Postulate 3: Measurement

Postulate 2 said the possible outcomes of measuring an observable are its eigenvalues. Postulate 3 completes the picture: it states with what probability each outcome occurs and what the state becomes after the measurement. This is where quantum mechanics becomes irreducibly probabilistic.

The measurement postulate

Postulate 3. Measuring an observable $\hat{A} = \sum_k a_k \hat{P}_k$ (with orthogonal projectors $\hat{P}_k$ onto the eigenspace of eigenvalue $a_k$ ) on a normalized state $|\psi\rangle$ yields outcome $a_k$ with probability
$p(a_k) = \langle\psi|\hat{P}_k|\psi\rangle,$
and immediately after the measurement the state becomes the normalized projection
$|\psi'\rangle = \frac{\hat{P}_k|\psi\rangle}{\sqrt{p(a_k)}}.$

These two clauses are the Born rule (the probability) and the projection postulate (the state update, sometimes called "collapse"). We devote the next two lessons to each in turn; here we establish the framework and check that it is self-consistent.

Projectors and completeness

For a non-degenerate eigenvalue the projector is simply $\hat{P}_k = |a_k\rangle\langle a_k|$ . The projectors are orthogonal and complete, meaning they sum to the identity:

\hat{P}_j \hat{P}_k = \delta_{jk}\hat{P}_k, \qquad \sum_k \hat{P}_k = \hat{I}.

Completeness is what makes the total probability equal to one:

\sum_k p(a_k) = \sum_k \langle\psi|\hat{P}_k|\psi\rangle = \langle\psi|\Big(\sum_k \hat{P}_k\Big)|\psi\rangle = \langle\psi|\hat{I}|\psi\rangle = \langle\psi|\psi\rangle = 1.

So a normalized state automatically produces a valid probability distribution over outcomes — the reason Postulate 1 demanded normalization.

Measurement in the computational basis

The most common measurement on a qubit register is in the computational basis, with projectors $\hat{P}_k = |k\rangle\langle k|$ . For a qubit $|\psi\rangle = \alpha|0\rangle + \beta|1\rangle$ ,

p(0) = \langle\psi|0\rangle\langle 0|\psi\rangle = |\langle 0|\psi\rangle|^2 = |\alpha|^2, \qquad p(1) = |\beta|^2.

If the result is $0$ , the post-measurement state is

|\psi'\rangle = \frac{|0\rangle\langle 0|\psi\rangle}{\sqrt{|\alpha|^2}} = \frac{\alpha}{|\alpha|}\,|0\rangle,

which is just $|0\rangle$ up to an irrelevant global phase. Subsequent measurements in the same basis return $0$ with certainty — the hallmark of collapse.

Repeatability and irreversibility

Two features distinguish quantum measurement from classical observation:

Repeatability. Immediately re-measuring the same observable gives the same eigenvalue with probability 1, because $\hat{P}_k|\psi'\rangle = |\psi'\rangle$ . The first measurement leaves the system in an eigenstate.
Irreversibility / disturbance. The projection generally destroys information about the original superposition. Measuring $\hat{Z}$ on $|+\rangle = \tfrac{1}{\sqrt2}(|0\rangle + |1\rangle)$ collapses it to $|0\rangle$ or $|1\rangle$ , erasing the relative phase that defined $|+\rangle$ . You cannot undo a measurement.

The role of this postulate

Postulate 3 is the link between the abstract state vector and the numbers an experiment records. It turns amplitudes into probabilities and tells us how observation feeds back on the system. The next two lessons make each half quantitative: the Born rule for computing $p(a_k)$ , and the projection postulate for computing $|\psi'\rangle$ .