The Born Rule
The Born rule maps a quantum state to measurement probabilities, connecting the abstract amplitude formalism to observable frequencies.
From amplitudes to probabilities
A pure quantum state is a normalized vector |psi> in a Hilbert space. Before measurement it carries complex amplitudes, not probabilities. The Born rule states that if we measure in an orthonormal basis {|k>}, the probability of outcome k is p(k) = |
General form with observables
For an observable A with spectral decomposition A = sum_a a P_a, where P_a projects onto the eigenspace of eigenvalue a, the probability of measuring a is p(a) =
Why squared modulus
Gleason's theorem shows that, in Hilbert spaces of dimension three or more, any consistent probability assignment to projectors that is additive on orthogonal outcomes must take the trace form Tr(P rho). The quadratic dependence on amplitude is therefore not an arbitrary choice but is forced by requiring a well-defined, basis-independent probability measure. The squaring is also what produces interference: cross terms between amplitudes survive because probability is a nonlinear function of the state.
import numpy as np
psi = np.array([1, 1j]) / np.sqrt(2) # normalized qubit
for k, ket in enumerate(np.eye(2)):
amp = ket.conj() @ psi
print(k, abs(amp)**2) # 0.5, 0.5The Born rule is a postulate: it is not derived from unitary evolution alone, and understanding its status is central to interpretations of quantum mechanics. What is settled is its operational content. Every prediction of quantum computing, from gate fidelities to sampling distributions, is a Born-rule statement about counts.
Relevance to computation
A quantum computer's output is sampled from the Born distribution of its final state in the measurement basis. Algorithm design is largely the art of shaping amplitudes so that this distribution concentrates on useful answers. Estimating an expectation value to precision epsilon by sampling requires on the order of 1/epsilon^2 repetitions, a direct consequence of the statistical nature of the rule.