Superposition & Measurement

A classical bit is either 0 or 1. A qubit can be in a superposition of both — written |ψ⟩ = α|0⟩ + β|1⟩, where α and β are complex numbers called probability amplitudes. The probability of measuring |0⟩ is |α|², and |1⟩ is |β|², with |α|² + |β|² = 1.

This isn't about not knowing the state — the qubit genuinely exists in both states simultaneously. Until you measure it.

Every single-qubit state can be parameterized by two angles: θ (how much |0⟩ vs |1⟩) and φ (the relative phase between them). Drag the sliders to explore the full space of qubit states. Try the preset buttons for the six cardinal states of the Bloch sphere.

Notice: Changing φ doesn't affect the measurement probabilities — both |+⟩ and |−⟩ give 50/50 outcomes when measured in the {|0⟩, |1⟩} basis. The phase only matters when you change your measurement basis, or when amplitudes interfere.

When you measure a qubit, the superposition collapses to a definite state. A state with 70% |0⟩ probability will sometimes give |0⟩ and sometimes |1⟩ — but each individual measurement is completely random. Click "Measure" to collapse the state, then prepare a new one to try again.

A coin under a cup is either heads or tails — we just don't know which. A quantum "coin" is genuinely in both states at once. The statistics look the same (both give 50/50), but the underlying reality is profoundly different. This distinction has real consequences for interference and entanglement.

Classical: Ignorance: The coin has a definite state; our uncertainty reflects lack of knowledge. No interference is possible.

Quantum: Superposition: The qubit genuinely has no definite value. This enables interference effects that classical randomness cannot produce.

Max Born discovered that |α|² gives the probability of each outcome. But you only see this statistically — prepare the same state thousands of times, measure each copy, and watch the frequencies converge to the predicted probabilities. Watch the convergence trace approach the theoretical value.

This is the deepest insight of quantum mechanics: nature doesn't add probabilities — it adds complex amplitudes, then squares. When two paths lead to the same outcome, their amplitudes add as vectors in the complex plane. Aligned phases give constructive interference (higher probability); opposite phases give destructive interference (lower probability).

The key formula: P = |α₁ + α₂|² ≠ |α₁|² + |α₂|². The cross terms (interference terms) are what make quantum mechanics different from classical probability. Try the "Destructive" preset to see two amplitudes that perfectly cancel, giving zero probability despite both paths being individually likely.