Years 11–12 · Week 3 of 12
Measurement and the Born Rule
Learning goals
By the end, you can…
- Calculate computational-basis probabilities from a normalised qubit state.
- Distinguish one random measurement outcome from a predicted probability distribution.
- Describe the state update associated with an ideal projective measurement.
- Compare theoretical probabilities with finite sampled frequencies.
What you already know
Connect to a familiar idea
A normalised state |ψ⟩ = α|0⟩ + β|1⟩ contains amplitudes. The Born rule turns those amplitudes into probabilities for a specified measurement.
- Normalised state vectors
- Squared magnitudes of real or complex numbers
- Experimental probability and relative frequency
Opening story
Start with something familiar
A fair coin model predicts a half chance of heads, but ten tosses need not contain exactly five heads. Quantum predictions have the same important separation between an ideal probability and a finite sample, although the quantum state and measurement process are not a classical coin.
Plain-English explanation
Build the idea carefully
From amplitudes to probabilities
For |ψ⟩ = α|0⟩ + β|1⟩, a computational-basis measurement has P(0) = |α|² and P(1) = |β|². This is the Born rule for this basis. Normalisation guarantees that the two probabilities add to one. Worked example: for |ψ⟩ = (|0⟩ + √3|1⟩)/2, the probabilities are P(0) = 1/4 and P(1) = 3/4. One shot still records only one outcome; it does not display the probability distribution.
One shot, many shots and state update
After an ideal projective computational-basis measurement, recording 0 leaves the qubit in |0⟩ and recording 1 leaves it in |1⟩. Measuring the same system again immediately in the same ideal basis repeats that outcome, unless another operation or disturbance intervenes. Shot counts fluctuate around theoretical probabilities. Repeating the entire preparation-and-measurement procedure estimates the distribution; each shot must begin with a freshly prepared copy of the intended state.
Try the model
Prepare, measure and compare
Choose a state, calculate its Born-rule probabilities, then run 1, 10, 100 and 1000 freshly prepared shots with an optional fixed seed.
Ready. Adjust a control, then run the model.
What this model shows: Random browser sampling illustrates finite-shot statistics; it does not reproduce every feature of laboratory hardware.
Text alternative for this interactive
Text alternative: expected counts for 100 shots are about 25 zeros and 75 ones, but nearby totals such as 22 and 78 are compatible with finite sampling.
Expected observation: For the state (|0⟩ + √3|1⟩)/2, larger shot samples tend towards 25% zero and 75% one, while a single shot is only zero or one.
Guided activity
Prediction before simulation
- Calculate P(0) and P(1) for three supplied normalised states.
- Predict approximate counts for 100 shots before pressing run.
- Run each circuit twice with different seeds and record the counts.
- Explain why differing samples can support the same probability model.
Evidence to collect: A calculation table showing squared magnitudes, predicted counts, two sampled histograms and a comparison that separates sampling variation from an incorrect theoretical prediction.
Glossary
Words to know
- Born rule
- The rule that obtains measurement probabilities from squared amplitude magnitudes.
- computational-basis measurement
- A measurement with possible qubit outcomes labelled 0 and 1.
- shot
- One preparation and measurement of a quantum circuit.
- relative frequency
- The number of times an outcome occurs divided by the total number of trials.
- projective measurement
- An idealised measurement associated with orthogonal outcome subspaces.
- state update
- The change in the state description conditioned on a measurement outcome.
- sampling variation
- Random differences between finite observed frequencies and ideal probabilities.
Short recap
Keep these ideas
- The Born rule uses squared amplitude magnitudes.
- One shot produces one outcome rather than a complete distribution.
- An ideal measurement outcome determines the corresponding post-measurement state.
- Repeated fresh preparations estimate probabilities through relative frequencies.
Knowledge check
7 clear questions
Choose an answer for immediate feedback. You may retry, and your best submitted score is kept.
Mathematical Extension Optional extension for Year 12
A bra such as ⟨ψ| is the conjugate transpose of the ket |ψ⟩, and putting a bra beside a ket forms an inner product. A projector maps a state onto a chosen subspace. For normalised |ψ⟩ and P₀ = |0⟩⟨0|, the probability of outcome 0 is ⟨ψ|P₀|ψ⟩. If outcome 0 occurs, the normalised post-measurement state is P₀|ψ⟩ divided by the square root of that probability.
Try this
Use P₀ = [[1, 0], [0, 0]] and P₁ = [[0, 0], [0, 1]] to calculate both probabilities and conditional post-measurement states for [1/2, √3/2]ᵀ.
Adult support Teacher and parent notes
Discuss
- Why must every shot begin with the intended preparation?
- What does a single measurement tell us about an unknown state?
- How can we decide whether a sample is broadly consistent with a prediction?
Answer guidance
Accept approximate shot counts, not only exact expected counts. Require students to identify the basis and to square magnitudes.
Offline activity
Use a biased spinner to model finite sampling, then explicitly list the ways the spinner analogy differs from a quantum measurement.
Safety
No physical quantum apparatus is used. Avoid language suggesting that human awareness causes the measurement outcome.
Sources and further reading
Checked references for this lesson
These sources support the lesson’s main scientific claims. Links open on the source organisation’s site.
- Quantum information IBM Quantum Learning · official learning module · checked 2026-08-02
- Introduction IBM Quantum Learning · official learning module · checked 2026-08-02
- The qubit in quantum computing Microsoft Learn · official documentation · checked 2026-08-02
Content review: Reviewed on 2026-08-02.