Measurement and the Born rule

In a classical program, reading a variable is free and repeatable: you can print it, print it again, and inspect every bit, and the value just sits there unchanged. Reading a qubit is nothing like that. A measurement returns a single bit, chosen at random, and in the act of returning it destroys the very superposition you were trying to inspect. This lesson is the rule that governs that read, and the wall it puts between you and the amplitudes.

Play first: measure once, then measure a thousand fresh copies

Set a state with the slider, then try the two buttons. Measure once collapses it: you get a single outcome, and measuring again just repeats that same value - the superposition is gone. Run 1000 shots does something different: it prepares a fresh copy of the state each time and measures that, so the tally slowly reveals the odds. Watch that the histogram closes in on the probabilities the slider predicted, while no single run ever showed you the amplitudes.

The gap between those two buttons is the whole lesson. One shot is one bit and a wrecked state. Many shots on identically prepared copies recover the distribution - but only the distribution, never the signs or phases inside the amplitudes, and only if you can make copies from a known recipe, which for an unknown state you cannot.

The Born rule

The rule connecting amplitudes to what you actually see is the Born rule. Measuring $|\psi ⟩=\alpha |0⟩+\beta |1⟩$ in the standard basis:

This is the bridge back to ordinary probability that the amplitudes lesson set up: square the magnitude and you are back among non-negative numbers that add to one. Normalization, $|\alpha {|}^2+|\beta {|}^2=1$, is exactly what guarantees these two probabilities cover all the possibilities. The squaring is not a convention we could swap out; it is derived from demanding that total probability stay one under the operations quantum mechanics allows.

Two consequences follow, and they are the load-bearing facts for everything later. First, measurement is irreversible: the pre-measurement amplitudes are gone, and no operation recovers them from the single bit you kept. Second, amplitudes are not observable. The best you can do is prepare many copies and estimate $|\alpha {|}^2$ from the frequencies, which is what the thousand-shot histogram did. That is a statistical shadow of one squared magnitude, not a readout of the state.

import numpy as np

# State |psi> = alpha|0> + beta|1>, here 0.6|0> + 0.8|1>.
alpha, beta = 0.6, 0.8
p0 = abs(alpha)**2   # 0.36  = P(0), the Born rule
p1 = abs(beta)**2    # 0.64  = P(1)

# A single measurement: one bit, sampled by (p0, p1). The state then collapses,
# so a repeat measurement of that same qubit returns the same bit.
one_shot = np.random.choice([0, 1], p=[p0, p1])

# 1000 fresh copies, each measured once, recover the distribution - not alpha, beta.
shots = np.random.choice([0, 1], size=1000, p=[p0, p1])
counts = np.bincount(shots)   # ~[360, 640], approaching [p0, p1] * 1000

Check yourself

The Born rule and collapse follow Quantum Country's measurement treatment^[1] and the measurement postulate in Nielsen & Chuang section 2.2.3^[2].