Every one of the N dots on the right starts prepared in the identical qubit superposition |ψ⟩ = cos(θ/2)|0⟩ + e^(iφ) sin(θ/2)|1⟩, orbiting a shared point — a picture of coherent superposition, not N different particles.
Born rule: P(0) = cos²(θ/2)
P(1) = sin²(θ/2)
Collapse: each trial samples one outcome from P(0)/P(1);
the state instantly becomes |0⟩ or |1⟩ — no in-between.
The small disk above is a genuine 2D (azimuthal-equidistant) projection of the Bloch sphere used for a single qubit: the north pole |0⟩ maps to the disk's center, the south pole |1⟩ maps to its rim, radial distance encodes the polar angle θ, and the angle around the disk encodes the phase φ. Dragging the dot is mathematically the same operation as dragging a point over the sphere's surface — nothing about the qubit's geometry is lost by flattening it this way.
Pressing Measure All N performs N *independent* measurements: every dot independently samples an outcome with Math.random() weighted by its own P(0), then snaps — collapses — to the |0⟩ well (blue, left) or the |1⟩ well (orange, right). No single measurement ever tells you θ; only the outcome frequency over many trials approaches the true probability, exactly as the law of large numbers predicts.
- θ slider / disk radius — sets the amplitudes and hence the true Born-rule probabilities (θ=90° is a fair 50/50 superposition, θ=0°/180° are the definite states |0⟩/|1⟩).
- φ slider / disk angle — the relative phase between the two amplitudes; it changes the superposition's orbital orientation but — crucially — never changes the measurement probabilities, since Born-rule probabilities only depend on |amplitude|².
- Re-prepare — returns every dot to the uncollapsed superposition cloud, ready for a fresh measurement.
- Auto-repeat — continuously re-prepares and re-measures the whole ensemble, accumulating trials so you can watch the running frequency converge on the theoretical curve.
Real-world relevance: this is exactly how a quantum-computer benchmark works — you cannot read out an amplitude directly, so you run the same circuit thousands of times and histogram the bitstrings, exactly as this ensemble does.