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Quantum Entanglement Simulator and CHSH Bell Inequality Calculator
Quick answer: predict the four Bell-test correlations and CHSH S value from a quantum-state model, or calculate S from your own correlations or coincidence counts. Optical distance and photon loss are optional link-performance estimates: entanglement has no universal maximum travel distance, and ordinary symmetric loss mainly reduces detected counts and confidence—not the normalized S value.
How to read this model
The calculator uses S = |E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′)|. Local hidden-variable models satisfy S ≤ 2 under the CHSH assumptions; quantum theory is bounded by 2√2. A result above 2 is meaningful only relative to its uncertainty and the experiment’s handling of loopholes.
For prediction with linear-polarization analyzers, the singlet model uses E = −Veffcos[2(a−b)], while |Φ+⟩ uses E = +Veffcos[2(a−b)]. Werner mixing contributes a separate p multiplier. Visibility describes correlation contrast; it is not itself an entanglement measure.
Distance correction: quantum mechanics supplies no universal maximum entanglement distance. Setting-independent symmetric attenuation cancels in normalized correlations when the surviving sample is representative, although it sharply reduces counts and widens uncertainty. The optional exp(−L/Ld) decay is a selectable phenomenological scenario—not a general law of fiber attenuation.
Worked examples
1. Ideal singlet, optimal angles
Inputs: |Ψ−⟩, V = 1; a = 0°, a′ = 45°, b = 22.5°, b′ = 67.5°.
Correlations: −0.7071, +0.7071, −0.7071, −0.7071.
Substitution: |−0.7071 − 0.7071 − 0.7071 − 0.7071| = 2.8284 = 2√2.
Meaning: maximum quantum prediction for this CHSH convention.
2. Werner state near the boundary
Inputs: Werner singlet p = 0.71, V = 1, same optimal angles.
Correlations: −0.5020, +0.5020, −0.5020, −0.5020.
Substitution: |−0.5020 − 0.5020 − 0.5020 − 0.5020| = 2.0082.
Meaning: barely above 2 because p is just above 1/√2; finite data need enough precision to establish a violation.
3. Loss versus selected decay
Inputs: 20 km, 0.2 dB/km, η = 0.60, 10⁶ pairs/s, singlet V = 0.97.
Link: 4 dB total loss, T = 63.10% per arm, detected fraction = 14.33%, about 143,319 coincidences/s.
Without visibility decay: S = 2√2 × 0.97 = 2.7436. With the optional Ld = 50 km assumption: Veff = 0.6502 and S = 1.8390.
Meaning: attenuation changes counts; only the separately selected visibility model changes predicted S.
CHSH and entanglement FAQ
What does a CHSH value above 2 mean?
Under the CHSH assumptions, a statistically significant |S| > 2 is incompatible with local hidden-variable models. Report uncertainty and check experimental loopholes before claiming a violation.
Does S ≤ 2 prove there is no entanglement?
No. An entangled state can fail this particular test because of the state, measurement angles, noise, loss-related sampling, or limited statistics. Bell nonlocality is a stronger condition than entanglement.
Why are 0°, 45°, 22.5°, and 67.5° used?
With this calculator’s linear-polarizer model and CHSH sign convention, they make the four ideal singlet terms add in magnitude and yield 2√2.
Does photon loss reduce S?
Ordinary setting-independent symmetric loss mainly reduces detected coincidences and statistical precision. It does not by itself reduce normalized S. Biased sampling, background, drift, or physical decoherence can change the measured correlations.
What is the difference between visibility and entanglement?
Visibility is measured correlation contrast. Entanglement is a property of the quantum state. Visibility, state fidelity, Werner mixing p, and detector efficiency are related in some experiments but are not interchangeable.
Can entanglement communicate faster than light?
No. Alice’s and Bob’s local outcomes are random. The correlation becomes visible only after they compare results through an ordinary classical channel.
How do I calculate E from coincidence counts?
For each setting pair, use E = (N++ + N−− − N+− − N−+) / (N++ + N+− + N−+ + N−−). Then combine the four E values using the displayed CHSH expression.
How far has entanglement been distributed experimentally?
There is no universal maximum. A prominent 2017 satellite experiment distributed entangled photons to ground stations 1,203 km apart and observed Bell-inequality violation. The practical limit depends on source, channel, detectors, protocol, and required confidence.
Model, assumptions, and sources
Updated: 17 July 2026 · Calculation review: 17 July 2026 (internal formula and interaction checks) · Publisher: Starlight Robotics. No individual credentialed reviewer is claimed. This educational calculator does not measure entanglement, certify a Bell test, correct accidental coincidences, or close detection, locality, freedom-of-choice, or memory loopholes.
Calculation validation
| Check | Expected | Page result |
|---|
| Ideal singlet, optimal angles | 2√2 = 2.828427 | 2.828 (rounding) |
| Werner p = 1/√2, V = 1, optimal angles | 2.000000 | 2.000 (rounding) |
| Equal counts in all four outcomes | E = 0 for each pair; S = 0 | 0.000 |
| 20 km, 0.2 dB/km | 4 dB total; 63.0957% per arm | 4.00 dB; 63.10% |
