A short, modular learning experience: Overview → Simulator → Bell test → Quiz.
Overview
Learning goals:
Understand what entanglement is and why the singlet state is important.
Use an interactive simulator to observe measurement correlations.
Run a Bell-CHSH test and interpret the violation of local realism.
Quick intuition
Entangled particles share correlations that cannot be explained by assigning independent properties to each particle. The singlet state is a common example: if Alice measures one qubit along an axis and obtains a result, Bob's measurement along the same axis will give the opposite result.
Theme: Dark scientific
Interactive Simulator
Visualize an entangled pair and measure them in chosen bases. Outcomes are shown as 0/1; underlying ±1 values are used for statistics.
θ₁ = 0°
θ₂ = 0°
State: Singlet (|01⟩ − |10⟩)/√2
Trials
0
Correlated (%)
—
Alice outcome
—
Bob outcome
—
Bell-CHSH quick runner
Enter two settings for Alice (A, A') and Bob (B, B'), then run many trials to estimate S (CHSH). Typical quantum settings can produce S > 2 (violation of local realism).
Estimated S: —
Bell's Inequality — CHSH
Bell-CHSH compares correlations between four measurement pairs: (A,B), (A,B'), (A',B), (A',B'). Compute expectation values E for each pair and combine as:
S = |E(A,B) − E(A,B')| + |E(A',B) + E(A',B')|
Classical local hidden variable theories satisfy S ≤ 2. Quantum mechanics predicts S up to 2√2 ≈ 2.828 for optimal choices.
Interpretation
When S > 2, the observed correlations cannot be explained by any local model where measurement outcomes are pre-assigned based only on local information. Experiments with photons and spins have repeatedly observed violations consistent with quantum predictions.
Practical notes
Real experiments must account for detector inefficiencies, timing (locality) loopholes, and noise.
The simulator here uses idealized sampling based on the singlet correlation model — it is educational, not experimental.
Quiz & Activities
Q1. If Alice and Bob measure the singlet along the same axis, what is the correlation of their outcomes?
Q2. What does a violation of Bell’s inequality (S > 2) imply?
Q3. In the singlet state, if Alice measures her qubit and finds it in |0⟩, what will Bob find when measuring in the same basis?
Q4. What physical property determines the correlation strength between two measurement directions?
Q5. Which expression correctly represents the quantum mechanical correlation in the polarization version of the Bell test?
Suggested activities
Set both angles equal — measure several pairs to see near 100% anti-correlation.
Change Bob's angle by 45° — plot the correlation vs angle difference and compare with the predicted curve.
Run the Bell test with canonical angles (0°,45°,22.5°,67.5°) and observe S > 2.
References & Further Reading
Einstein, Podolsky, Rosen — "Can Quantum-Mechanical Description of Physical Reality Be Considered Complete?" (1935).
Bell, J. S. — "On the Einstein Podolsky Rosen paradox" (1964).
Aspect et al. — Experimental tests of Bell's inequalities (1980s onward).