Quantum Entanglement Distance Calculator

How this works

• CHSH formula: For a singlet, E(a,b) = -cos(2(a-b)). S = |E(a,b) - E(a,b') + E(a',b) + E(a',b')|. Quantum max is 2√2 ≈ 2.828.
• Visibility: Imperfections reduce correlations. We model visibility as V = V0 · exp(-L / Ld). S scales as 2√2 · V.
• Loss and coincidences: Each arm sees fiber loss 10^(-α·L/2 / 10). Coincidences ≈ rate · (T · η)² where η is detector efficiency.
• Violation threshold: S > 2 ↔ V > 1/√2 ≈ 0.707. The “max distance” solves that cutoff.
• Spooky action note: Changing one polarizer flips the distant statistics only in the joint correlations; single-arm stats stay random.

Why this matters (and how to read the numbers)

Entanglement is the headline act of quantum mechanics: two particles share correlations that exceed anything classical physics allows. The CHSH inequality gives a clean test: any local hidden-variable model must obey S ≤ 2, while quantum mechanics predicts up to S = 2√2 ≈ 2.828 for an ideal singlet. This calculator turns the abstract math into a tactile checklist: you can dial in distance, optical loss, decoherence, detector efficiency, and polarization angles, then see whether your scenario still beats the classical limit. It also returns an approximate coincidence rate so you can gauge how long you would need to collect statistically convincing data.

The visibility slider captures how “clean” your entangled state is. Real sources suffer from multi-pair noise, polarization drift, or imperfect alignment. In the model, visibility decays exponentially with distance over a decoherence length L_d. The rule of thumb is simple: Bell violation needs visibility > 1/√2 ≈ 0.707. The “Max distance” readout solves for where your initial visibility V₀ would fall to that threshold; beyond it, the S value drops below 2 and the violation disappears. Changing L_d shows how sensitive long links are to dephasing and timing jitter.

Loss is the second killer. Fiber attenuation cuts the photon flux as 10^-αL/10 per arm (here we assume symmetric arms of length L/2 each). Detectors then shave off more counts according to their efficiency η. The coincidence rate scales as (T·η)², so small improvements in efficiency pay double dividends. Play with the presets: a short lab bench shows a high S with huge coincidence rates; 20 km of telecom fiber still violates Bell if visibility and detectors are decent; a 100 km free-space link trades lower loss for pointing and turbulence challenges, captured here by a larger decoherence length but modest efficiencies.

The angles box defaults to 0°, 45°, 22.5°, 67.5°—the classic CHSH optimum for a singlet. If you change them to 0°, 90°, 45°, 135° you will see S shrink toward the classical limit, illustrating how geometry matters. The status line under the buttons spells out whether the current S exceeds 2 and how close you sit to the boundary. The bar below the outputs provides a quick visual of “spooky action”: at 100% the bar hits the classical limit; above 141% (≈ 2√2 / 2) you are in the fully quantum regime; if it shrinks below 100% you have drifted into classical territory.

Use this page as a teaching aid: show that single-arm statistics remain random even while joint correlations look coordinated, and emphasize that no superluminal signal is sent. For students, it connects the algebra of CHSH with experimental realities—loss budgets, detector specs, and run times. For engineers, it offers a quick sanity check when sketching quantum key distribution links or table-top demos. Because everything runs locally, you can tweak parameters freely without sending data anywhere. Experiment with aggressive losses or low visibilities to see how easy it is to “lose” entanglement in practice, and let the distance estimate guide what upgrades (better detectors, shorter links, lower-loss fiber) matter most for keeping your Bell violation alive.