Years 11–12 · Week 7 of 12
Multi-Qubit States and Tensor Products
Learning goals
By the end, you can…
- List the computational-basis states of two and three qubits.
- Calculate simple tensor products of one-qubit state vectors.
- Distinguish separable joint states from states that cannot be factored.
- Apply CNOT to computational-basis states using stated qubit ordering.
What you already know
Connect to a familiar idea
One qubit needs two amplitudes. Combining qubits requires one state description for the whole register, not a separate list of independent states in every case.
- One-qubit state vectors
- Basic vector multiplication
- Single-qubit circuits and basis labels
Opening story
Start with something familiar
Two two-choice questions have four joint answer labels: 00, 01, 10 and 11. A two-qubit state likewise uses four computational-basis amplitudes, but quantum joint states can include relationships that no pair of separate one-qubit states can describe.
Plain-English explanation
Build the idea carefully
Tensor products build joint state spaces
For displayed labels |q₁q₀⟩, the two-qubit computational basis is |00⟩, |01⟩, |10⟩ and |11⟩. The course simulator treats q₀ as the rightmost, least-significant bit of a basis label; stating this convention prevents ambiguous circuit predictions. The tensor product combines state spaces. If |a⟩ = [a₀,a₁]ᵀ and |b⟩ = [b₀,b₁]ᵀ, then |a⟩⊗|b⟩ = [a₀b₀,a₀b₁,a₁b₀,a₁b₁]ᵀ in the corresponding basis order.
Separable states, CNOT and dimension growth
Worked example: |1⟩⊗|0⟩ = |10⟩ = [0,0,1,0]ᵀ. A state is separable if it can be written as a tensor product of individual states. Some valid joint states cannot be factored and are entangled. CNOT flips its target exactly when its control is 1 in the computational basis. With control q₁ and target q₀, |10⟩ maps to |11⟩. An n-qubit pure state has 2ⁿ amplitudes, which explains why general classical state-vector simulation becomes demanding as n grows.
Try the model
Two- and three-qubit state viewer
Choose one-qubit input states, expand their tensor product, then add a CNOT with a selected control and target 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: Basis labels are displayed as q(n−1)…q₀ with q₀ on the right; the control and target must be different qubits.
Text alternative for this interactive
Text alternative: expand each entry of the first vector by multiplying it by every entry of the second, then use the stated basis-label order.
Expected observation: The tensor product |1⟩⊗|0⟩ occupies only |10⟩, and a CNOT with control q₁ and target q₀ maps that basis state to |11⟩.
Guided activity
Expand, label and transform
- Calculate |0⟩⊗|+⟩ and |+⟩⊗|1⟩ by hand.
- Label every vector entry using the simulator's basis convention.
- Apply CNOT to each computational-basis input and build its truth table.
- Predict the number of amplitudes required for three, four and ten qubits.
Evidence to collect: Two correctly expanded tensor products, a four-row CNOT truth table with control and target named, and the values 8, 16 and 1024 for the requested state-vector dimensions.
Glossary
Words to know
- tensor product
- The operation used to combine the state spaces of quantum systems.
- joint state
- One state description for a compound system.
- computational basis
- The bit-string-labelled basis used for a qubit register.
- separable state
- A joint pure state that factors into individual subsystem states.
- CNOT
- A two-qubit gate that flips its target when its control is 1.
- control qubit
- The qubit whose computational-basis value conditions a controlled gate.
- target qubit
- The qubit transformed by a controlled gate.
Short recap
Keep these ideas
- Two qubits require four basis amplitudes and three qubits require eight.
- Tensor products construct separable joint states from subsystem states.
- Not every joint quantum state can be factored.
- CNOT predictions require an explicit basis order and distinct control and target qubits.
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 n qubits the state space has dimension 2ⁿ. Expand |+⟩⊗|0⟩ and apply a CNOT whose control is the plus-state qubit. Test whether the resulting vector can be written as [a,b]ᵀ⊗[c,d]ᵀ.
Try this
Set up the factor equations ac, ad, bc and bd for the four amplitudes. Show why the resulting Bell-state vector cannot satisfy them all for normalised one-qubit factors.
Adult support Teacher and parent notes
Discuss
- Why does basis ordering matter even when the physics is convention-independent?
- What condition makes a joint pure state separable?
- Why does adding one qubit double a general state vector's length?
Answer guidance
Require students to state which qubit is leftmost and which is the control. Treat a different consistent endianness convention as valid only if every label and transformation follows it.
Offline activity
Use a four-cell grid and amplitude cards to construct tensor products and perform the CNOT permutation without a device.
Safety
No physical equipment is needed. Avoid claiming that state-vector dimension alone gives a practical speed-up or a current hardware capability.
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
- Quantum Computation and Quantum Information Cambridge University Press · textbook publisher page · checked 2026-08-02
Content review: Reviewed on 2026-08-02.