Experimental Tests and Violation of Bell Inequalities

For thirty years after Einstein, Podolsky and Rosen argued that quantum mechanics must be incomplete, the debate sat frozen in metaphysics — until John Bell showed in 1964 that the disagreement had teeth. Any theory built on local realism, in which each particle carries pre-set answers to every measurement decided at the source, obeys a numerical bound on how strongly two distant measurements can correlate. Quantum mechanics, using entangled pairs, predicts correlations that punch straight through that bound. The question of which description Nature actually obeys stopped being philosophy and became an experiment. This is the article about that experiment — six decades of it. You already know the inequality itself: "Bell's Theorem and the CHSH Inequality" derives the local-realistic ceiling $|S|\le 2$ and the quantum Tsirelson value $2\sqrt 2$ . Here the target is the laboratory: how you turn an entangled source and two rotatable analyzers into a single measured number $S$ , and — far harder — how you close the escape hatches. A determined local realist has three of them: let the detectors talk (the locality loophole), throw away inconvenient events (the detection loophole), or let the settings be foreseen (the freedom-of-choice loophole). A convincing test must slam all three shut at once. The payoff of mastering this is a rare thing in physics: watching an abstract theorem about "elements of reality" get settled by photon counters and diamond spins, culminating in the 2015 loophole-free results and the 2022 Nobel Prize. It is also intensely practical — a measured Bell violation is now a security certificate for quantum cryptography.

Key takeaways

A Bell test compresses an entire worldview into one number: local realism caps the CHSH correlator at $|S|\le 2$ , quantum mechanics reaches $|S|=2\sqrt 2 \approx 2.828$ .
The experimenter's whole job is closing loopholes — locality (space-like separation + fast switching), detection (efficiency $\eta$ above threshold), and freedom-of-choice (independent random settings).
Pioneers Freedman–Clauser (1972) and Aspect (1981–82) saw violations but left loopholes open; the static Aspect experiment reached $S=2.697\pm0.015$ , the locality-closing one $S=2.404\pm0.080$ .
Loophole-free tests arrived in 2015: Delft NV-centre spins ( $S=2.42\pm0.20$ , $1.3\,$ km apart) and the Vienna/NIST high-efficiency photon experiments closed locality and detection together.
The detection loophole sets a hard efficiency bar: $\eta>2(\sqrt2-1)\approx 0.828$ for maximally entangled pairs, relaxed to $\eta>2/3\approx0.667$ by Eberhard's partially-entangled trick.
Key idea: a single measured number $S$ separates two worldviews — $|S|\le 2$ is the absolute ceiling for any locally realistic theory, and loophole-free experiments measure $|S|>2$ with overwhelming significance, so Nature is not locally realistic.

The core insight

A single measured number $S$ separates two worldviews: $|S|\le 2$ is the absolute ceiling for any local realistic theory, while entangled quantum systems reach $|S|=2\sqrt2$ — and loophole-free experiments measure $|S|>2$ with high significance, so Nature is not locally realistic.

Where this lives today

The Bell test has completed the journey from philosophy to engineering. Because no local-realistic device — and hence no classically programmed box — can produce $S>2$ , a measured Bell violation is now used as a hardware-independent security certificate. Device-independent quantum key distribution and certified random-number generation rest on exactly this: you need not trust the internals of your detectors, only the observed value of $S$ . The 2018 "Big Bell Test" even crowd-sourced setting choices from a hundred thousand human volunteers to attack the freedom-of-choice loophole, and cosmic Bell tests have used the colours of photons emitted by quasars billions of years ago to make the settings. The frontier now is turning loophole-free violation into a routine, networked resource — entangling distant quantum-network nodes so that a certified Bell violation can be generated on demand between cities, the substrate of a future quantum internet whose security is guaranteed by the same inequality Bell wrote down in 1964.

The physical picture

Picture Alice and Bob, far apart, each handed one particle of an entangled pair. Each freely turns a dial to one of two settings and records a $\pm1$ outcome. Local realism makes a very specific promise: every particle left the source carrying a complete instruction sheet — the answer it will give for either of Alice's dial positions, and likewise for Bob's — and nothing that happens at Alice's end can rewrite Bob's sheet. Bell's insight was that these instruction sheets, no matter how cunningly correlated at the source, cannot make all four setting-combinations agree as strongly as an entangled quantum pair does. The gap between "as strong as any instruction sheet allows" and "as strong as quantum mechanics delivers" is a number you can measure. That number is the CHSH quantity $S$ , built from four correlation measurements. For a quantum singlet the two outcomes correlate as $-\cos\theta$ in the relative dial angle, and at a clever spacing of $22.5^\circ$ the four terms reinforce to $2\sqrt2$ rather than the classical ceiling of $2$ . The "extra" correlation is not stored on either particle — it is a property of the joint state that no separable description reproduces. This is exactly the entanglement resource that "The Bell States and Maximally Entangled Pairs" identifies as maximally entangled.

Figure

Alice and Bob measure an entangled pair

Loopholes are the escape routes a stubborn local realist would use to keep the classical story alive. Slow settings let a light-speed signal carry Alice's choice to Bob's particle. Inefficient detectors let a local model quietly discard the pairs that would spoil its fake violation. Correlated settings let the hidden variable "know" the dials in advance. Each loophole is an auxiliary assumption that, if false, would let $|S|>2$ arise from a local model after all — which is why the decisive experiments had to block every route simultaneously, and why they took until 2015.

Figure

Where measured S-values land

The core equations

The whole experiment funnels into one linear combination of four measured correlators. A correlation function $E(a,b)$ is just the average of the product of the two $\pm1$ outcomes when Alice uses setting $a$ and Bob uses $b$ ; you estimate it from coincidence counts. The CHSH combination pits two of Alice's settings against two of Bob's in a specific $+,-,+,+$ pattern that is engineered to be maximally sensitive to the quantum–classical gap.

S = E(a,b) - E(a,b') + E(a',b) + E(a',b') \qquad (1)

where: $E(a,b)$ — the correlation $\langle A_a B_b\rangle$ of $\pm1$ outcomes for Alice's setting $a$ and Bob's setting $b$ ; $a,a'$ — Alice's two analyzer settings; $b,b'$ — Bob's two settings. Read physically: three terms enter with a plus sign and one with a minus, so a theory that could make three correlators near $+1$ and the fourth near $-1$ would maximize $S$ — the tension is that no local model can do this for all four at once. In the laboratory each correlator is reconstructed from raw coincidence counts, and this bookkeeping is the experimental heart of the article. If $N_{++}$ is the number of pairs where both detectors registered $+1$ , and so on, then the estimator is a simple ratio.

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

where: $N_{\pm\pm}$ — coincidence counts for the four outcome combinations at settings $(a,b)$ . Read physically: agreements ( $++$ , $--$ ) push $E$ toward $+1$ , disagreements toward $-1$ ; a perfectly anti-correlated pair gives $E=-1$ . The denominator is where the detection loophole hides — if you only count pairs both detectors saw, you have silently assumed the unseen pairs behave like the seen ones (fair sampling). The three theoretical ceilings that experiment tries to distinguish sit in a strict hierarchy. It is worth having them side by side before touching a single detector.

bound	value of $\|S\|_{\max}$	who obeys it
local realism	$2$	any local hidden-variable theory
quantum mechanics	$2\sqrt2 \approx 2.828$	entangled pairs at optimal angles (Tsirelson)
no-signalling	$4$	Popescu–Rohrlich boxes (not realized in Nature)

The local-realistic ceiling and the quantum value are derived in "Bell's Theorem and the CHSH Inequality"; here we take $|S|\le 2$ and $2\sqrt2$ as given and ask what it costs to measure the gap between them cleanly. The singlet correlation that produces the quantum value is the standard entangled-pair result.

E_{\mathrm{QM}}(a,b) = -\cos(\theta_a - \theta_b) \qquad (3)

where: $\theta_a,\theta_b$ — the analyzer orientation angles for Alice and Bob; the correlation depends only on the relative angle. Read physically: aligned analyzers ( $\theta_a=\theta_b$ ) give perfect anti-correlation $-1$ ; crossed at $90^\circ$ give $0$ . Feeding this into (1) at the CHSH angles produces $2\sqrt2$ , as the first derivation shows.

Derivation AThe quantum CHSH value for the singlet

We want to show that an entangled singlet, measured at the right four analyzer angles, produces $|S|=2\sqrt2$ — the number every experiment is chasing. The strategy is to insert the singlet correlation $E=-\cos(\theta_a-\theta_b)$ into the CHSH combination with equally spaced settings, collapse the four terms into a single trigonometric function of the spacing, and maximize it. The one thing to watch is signs: the CHSH pattern is $+,-,+,+$ , and getting the relative angles to line up so that all four terms reinforce is the whole trick that beats the classical bound.

Step 1. Choose settings spaced by an angle $\theta$ : take Alice at $\theta_a=0,\ \theta_{a'}=2\theta$ and Bob at $\theta_b=\theta,\ \theta_{b'}=3\theta$ . Then the four relative angles are $\theta_a-\theta_b=-\theta$ , $\theta_a-\theta_{b'}=-3\theta$ , $\theta_{a'}-\theta_b=\theta$ , $\theta_{a'}-\theta_{b'}=-\theta$ .

Step 2. Insert $E=-\cos(\Delta\theta)$ into (1), using $\cos$ even:

S = -\cos\theta - \big(-\cos 3\theta\big) + \big(-\cos\theta\big) + \big(-\cos\theta\big) = -3\cos\theta + \cos 3\theta

Why this is the clean form: the three plus-terms all carry the same relative angle $\theta$ and the minus-term carries $3\theta$ ; the $+,-,+,+$ sign pattern turns into a compact $-3\cos\theta+\cos3\theta$ .

Step 3. Maximize $|S(\theta)|=|-3\cos\theta+\cos3\theta|$ . Differentiate and set to zero: $3\sin\theta - 3\sin3\theta = 0$ , solved by $\theta = 45^\circ$ (and $135^\circ$ ).

Step 4. Evaluate at $\theta=45^\circ$ : $\cos45^\circ = 1/\sqrt2$ and $\cos135^\circ = -1/\sqrt2$ , so $S = -3(1/\sqrt2)+(-1/\sqrt2) = -4/\sqrt2 = -2\sqrt2$ . The magnitude is $2\sqrt2\approx2.828$ — a $41\%$ excess over the classical ceiling of $2$ , sitting at settings spaced by $22.5^\circ$ between Alice's and Bob's nearest dials. Verified: sympy expansion of (1) with $E=-\cos\Delta\theta$ gives exactly $-3\cos\theta+\cos3\theta$ ; global maximization over four independent angles returns $|S|=2.8284=2\sqrt2$ .

The detection loophole deserves its own equation, because it is the one that gated the whole field for decades. If detectors miss a fraction of particles and you keep only coincidences under the fair-sampling assumption, a local model gets a free parameter — which events to register — and can sculpt the surviving sample to fake a violation. Closing it requires a genuinely high efficiency $\eta$ .

\eta_{\min}^{\text{(max.\,ent.)}} = \frac{2}{\sqrt2+1} = 2(\sqrt2-1) \approx 0.828 \qquad (4)

where: $\eta$ — the single-particle detection efficiency (probability a present particle is registered). Read physically: for a maximally entangled pair you need to catch better than $83\%$ of particles on each side, or a local model survives. This is a brutal bar for photons, which is why the first loophole-free photon tests waited for superconducting-nanowire detectors.

Go deeper · Derivation B · 1 minThe detection-efficiency threshold and Eberhard's escape

Derivation BThe detection-efficiency threshold and Eberhard's escape

We want the minimum detector efficiency $\eta$ below which the detection loophole cannot be closed, and why Philip Eberhard's 1993 trick with non-maximally entangled states lowers that bar from $0.83$ to $2/3$ . The route is to model non-detection as a genuine third outcome rather than a discarded event, so that fair sampling is never assumed, and then ask how much of the quantum violation survives the losses. The subtlety to watch: the maximally entangled state is not optimal once losses enter — a lopsided state hides its "no-click" events more cleverly.

Step 1. Treat each side as having three outcomes: $+1$ , $-1$ , and "no detection." A local model may condition all three on the setting and the hidden variable, so postselecting on coincidences no longer sanitizes the data.

Step 2. For a maximally entangled pair, requiring the loophole-free inequality to still be violated forces the efficiency above $\eta > 2/(\sqrt2+1) = 2(\sqrt2-1) \approx 0.828$ . Below this, non-detection events give a local model enough slack to reproduce the counts.

Step 3. Eberhard's inequality counts non-detections explicitly and is optimized over the degree of entanglement. A nearly-product state (small entanglement) concentrates its detectable events where they matter, and the threshold drops.

Step 4. In the limit of vanishing entanglement the required efficiency falls to $\eta > 2/3 \approx 0.667$ . This is the route the 2015 Vienna and NIST photon experiments took — trading maximal correlation strength for a survivable efficiency bar. Verified: $2/(\sqrt2+1)=0.82843$ and $2/3 = 0.66667$ recomputed; both are the standard CHSH and Eberhard thresholds (Eberhard, PRA 47, R747, 1993).

A last equation guards against the most common misreading — that a Bell violation lets you signal faster than light. It does not, and the reason is a one-line partial trace.

\rho_A = \mathrm{Tr}_B\,|\Psi^-\rangle\langle\Psi^-| = \tfrac12\,\mathbb{1}, \qquad P(A=\pm1\,|\,a,b) = \tfrac12 \ \text{(independent of } b) \qquad (5)

where: $\rho_A$ — Alice's reduced density matrix after tracing out Bob. Read physically: Alice's local outcome is a fair coin for every setting, and nothing Bob does changes it — so no message crosses. The strong correlation is real but only appears when the two records are later compared, a step that itself travels no faster than light.

Limiting cases

case	result	meaning
local realism (LHV)	$\lvert S\rvert \le 2$	absolute classical ceiling; every factorizable hidden-variable model obeys it
quantum mechanics	$\lvert S\rvert \le 2\sqrt2 \approx 2.828$	Tsirelson bound, saturated by a maximally entangled pair at optimal angles
no-signalling (PR boxes)	$\lvert S\rvert \le 4$	algebraic maximum allowed by no-signalling alone; quantum theory does not reach it
detection loophole, max. entangled	$\eta > 0.828$	minimum efficiency to close fair-sampling with symmetric CHSH
detection loophole, Eberhard	$\eta > 2/3$	relaxed threshold using partially entangled states
aligned analyzers ( $\Delta\theta=0$ )	$E=-1$	perfect anti-correlation; contributes no violation on its own

Worked examples

Example 1 — Reconstruct $S$ from coincidence counts and decide the verdict. A tabletop experiment records the following coincidence counts (per setting pair, arbitrary integration time):

Step 1. At $(a,b)$ : $N_{++}=90,\ N_{--}=90,\ N_{+-}=10,\ N_{-+}=10$ . Apply equation (2): $E(a,b) = (90+90-10-10)/(90+90+10+10) = 160/200 = \mathbf{+0.80}$ .

Step 2. At $(a,b')$ : $N_{++}=10,\ N_{--}=10,\ N_{+-}=90,\ N_{-+}=90$ , giving $E(a,b') = (20-180)/200 = \mathbf{-0.80}$ .

Step 3. At $(a',b)$ : same as the first pair, $E(a',b)=\mathbf{+0.80}$ . At $(a',b')$ : also $\mathbf{+0.80}$ .

Step 4. Assemble with the CHSH sign pattern of equation (1): $S = 0.80 - (-0.80) + 0.80 + 0.80 = 0.80+0.80+0.80+0.80 = \mathbf{3.20}$ .

Step 5. Compare against the ceilings. $3.20 > 2$ , so local realism is violated; and $3.20 > 2\sqrt2 = 2.828$ , which is impossible for real quantum data — flagging that these idealized counts exceed the Tsirelson bound and could only come from a super-quantum (PR-box) toy model, not a genuine experiment. A real singlet at optimal angles would give $E=\pm 1/\sqrt2 = \pm0.707$ per term and $S=2.828$ .

Example 2 — The detection-efficiency budget for a photon experiment. You want a loophole-free CHSH test with maximally entangled photon pairs.

Step 1. The maximally-entangled threshold is $\eta_{\min} = 2(\sqrt2-1)$ . Compute: $\sqrt2 = 1.41421$ , so $\eta_{\min} = 2(0.41421) = \mathbf{0.8284}$ .

Step 2. Superconducting-nanowire single-photon detectors reach system efficiencies around $0.90$ . Since $0.90 > 0.8284$ , a symmetric CHSH test is feasible — barely — with margin $0.90-0.83 = 0.07$ .

Step 3. If instead your detectors sit at $\eta = 0.75$ , you are below the symmetric threshold and must switch to Eberhard's inequality with a non-maximally entangled state, whose bar is $2/3 = \mathbf{0.667}$ . Since $0.75 > 0.667$ , the Eberhard route now works where the symmetric one failed.

Step 4. The lesson in one number: dropping from maximal entanglement to the Eberhard scheme buys you $0.828-0.667 = \mathbf{0.161}$ of efficiency headroom — the difference between an impossible and a feasible experiment, and precisely the trade the 2015 photon groups made.

Origins · 1 minAlain Aspect and how this was discovered

Alain Aspect

born 1947 · French physicist; Orsay experiments (1981–82) that pushed Bell tests toward closing the locality loophole; shared the 2022 Nobel Prize in Physics

Aspect's thesis experiments at Orsay were the first to change analyzer settings while the photons were in flight, using acousto-optic switches to attack the locality loophole head-on. His static high-precision run reached $S=2.697\pm0.015$ , and the time-varying (locality-directed) run gave $S=2.404\pm0.080$ . Four decades later the 2022 Nobel Prize recognized him alongside John Clauser and Anton Zeilinger for turning Bell's theorem into experimental fact.

History

1964

— John Bell proves that local hidden-variable theories obey an inequality quantum mechanics can violate

1969

— Clauser, Horne, Shimony and Holt cast Bell's result as the experimentally testable CHSH inequality

1972

— Freedman and Clauser perform the first Bell test with calcium-cascade photons, seeing a clear violation

1982

— Aspect, Grangier and Roger reach S = 2.697 ± 0.015; Aspect, Dalibard and Roger use time-varying analyzers (S = 2.404 ± 0.080) to attack the locality loophole

1993

— Eberhard shows partially entangled states lower the detection-efficiency threshold to two-thirds

1998

— Weihs and Zeilinger enforce strict Einstein locality with space-like separation and fast random switching

2015

— Hensen (Delft), Giustina (Vienna) and Shalm (NIST) report the first loophole-free Bell violations

2022

— Aspect, Clauser and Zeilinger share the Nobel Prize in Physics for entangled-photon Bell tests

Going further

References. Griffiths & Schroeter, Introduction to Quantum Mechanics, 3rd ed., §12.2 (Bell's theorem) · Sakurai & Napolitano, Modern Quantum Mechanics, 3rd ed., §3.10 · Clauser, Horne, Shimony, Holt, Phys. Rev. Lett. 23, 880 (1969) · Freedman & Clauser, Phys. Rev. Lett. 28, 938 (1972) · Aspect, Grangier, Roger, Phys. Rev. Lett. 49, 91 (1982); Aspect, Dalibard, Roger, Phys. Rev. Lett. 49, 1804 (1982) · Eberhard, Phys. Rev. A 47, R747 (1993) · Weihs et al., Phys. Rev. Lett. 81, 5039 (1998) · Hensen et al., Nature 526, 682 (2015); Giustina et al., Phys. Rev. Lett. 115, 250401 (2015); Shalm et al., Phys. Rev. Lett. 115, 250402 (2015).

Provenance. Built from the AllQFT base article, corrected and verified: the singlet CHSH form $S=-3\cos\theta+\cos3\theta$ peaking at $2\sqrt2$ (sympy), the efficiency thresholds $2(\sqrt2-1)\approx0.828$ and $2/3$ , the PR-box ceiling $4$ , and the no-signalling partial trace $\rho_A=\tfrac12\mathbb{1}$ were all recomputed. The base attributed $S\approx2.70\pm0.05$ to Aspect's time-varying experiment; corrected here — the static Aspect–Grangier–Roger run gave $S=2.697\pm0.015$ , while the time-varying Aspect–Dalibard–Roger run gave $S=2.404\pm0.080$ .

Builds on: Bell's Theorem and the CHSH Inequality · The EPR Paradox and Local Realism · The Bell States and Maximally Entangled Pairs · Hidden Variable Theories. Leads to: The Kochen–Specker Theorem and Contextuality · The GHZ State and All-or-Nothing Nonlocality · device-independent quantum cryptography.

If quantum mechanics only reaches 2√2 but no-signalling allows 4, why doesn't Nature use the stronger boxes?

This is a genuine open question, sometimes called the Tsirelson problem or "why 2√2?". Popescu–Rohrlich boxes would violate CHSH all the way to the algebraic maximum of $4$ while still forbidding signalling, yet they are never observed. Several principles — information causality, macroscopic locality, and limits on communication complexity — single out the quantum bound $2\sqrt2$ as special, but there is no fully agreed first-principles reason. Nature apparently stops at Tsirelson's value, and understanding exactly why is still active research.

How is the correlation E(a,b) actually measured — where do the ±1 outcomes come from?

For polarization-entangled photons, each analyzer is a polarizing beam splitter followed by two detectors: a click in the "transmitted" port is $+1$ , a click in the "reflected" port is $-1$ . You accumulate coincidence counts $N_{++},N_{+-},N_{-+},N_{--}$ over many pairs at a fixed setting pair, then plug them into $E=(N_{++}+N_{--}-N_{+-}-N_{-+})/N_{\text{tot}}$ . Repeat for all four setting combinations, then assemble $S$ . For NV-centre spins the " $\pm1$ " is a deterministic spin-state readout, which is why that platform sidesteps the detection loophole entirely.

Why did loophole-free tests take until 2015 when violations were seen in 1972?

Because seeing a violation and closing every loophole are very different bars. The 1972 and 1982 photon experiments used low-efficiency detectors (fair-sampling assumption open) and relatively slow settings (locality assumption partly open). Closing locality needs space-like separation with genuinely fast, random switching; closing detection needs efficiency above $\sim0.83$ (or the Eberhard trick above $2/3$ ). No platform could do both at once until superconducting-nanowire detectors and long-distance spin-entanglement matured around 2015. Each earlier experiment closed one loophole while leaving another open — a determined local realist could always retreat to the open one.

Does closing the loopholes prove quantum mechanics is correct?

Not directly. A loophole-free Bell violation refutes all local hidden-variable theories — it rules out the entire class of locally realistic explanations, whatever their internal machinery. Quantum mechanics is the theory that predicts and matches the observed violation, so the result is powerful confirmation, but strictly the experiment falsifies local realism rather than proving quantum theory. Any successor theory must also be non-local (or non-realistic) and reproduce $S>2$ .

What exactly is the freedom-of-choice loophole, and how can quasar light close it?

The freedom-of-choice (or measurement-independence) loophole is the worry that the analyzer settings might not be truly independent of the hidden variables — perhaps a common cause in the past correlated the source and the setting choices. To make this conspiracy implausible, cosmic Bell tests use the arrival colour of photons from quasars billions of light-years away to pick the settings. Any common cause would have to reach back billions of years into the past light-cone, which strains superdeterminism to a degree most physicists find untenable.

Why does a non-maximally entangled state tolerate lower detector efficiency (Eberhard)?

Counterintuitively, the maximally entangled state is not optimal once detector losses are counted honestly. Eberhard showed that a lopsided state — close to a product state — concentrates its "useful" detectable events in a way that keeps the loophole-free inequality violated even when many particles are missed. The price is a smaller raw violation, but the payoff is that the required efficiency drops from $0.828$ to $2/3$ . It is a genuine trade of correlation strength for loss-tolerance, and it is what made the 2015 photon experiments possible.

Is the CHSH inequality the only Bell inequality?

No — CHSH is the simplest and most-used two-setting, two-outcome inequality, but there is a whole zoo. Bell's original 1964 inequality assumed perfect anti-correlation; CHSH (1969) removed that idealization for real detectors. The Clauser–Horne (CH) and Eberhard forms are tailored to the detection loophole; multi-setting inequalities (e.g. $I_{3322}$ ) and multi-party inequalities (Mermin's for GHZ states) probe richer structure. CHSH is popular because it is optimal for the two-qubit singlet and needs only four correlation measurements.