Years 11–12 · Week 2 of 12
State Vectors and Normalisation
Learning goals
By the end, you can…
- Use ket and column-vector notation for |0⟩, |1⟩ and a general qubit state.
- Distinguish probability amplitudes from measurement probabilities.
- Test and, when appropriate, normalise a two-entry real state vector.
- Explain why global phase does not change measurement predictions.
What you already know
Connect to a familiar idea
A qubit state predicts measurement statistics. This lesson gives that state a precise vector form, beginning with real amplitudes before placing complex-number details in the extension.
- Squares, square roots and Pythagoras' theorem
- Two-entry column vectors
- Classical probability
Opening story
Start with something familiar
A navigation arrow needs both horizontal and vertical components, but not every pair of numbers has unit length. A qubit state vector also has two components and must satisfy a length condition called normalisation.
Plain-English explanation
Build the idea carefully
Basis vectors and amplitudes
The computational-basis states are |0⟩ = [1, 0]ᵀ and |1⟩ = [0, 1]ᵀ. A pure qubit state is |ψ⟩ = α|0⟩ + β|1⟩ = [α, β]ᵀ, where α and β are probability amplitudes. For real amplitudes, normalisation requires α² + β² = 1. The computational-basis probabilities are then P(0) = α² and P(1) = β². Amplitudes may be negative even though probabilities cannot be negative.
Normalisation and global phase
Worked example: [3/5, 4/5]ᵀ is normalised because 9/25 + 16/25 = 1. It predicts probabilities 9/25 for 0 and 16/25 for 1. The vector [3, 4]ᵀ is not a valid normalised state until divided by its length 5. Multiplying every amplitude by the same phase factor does not change any physical prediction for the isolated state. In the real-amplitude examples, [α, β]ᵀ and [−α, −β]ᵀ differ only by global phase; relative signs, such as [α, β]ᵀ versus [α, −β]ᵀ, can matter.
Try the model
Amplitude and normalisation editor
Enter two real amplitudes or move the sliders. Check normalisation, normalise a non-zero vector, and compare amplitudes with probabilities.
Ready. Adjust a control, then run the model.
What this model shows: This editor begins with real amplitudes. The Mathematical Extension adds complex amplitudes and phase.
Text alternative for this interactive
Text alternative: divide each entry of a non-zero vector by √(a² + b²). For [3, 4]ᵀ the length is 5, so the normalised entries are 3/5 and 4/5.
Expected observation: Normalising [3, 4]ᵀ produces [0.6, 0.8]ᵀ and predicted probabilities 0.36 and 0.64, which add to one.
Guided activity
Build and check three states
- Test [1, 0]ᵀ, [1/√2, 1/√2]ᵀ and [3, 4]ᵀ.
- Calculate each squared length before using the editor.
- Normalise any non-zero vector that fails the check.
- Record the resulting measurement probabilities and verify their total.
Evidence to collect: A table containing each original vector, its squared length, any normalised vector and non-negative probabilities whose total is one within rounding tolerance.
Glossary
Words to know
- ket
- A symbol such as |ψ⟩ used to denote a quantum state vector.
- computational basis
- The reference basis consisting of |0⟩ and |1⟩ for one qubit.
- state vector
- An ordered list of amplitudes representing a pure quantum state.
- amplitude
- A state-vector component whose squared magnitude contributes a probability.
- normalisation
- The condition that the total squared magnitude of a state vector equals one.
- global phase
- A common phase factor multiplying every amplitude, with no observable effect by itself.
- relative phase
- A phase difference between components that can affect interference.
Short recap
Keep these ideas
- |0⟩ and |1⟩ form the computational basis for one qubit.
- A valid pure-state vector has total squared magnitude one.
- The vector [3/5, 4/5]ᵀ predicts probabilities 9/25 and 16/25.
- Global phase does not alter predictions, but relative phase can alter later interference.
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
For a complex number z = a + bi, define |z|² = a² + b². A general qubit obeys |α|² + |β|² = 1. Multiplying the whole state by e^(iγ) adds global phase, whereas a factor e^(iφ) on only one component changes relative phase.
Try this
Check that [1/√2, i/√2]ᵀ is normalised. Compare it with [1/√2, 1/√2]ᵀ in the computational basis, then predict why a later basis change can distinguish them.
Adult support Teacher and parent notes
Discuss
- Why can an amplitude be negative while a probability cannot?
- What operation turns [3, 4]ᵀ into a unit vector?
- How is a relative minus sign different from multiplying the whole vector by −1?
Answer guidance
Students should square magnitudes, not raw complex values, and should state that the zero vector cannot be normalised.
Offline activity
Plot real amplitude pairs on coordinate paper. Mark the unit circle and connect each valid pair to its two computational-basis probabilities.
Safety
The activity is mathematical and simulation-based. Remind students that the coordinate picture represents states, not a qubit's physical path.
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
- The qubit in quantum computing Microsoft Learn · official documentation · checked 2026-08-02
- Quantum Computation and Quantum Information Cambridge University Press · textbook publisher page · checked 2026-08-02
Content review: Reviewed on 2026-08-02.