The Born Rule

The Born rule states that the probability of a measurement outcome equals the squared magnitude of its amplitude.

Amplitudes to probabilities

The Born rule is the bridge between the quantum state and observable statistics. For a state |psi> = a|0> + b|1>, the probability of measuring 0 is |a|^2 and of measuring 1 is |b|^2. More generally, the probability of outcome x is ||^2, the squared magnitude of the amplitude for that outcome.

Why squared magnitude

Amplitudes are complex and can cancel; probabilities are real, non-negative, and sum to one. Taking the squared magnitude produces valid probabilities while discarding overall phase. The squaring is also what makes interference visible: two amplitudes are summed first, then squared, so cross terms appear that pure probability addition would miss.

General form

For a density matrix rho and a measurement with projector P_x, the rule reads prob(x) = Tr(P_x rho). This covers mixed states and general measurements and reduces to ||^2 for a pure state. It is the single formula that connects theory to every experiment.

Consequences

• Global phase drops out, since |e^{i alpha} a|^2 = |a|^2
• Normalisation = 1 guarantees probabilities sum to one
• Orthogonal states are perfectly distinguishable (overlap zero)

Status of the rule

The Born rule is a postulate of quantum mechanics, not derived from the others within the standard formulation, though attempts to justify it (Gleason's theorem, decision-theoretic arguments) exist. Empirically it is among the most precisely confirmed statements in physics. Every prediction a quantum algorithm makes about its output distribution ultimately rests on it.