The Measurement Postulate

Quantum mechanics comes with a set of postulates — formal rules that tell you how systems evolve and what you observe when you look at them. The measurement postulate is the one that connects the abstract state vector to the concrete numbers that appear on a detector.

What measurement does

A qubit's state can always be written as a superposition of the two computational basis states:

$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle,$

where $\alpha$ and $\beta$ are complex amplitudes satisfying $|\alpha|^2 + |\beta|^2 = 1$.

When you measure this qubit in the computational basis, two things happen simultaneously:

1. You get a definite classical outcome, either 0 or 1, chosen at random. The probability of each outcome is given by the Born rule:

   $P(\text{outcome }0) = |\alpha|^2, \qquad P(\text{outcome }1) = |\beta|^2.$

2. The state collapses. Immediately after the measurement the qubit is no longer in the superposition $\alpha|0\rangle + \beta|1\rangle$. It is now in the basis state that matches the outcome: $|0\rangle$ if you saw 0, or $|1\rangle$ if you saw 1.

The two effects are inseparable. Measurement is irreversible — the prior amplitudes are gone — and the outcome is genuinely random (not just unknown to you).

Before vs. after

It is worth being precise about the timeline:

• Before measurement: the qubit holds a superposition. Both $|0\rangle$ and $|1\rangle$ components are physically present; interference effects are possible.
• During measurement: an interaction with a macroscopic device forces a single outcome according to the Born rule probabilities.
• After measurement: the qubit is in the corresponding basis state. If you measure again immediately, you get the same outcome with certainty — the first measurement has already collapsed the state.

This last point is the hallmark of projective measurement: once the collapse has happened, re-measuring gives no new information.

Try it

Measure the qubit while it is in $|0\rangle$ (the default starting state). Because $|\alpha|^2 = 1$ and $|\beta|^2 = 0$, the distribution should be entirely on outcome 0.

Switch to the Probabilities tab. You should see a single bar at 0 with height 1 and an empty bar at 1 — exactly what the Born rule predicts for a definite state.

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