The qubit's state is |ψ⟩ = α|0⟩ + β|1⟩, with complex amplitudes α, β satisfying |α|² + |β|² = 1. The left phasor plots α, the right plots β, rotating relative to each other at rate ω — that phase drift is what a real energy splitting between |0⟩ and |1⟩ produces. Coupling to an uncontrolled environment doesn't change the populations |α|², |β|², but it randomizes the relative phase between them, so the off-diagonal density-matrix term ρ₀₁ = αβ* decays exponentially at rate γ. That decay — not a change in P(0)/P(1) — is decoherence: a coherent superposition curdling into a classical either/or mixture with the same odds.
ρ₀₁(t) = α·β* · e^(−γt)
purity = Tr(ρ²) = |α|⁴ + |β|⁴ + 2|ρ₀₁|²
P(outcome k) = |⟨k|ψ⟩|² (Born rule)
- θ — sets the initial split: α = cos(θ/2), β = sin(θ/2), so θ=90° is an even 50/50 superposition.
- ω — relative phase precession speed between the two phasors; watch the interference fringe on the coherence strip oscillate as they wind past each other.
- γ — environment coupling strength; higher γ collapses the coherence strip toward zero faster, after which repeated measurements behave exactly like a classical coin biased by P(0)/P(1).
- Measure — samples one outcome via the Born rule and collapses the state to that basis vector; coherence resets to full since a freshly collapsed state is pure again.
Run Auto-measure for a while and watch the outcome tally converge toward the P(0):P(1) ratio shown above — that convergence, over many identical trials, is the empirical meaning of "probability" in quantum mechanics.