Measurement

Learning objectives

• Treat measurement as a basis-dependent physical operation.
• Apply the Born rule to computational-basis and X-basis measurements.
• Distinguish repeated preparation from repeated measurement of one system.
• Describe the post-measurement state in the ideal projective model.

A scene about asking a question

Alice stands before the observatory instrument again. The guide turns a ring around the dial before reading it. Alice notices that the same preparation can give different statistics depending on how the ring is aligned. The guide says: measurement is not merely looking. It is asking a physical question defined by a basis.

This chapter makes that statement precise for one qubit.

Measurement as a physical operation

In a projective measurement, we choose a measurement basis: a set of mutually exclusive alternatives represented by orthonormal basis states. The apparatus returns one classical outcome associated with one basis vector. It also changes the state assignment for that system.

For computational-basis measurement, the possible outcomes are 0 and 1, associated with ( |0\rangle ) and ( |1\rangle ). For X-basis measurement, the possible alternatives are ( |+\rangle ) and ( |-\rangle ). These are different questions.

Computational basis

• Alternatives: |0⟩ and |1⟩.
• Often called Z-basis measurement.
• Directly reads the standard circuit output bit.

X basis

• Alternatives: |+⟩ and |−⟩.
• Can be implemented conceptually by H then computational-basis measurement.
• Reveals phase information hidden from Z-basis probabilities.

The state is not enough to define probabilities; the selected measurement basis is part of the question.

Basis comparison

|0>|+>

Z / computational

050%

150%

X / H then Z

+100%

-0%

Computational-basis measurement and the Born rule

For

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

computational-basis measurement gives

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

Consider the state required throughout this chapter:

$$|\psi\rangle = \sqrt{0.7}|0\rangle + \sqrt{0.3}|1\rangle.$$

The probabilities are 70% for outcome 0 and 30% for outcome 1 when measured in the computational basis.

0NaN%

1NaN%

These frequencies emerge over many independent preparations and measurements, not by repeatedly extracting information from one unchanged qubit.

Measurement experiment simulator

Prepared state|+>sqrt(0.7)|0> + sqrt(0.3)|1>|0>Shots

Run experiment

Theoretical

070%

130%

Observed

00%

10%

Checkpoint

For |ψ⟩ = √0.7|0⟩ + √0.3|1⟩, what should many independent computational-basis trials approach?

Exactly 70 zeros followed by exactly 30 ones every 100 trials.Approximately 70% outcome 0 and 30% outcome 1.Both outcomes on every trial.

Check answerTry again

Post-measurement state

In the ideal projective model, once a computational-basis measurement returns 0, the post-measurement state is ( |0\rangle ). If the result is 1, the post-measurement state is ( |1\rangle ).

This is why repeated preparation and repeated measurement must be separated:

• Repeated independent preparation means creating the same initial state again and measuring each fresh system.
• Repeated measurement of one system means measuring after the first measurement has already changed the state.

If a single qubit prepared as ( \sqrt|0\rangle + \sqrt|1\rangle ) is measured and outcome 0 occurs, an immediate second computational-basis measurement ideally returns 0 with probability 1. It does not recreate the original 70/30 distribution.

1. 1

   Prepare |ψ⟩ freshly.

2. 2

   Measure once and get 0 or 1.

3. 3

   The post-measurement state matches that outcome for an immediate repeat in the same basis.

The 70/30 distribution is a statement about many independent preparations, not many reads of one unchanged qubit.

Basis dependence and X-basis measurement

Measurement outcomes depend on the selected basis. The X basis is

$$|+\rangle = \frac{|0\rangle + |1\rangle}{\sqrt{2}}, \qquad |-\rangle = \frac{|0\rangle - |1\rangle}{\sqrt{2}}.$$

Conceptually, measuring in the X basis can be implemented by applying H and then measuring in the computational basis. This works because

$$H|+\rangle = |0\rangle, \qquad H|-\rangle = |1\rangle.$$

For example, ( |+\rangle ) gives 50/50 outcomes in the computational basis, but an X-basis measurement returns the (+) outcome with probability 1. The basis defines which alternatives are being distinguished.

Statistical interpretation

A probability such as (P(0)=0.7) is not a promise about a short sequence of trials. In 10 shots, one might see 6 zeros or 8 zeros. In 10,000 independent preparations, the observed fraction is expected to be closer to 0.7. The theory predicts the distribution, not the exact order of outcomes.

Check your understanding

How can an X-basis measurement be implemented conceptually?

Apply H, then measure in the computational basis.Measure repeatedly until both outcomes appear.Ignore phase and use only |α|² and |β|².

Check answerTry again

Check your understanding

After measuring |ψ⟩ in the computational basis and obtaining 1, what is the ideal post-measurement state?

The original |ψ⟩.|1⟩.A hidden 70/30 mixture.Both |0⟩ and |1⟩.

Check answerTry again

Summary

Summary

• Measurement is a basis-dependent physical operation, not passive inspection.
• The Born rule converts amplitudes in the selected basis into probabilities.
• For ( \sqrt|0\rangle + \sqrt|1\rangle ), repeated independent computational-basis trials approach 70% outcome 0 and 30% outcome 1.
• After an ideal projective measurement, the post-measurement state corresponds to the observed result.
• X-basis measurement can be understood as applying H and then measuring in the computational basis.

References and further study

• Michael A. Nielsen and Isaac L. Chuang, Quantum Computation and Quantum Information.
• John Preskill, Lecture Notes for Physics 229: Quantum Information and Computation.
• Qiskit Textbook, measurement and single-qubit basis-change sections.