Measurement

Learning Objectives

• State the Born rule for computational-basis measurement.
• Explain post-measurement state update in a single trial.
• Distinguish repeated preparation from repeated measurement of one system.

Scene

Alice reaches a gate that opens only when a detector records a definite answer. The guide clarifies: measurement is a physical interaction with a chosen basis, not a passive look at a hidden classical note.

Measurement basis and Born rule

For a qubit measured in the computational basis $\{|0\rangle, |1\rangle\}$:

$$|\psi\rangle = \alpha|0\rangle + \beta|1\rangle,\qquad P(0)=|\alpha|^2,\quad P(1)=|\beta|^2.$$

Post-measurement state

In one run:

• If outcome is 0, the post-measurement state is |0⟩.
• If outcome is 1, the post-measurement state is |1⟩.

This update is sometimes called collapse. It is a statement about the state assigned after obtaining an outcome in that run.

Numeric example

|ψ⟩ = sqrt(0.7)|0⟩ + sqrt(0.3)|1⟩

Use repeated preparation to estimate probabilities experimentally.

If we repeatedly prepare this same state and then measure in the computational basis, we expect approximately 70% zeros and 30% ones over many trials.

Repeated preparation vs repeated measurement

• Repeated preparation: prepare the same initial state each trial, then measure once. This estimates Born probabilities.
• Repeated measurement on one already-measured system: after the first projective measurement, immediate repetition in the same basis returns the same outcome with probability 1.

Check Your Understanding

Why do we repeat preparation when estimating quantum probabilities?

Because one measurement already gives the full probability distribution.Because one outcome is sampled per trial; distributions appear across many identically prepared trials.Because quantum theory is invalid for single runs.

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Summary

Measurement returns one basis outcome per trial, with probabilities set by the Born rule. Post-measurement states depend on the recorded outcome. Careful distinction between single-shot outcomes and ensemble statistics is essential for correct quantum reasoning.