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.
| Setting | N++ | N+− | N−+ | N−− | Calculated E |
|---|---|---|---|---|---|
| (a,b) | — | ||||
| (a,b′) | — | ||||
| (a′,b) | — | ||||
| (a′,b′) | — |
Loss is not decoherence. This symmetric-link model uses attenuation to estimate detected counts only. Exponential visibility decay is off by default and must be selected as a phenomenological assumption.
| Setting pair | Correlation E | Estimated samples |
|---|---|---|
| E(a,b) | — | — |
| E(a,b′) | — | — |
| E(a′,b) | — | — |
| E(a′,b′) | — | — |
S = |E(a,b) − E(a,b′) + E(a′,b) + E(a′,b′)|
Prediction mode only.
αL across the two equal arms.
10−αL/20.
(Tη)² under the symmetric model.
Detected pairs per second.
Across all four settings.
Uncertainty assumes independent ±1 outcomes, fixed settings, fair sampling, negligible background/accidentals, and no drift or setting bias. It is a planning estimate, not a loophole-free statistical analysis.
The curve shows the selected prediction model. Markers identify the four setting pairs.
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.
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.
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.
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.
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.
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.
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.
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.
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.
No. Alice’s and Bob’s local outcomes are random. The correlation becomes visible only after they compare results through an ordinary classical channel.
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.
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.
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.
| 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% |
