Years 7–10 · Week 8 of 12
Multiple Qubits and CNOT
Learning goals
By the end, you can…
- Label two-qubit computational-basis states as 00, 01, 10 and 11.
- Trace all four basis inputs through a CNOT truth table.
- Identify the control and target roles in CNOT.
- Distinguish separable correlation from entanglement at an introductory level.
What you already know
Connect to a familiar idea
A one-qubit state needs two basis amplitudes. Two qubits have four computational-basis labels, and a general pure state assigns an amplitude to each of |00⟩, |01⟩, |10⟩ and |11⟩.
- One qubit uses basis states |0⟩ and |1⟩.
- X exchanges 0 and 1.
- A truth table checks every basis input.
Opening story
Start with something familiar
A circuit has two wires. The first qubit acts as a control: when it is 0, the second qubit is left alone; when it is 1, the second qubit is flipped. Testing all four basis inputs reveals the complete CNOT rule.
Plain-English explanation
Build the idea carefully
A four-state basis for two qubits
Two qubits have four computational-basis states—|00⟩, |01⟩, |10⟩ and |11⟩—because each qubit contributes two basis labels. For a CNOT with the first qubit as control and the second as target, the target flips only when the control is 1.
Control, target and correlation
The ideal basis mapping is 00→00, 01→01, 10→11 and 11→10 when labels are written control first. CNOT can copy a known computational-basis value into a target initially at 0, but it cannot clone an arbitrary unknown quantum state. Correlated outcomes can occur in ordinary classical mixtures or separable quantum states. For a pure joint state, entanglement means the state cannot be written as one state for each qubit. More generally, a separable mixed state may be a probabilistic mixture of product states.
Try the model
Two-wire state and CNOT explorer
Choose each basis input, apply CNOT and verify the truth table; then try H on the control followed by CNOT and inspect the joint state.
Qubit 0 is the least-significant state-vector bit. Displayed basis labels read q(n−1)…q0.
Ready. Adjust a control, then run the model.
What this model shows: An ideal two-qubit state-vector model with the control bit written first.
Text alternative for this interactive
Two-qubit transition table Select one of four basis inputs and read the CNOT output, then inspect an accessible Bell-state probability table.
Expected observation: CNOT flips the target only for control 1, and H followed by CNOT from |00⟩ produces joint probabilities only for 00 and 11.
Guided activity
Prove the CNOT rule by cases
- Predict the output for each computational-basis input.
- Run all four cases and correct any prediction.
- Group the rows by control value and describe the target rule.
- Run H then CNOT from 00 and record the non-zero joint outcomes.
Evidence to collect: A complete CNOT truth table plus a joint probability table showing only 00 and 11 after the ideal H–CNOT preparation.
Glossary
Words to know
- joint state
- A state describing a combined system of two or more subsystems.
- computational basis
- For two qubits, the basis |00⟩, |01⟩, |10⟩ and |11⟩.
- CNOT
- A two-qubit gate that conditionally applies X to a target.
- control
- The qubit whose basis value determines the CNOT action.
- target
- The qubit conditionally flipped by CNOT.
- correlation
- A statistical relationship between outcomes.
- entanglement
- A joint state that is not separable; a separable state is a product state or a probabilistic mixture of product states.
Short recap
Keep these ideas
- Two qubits require four computational-basis labels.
- CNOT has distinct control and target roles and a testable four-row basis mapping.
- Correlation alone is not enough to establish entanglement.
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, represent two-qubit probabilities as a four-entry vector ordered 00, 01, 10, 11. Compare the product state |+0⟩ with the Bell state (|00⟩ + |11⟩)/√2.
Try this
Mark which basis entries are non-zero, then explain why the Bell state cannot be obtained by independently choosing one fixed pure state for each qubit.
Adult support Teacher and parent notes
Discuss
- State the bit-order convention every time a truth table is introduced.
- Do not use matching outcomes alone as a definition or proof of entanglement.
Answer guidance
Students should give all four CNOT mappings and identify which written bit is control and which is target.
Offline activity
Pairs hold control and target cards while a third student applies the conditional rule for each basis input.
Safety
No special hazards; use cards or the browser simulation.
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
- Circuits IBM Quantum Learning · official learning module · checked 2026-08-02
- Entanglement and correlations Microsoft Learn · official documentation · checked 2026-08-02
Content review: Reviewed on 2026-08-02.