Years 7–10 · Week 6 of 12
Qubits, Amplitudes and the Bloch Sphere
Learning goals
By the end, you can…
- Identify |0⟩ and |1⟩ as computational-basis states.
- Relate probability amplitudes to measurement probabilities conceptually.
- Use normalisation to check that total probability is one.
- Interpret the Bloch sphere as a representation of one pure qubit state.
What you already know
Connect to a familiar idea
A classical bit has one of two logical values. A qubit measurement in the computational basis also returns 0 or 1, but before measurement its pure state can contain amplitudes for both basis states and a relative phase.
- Probabilities for all outcomes total one.
- A bit has two computational labels.
- Coordinates can represent information without being a physical location.
Opening story
Start with something familiar
Two qubit preparations both predict 50% for 0 and 50% for 1 in an immediate computational-basis measurement. Later gates can still make them behave differently because their relative phases are different.
Plain-English explanation
Build the idea carefully
From amplitudes to probabilities
The computational basis uses |0⟩ and |1⟩ as two reference states for a qubit. A pure qubit state assigns a generally complex probability amplitude to each basis state; the squared magnitude of an amplitude gives the corresponding computational-basis measurement probability.
Reading a Bloch-sphere representation
Normalisation requires the two probabilities to sum to one, so a valid state cannot assign arbitrary independent probabilities. Relative phase affects how amplitudes interfere after later gates, even when two states have the same immediate 0/1 probabilities. The Bloch sphere maps every pure single-qubit state to a point on a sphere, apart from physically irrelevant global phase; it is a mathematical representation, not the qubit's shape or position.
Try the model
Single-qubit state explorer
Move the state point using labelled sliders, then compare the sphere, state description and predicted 0/1 probabilities.
Ready. Adjust a control, then run the model.
What this model shows: A mathematical pure-state visualisation with a fully operable two-dimensional probability-and-phase alternative.
Text alternative for this interactive
Probability and phase controls Use labelled buttons for |0⟩, |1⟩, |+⟩ and |−⟩ and read an accessible table of probabilities and relative phase.
Expected observation: States at the poles predict definite computational-basis outcomes, while equator states predict equal 0/1 probabilities but can have different relative phases.
Guided activity
Same probabilities, different phase
- Record P(0) and P(1) for |0⟩ and |1⟩.
- Compare the |+⟩ and |−⟩ presets.
- Note what is the same and what differs between |+⟩ and |−⟩.
- Predict why a later gate might distinguish them.
Evidence to collect: A table showing that |+⟩ and |−⟩ have the same computational-basis probabilities but different relative phases.
Glossary
Words to know
- qubit
- A quantum information unit described by a two-dimensional state space.
- computational basis
- The reference states |0⟩ and |1⟩ used for standard readout.
- probability amplitude
- A number whose squared magnitude contributes a measurement probability.
- normalisation
- The condition that probabilities over all outcomes sum to one.
- relative phase
- A phase relationship between state components that can affect interference.
- Bloch sphere
- A sphere-shaped representation of pure states of one qubit.
- global phase
- A common phase factor that does not change measurement predictions.
Short recap
Keep these ideas
- A qubit state contains amplitudes for computational-basis states.
- Squared amplitude magnitudes give probabilities that must total one.
- The Bloch sphere represents a pure single-qubit state and is not a physical ball.
Knowledge check
6 clear questions
Choose an answer for immediate feedback. You may retry, and your best submitted score is kept.
Go Further Optional extension for Years 9–10
For Years 9–10, write |ψ⟩ = α|0⟩ + β|1⟩ and use |α|² + |β|² = 1. Treat α and β as amplitudes without requiring complex-number calculations.
Try this
Check whether several supplied probability pairs can be normalised, then identify which additional phase information is missing.
Adult support Teacher and parent notes
Discuss
- Keep the distinction between state representation and physical implementation visible.
- Use |+⟩ and |−⟩ to show why relative phase matters beyond immediate probabilities.
Answer guidance
Students should connect squared magnitudes with probabilities and state that the Bloch sphere applies to one pure qubit.
Offline activity
Use state cards containing probability bars and phase arrows; match cards with equal probabilities but different phase.
Safety
No special hazards; the activity uses mathematical models only.
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.
- The qubit in quantum computing Microsoft Learn · official documentation · checked 2026-08-02
- Quantum information IBM Quantum Learning · official learning module · checked 2026-08-02
Content review: Reviewed on 2026-08-02.