Measurement

Measure a qubit in the standard basis and you get a single bit, 0 or 1. The odds are fixed by the state: the outcome 0 has probability |α|² and the outcome 1 has probability |β|², where α and β are the two amplitudes. This is the Born rule. One measurement never shows you α or β themselves; it shows one classical result, drawn at random.

Then the state changes. In the textbook rule, when an outcome occurs the state becomes the normalised part of the old state that agrees with it. For a qubit measured along Z that is |0⟩ or |1⟩, and on the sphere the arrow jumps to a pole. Press Measure in the panel “Measure one qubit, then measure it again” and watch it: the arrow lands on the axis you measured along, the chips record the axis and outcome of each press, and the line above them says how likely that outcome was. For a measurement of one part of a larger system, IBM’s course describes the same step as collapsing as far as the outcome requires and then normalising.

A second measurement along the same axis therefore has nothing left to decide: the state is already one of its two answers, so it repeats the first result every time. Preskill’s notes make the general point that orthogonal measurements are the repeatable kind, and that the more general measurements need not be. Measure along a different axis and the state is no longer an answer to that question. From |0⟩ an X measurement is a fair coin, because |0⟩ has equal weight on |+⟩ and |−⟩, and the qubit then sits on the X axis. Measure along Z once more and it is a fair coin again: the first Z result was erased by the X measurement in between.

IBM’s single-systems lesson shows why the basis matters. The states |+⟩ and |−⟩ give identical odds in the Z basis, so a Z measurement says nothing about which was prepared; applying a Hadamard first turns the difference into a certain 0 or 1. On the page the basis buttons do that rotation for you, and the chance of the first outcome along each axis is (1 + the arrow’s coordinate on that axis) / 2, which the page recomputes from the state.

Because a measurement uses the state up, you cannot find the odds by measuring one qubit over and over. You prepare it afresh each time, which the panel “Estimate the odds from many copies” does for a state you chose, and count. After N shots the share of 1s is an estimate of the chance of 1, with a typical error of √(p(1 − p)/N). That error shrinks like one over the square root of N: four times the shots halves it, a hundred times the shots cuts it by ten. Move the marker along the green line with the shots slider and the line settles into the grey band around the true value. Taking the square root of the estimated chances gives the size of each amplitude, but a Z count carries no sign or phase; the Bloch sphere page shows what the other bases add.

The Bloch sphere, where the basis changes what a histogram can see →

A caution about the word “collapse”. Everything above is the textbook rule: odds from the Born rule, then the updated state. Whether that update is a physical change, a change in what is known, or something that never happens at all in a larger description is not something the rule settles, and the interpretations of quantum mechanics answer it differently. Stanford’s encyclopedia sorts them: some read collapse as an update of knowledge, some as a real physical process, some replace it with a modified dynamics, and Everettian views drop the postulate. This page uses the rule and takes no side.

The shots on the page come from a seeded random generator, so the same presses give the same results. With more than one qubit, measuring one of them collapses only part of the state; that is for a later page.