A flat, purely 2D companion to the 3D Bell-state visualizer: rotate Alice's and Bob's measurement bases independently, sample real Born-rule outcomes one at a time or in batches of 200, and run a genuine Monte-Carlo CHSH test whose sampled statistic climbs toward the Tsirelson bound of 2√2.
The 3D scene fixes the CHSH angles and animates a beam. Here you directly drag Alice's and Bob's measurement-basis dials to any angle, watch the exact Born-rule probabilities recompute live via a real 2-qubit basis rotation, and see a sampled histogram converge to those probabilities as trial count grows.
Pick a Bell state, set Alice's and Bob's basis angles with the sliders, then Measure once or run 200 trials to build the histogram. Press CHSH Test to sample 400 trials at each of the four standard angle pairs (0°, 45° for Alice; 22.5°, 67.5° for Bob) and read the resulting S statistic against both bounds.
Every probability comes from rotating the actual 4-amplitude state vector by (U(a)⊗U(b)) and squaring the resulting amplitudes — the same projective measurement math used in real Bell tests, not a scripted animation.
This simulation keeps the full 4-amplitude two-qubit state vector for one of the four Bell states and lets you choose an arbitrary measurement basis angle for Alice and for Bob independently. Rotating a basis by angle θ changes the qubit's measurement eigenvectors to cos(θ/2)|0⟩+sin(θ/2)|1⟩ and −sin(θ/2)|0⟩+cos(θ/2)|1⟩; applying that rotation to both qubits and squaring the resulting amplitudes gives the exact joint outcome probabilities via the Born rule.
Sampling many trials at fixed angles builds a histogram that converges to those exact probabilities — real statistical convergence, not a scripted result. The CHSH test runs 400 sampled trials at each of four angle pairs (Alice at 0°/45°, Bob at 22.5°/67.5°) and combines the resulting correlations E(a,b) into S = |E(a,b)−E(a,b′)+E(a′,b)+E(a′,b′)|, which for these angles and any maximally entangled Bell state approaches the Tsirelson bound 2√2≈2.828, exceeding the classical limit of 2.
How is this different from the 3D Quantum Entanglement simulation?
The 3D version renders an animated WebGL scene with fixed CHSH angles. This 2D version is an interactive lab: you drag the measurement-basis angle for each qubit yourself, see the exact quantum probabilities recompute in real time, and watch sampled trial statistics converge to them — a different mechanic built around basis rotation rather than a fixed animated demonstration.
What does rotating the measurement basis mean physically?
Instead of always measuring "spin up vs spin down" along a fixed axis, you can measure along any axis in the qubit's plane. The two projected outcomes for that axis still add up to probability 1, but the correlation between Alice's and Bob's outcomes depends on the angle between their chosen axes — precisely what Bell tests exploit.
Why do the histogram and exact bars sometimes differ?
The exact bars are the theoretical Born-rule probabilities at the current angles. The histogram is built from actual random sampling of those probabilities, so with few trials it fluctuates; run more trials and it converges to the exact bars, exactly as real statistics would.
The Born rule says the probability of a measurement outcome equals the squared magnitude of its quantum amplitude. After rotating the two-qubit state into the chosen measurement basis, each of the four joint outcomes' probability is literally the square of its resulting amplitude — computed fresh every time you move a slider.
At Alice angles 0°/45° and Bob angles 22.5°/67.5°, the correlation function for a maximally entangled state is optimally spread across the four terms of the CHSH combination, reaching the Tsirelson bound 2√2≈2.828 — the maximum any quantum state can achieve, still below the algebraic maximum of 4 but above the classical local-hidden-variable limit of 2.
Both are shown. "S" is computed from 400 simulated random measurement trials per angle pair, so it fluctuates trial to trial like a real experiment. "exact S" is the closed-form value from the state vector, shown for comparison so you can see the sampled estimate converging toward it.