Einstein, Podolsky and Rosen argue that quantum mechanics is an incomplete description of a locally real world

1964

— Bell derives his inequality, converting the EPR debate into a testable prediction

1969

— Clauser, Horne, Shimony and Holt cast Bell's result in the experiment-ready CHSH form

1982

— Aspect, Dalibard and Roger use fast-switched analyzers to address the locality loophole

2015

— three groups close the locality and detection loopholes together, in diamond spins and in photons

Going further

Texts. Bell, Speakable and Unspeakable in Quantum Mechanics, 2nd ed., Cambridge (2004) — the collected foundational papers · Nielsen & Chuang, Quantum Computation and Quantum Information, §2.6 (the CHSH game) · Foot, Atomic Physics, Oxford (2005), on entanglement and Bell tests · Loudon, The Quantum Theory of Light, 3rd ed., OUP (2000) · Saleh & Teich, Fundamentals of Photonics, Wiley, on entangled-photon optics · the review by Brunner, Cavalcanti, Pironio, Scarani & Wehner, Rev. Mod. Phys. 86, 419 (2014), for Bell nonlocality as a resource.

Provenance. A verifier should recompute: (1) the CHSH bound $|S|\le2$ from the factorized local model; (2) $E(a,b)=-\cos(2(a-b))$ and the resulting $S=2\sqrt2$ at $0^\circ,45^\circ,22.5^\circ,67.5^\circ$ ; (3) Tsirelson's bound $2\sqrt2$ ; (4) the threshold $2/(1+\sqrt2)\approx0.828$ ; (5) the visibility relation $S=2\sqrt2\,V$ with $V_{\mathrm{crit}}=1/\sqrt2$ . Anchor numbers: Aspect's two-channel $S=2.697\pm0.015$ (PRL 49, 91, 1982); loophole-free tests, Hensen et al., Nature 526, 682 (2015) and Giustina et al. & Shalm et al., PRL 115, 250401/250402 (2015); Nobel Prize 2022. Foundational: Bell, Physics 1, 195 (1964); CHSH, PRL 23, 880 (1969); Cirel'son, Lett. Math. Phys. 4, 93 (1980); Eberhard, Phys. Rev. A 47, R747 (1993); Weihs et al., PRL 81, 5039 (1998).

Prerequisites. Quantization of the Electromagnetic Field · Fock States and the Photon Number Basis · Single-Photon Sources · Coherence Functions and the Degree of Coherence. Leads to. Photon Antibunching · The Hanbury Brown–Twiss Experiment · Device-Independent Quantum Key Distribution.

Ask the tutor: Ask me to walk you through why the CHSH minus sign is exactly what makes all four correlations add to $2\sqrt2$ at the magic angles.

What is the difference between Bell's original inequality and CHSH?

Bell's 1964 inequality assumed perfect anti-correlation at aligned settings — true of the ideal singlet but never of a real source — which made it hard to test directly. Clauser, Horne, Shimony and Holt reworked it in 1969 into a form using four correlation functions with two settings per side and no assumption of perfect correlation. The CHSH bound $|S|\le2$ is what experiments actually measure, because it tolerates finite visibility and imperfect alignment while still separating every local hidden-variable theory ( $\le2$ ) from quantum mechanics ( $\le2\sqrt2$ ). In practice "Bell test" almost always means a CHSH test.

Why the factor of two in E equals minus cosine of twice the angle?

It is the geometry of polarization. A polarizer rotated by $90^\circ$ fully blocks light a $0^\circ$ polarizer passed, so polarization states $90^\circ$ apart are orthogonal — whereas for a spin-1/2 particle you must rotate a full $180^\circ$ to reach the orthogonal state. Mapping polarization angles onto the spin formalism therefore doubles the angle, and the singlet correlation $-\cos\theta$ of spin becomes $-\cos(2(a-b))$ for photons. This is why the magic angles are spaced by $22.5^\circ$ rather than the $45^\circ$ you would use for spins.

If nothing can signal faster than light, in what sense is this nonlocal?

Bell nonlocality means correlations that no shared past cause acting locally can explain. The two outcomes are more strongly linked than any pre-agreed instruction set allows, yet each side's own statistics are perfectly random and independent of the far setting, so the link carries no message. "Nonlocal" here means precisely that the joint distribution has no local hidden-variable model, even though the marginals respect no-signalling. It is a failure of local causality in Bell's sense, not a hidden telephone line — you cannot transmit with it, only certify that the correlations are genuinely quantum.

What exactly are the two loopholes, and why did closing both take until 2015?

The locality loophole is that a subluminal influence could pass from Alice's setting to Bob's outcome; you close it by choosing settings randomly and completing each measurement within a spacelike interval. The detection loophole is that undetected photons form a biased sub-ensemble faking violation; you close it with per-photon efficiency above $\approx83\%$ . Each was shut separately decades earlier — Aspect and Weihs addressed locality, high-efficiency ion experiments addressed detection — but doing both at once, with fast switching and above-threshold detectors over a large separation, needed the superconducting detectors and long optical links that matured around 2015.

Does a PR box exist, or is S equal to 4 just a curiosity?

A Popescu–Rohrlich box is a hypothetical device reaching the algebraic maximum $S=4$ while still respecting no-signalling, so it is not forbidden by relativity alone. Yet no PR box has ever been found, and quantum mechanics stops firmly at $2\sqrt2$ . The gap between $2\sqrt2$ and $4$ is a deep open question: something beyond mere no-signalling limits nature to Tsirelson's bound, and candidate principles (information causality, macroscopic locality) have been proposed to explain why. So $S=4$ is not idle — it marks the frontier between what relativity permits and what quantum mechanics actually delivers.

How is a Bell violation used to certify randomness or secure a key?

Because $S>2$ cannot be produced by any local deterministic process, an eavesdropper who pre-programmed the boxes could never make them violate CHSH — so a measured violation guarantees the outcomes were not fixed in advance and contain genuine private randomness. Device-independent protocols treat the boxes as black boxes and derive a security bound from the observed $S$ alone, without trusting the hardware. The closer $S$ is to $2\sqrt2$ , the more certified random bits per run and the more tolerable eavesdropper information. This makes loophole-free Bell tests not just foundational milestones but the trust anchor for a class of quantum-cryptographic devices.