Years 11–12 · Week 8 of 12
Bell States, Entanglement and Bell Tests
Learning goals
By the end, you can…
- Derive a Bell state from H followed by CNOT using a stated qubit order.
- Explain why the Bell state cannot be factored into individual qubit states.
- Predict computational-basis outcomes and individual-qubit statistics.
- Describe carefully what Bell-inequality violations and no-signalling mean.
What you already know
Connect to a familiar idea
Some joint states cannot be written as independent subsystem states. This lesson studies the Bell state (|00⟩ + |11⟩)/√2 as the simplest example.
- Tensor products and two-qubit basis states
- H and CNOT gates
- Repeated-shot probability distributions
Opening story
Start with something familiar
Two datasets can show correlation, but correlation alone does not tell us its physical origin. Bell experiments compare correlations across carefully chosen measurement settings to test broad classes of local hidden-variable models.
Plain-English explanation
Build the idea carefully
Preparing and analysing a Bell state
Using displayed order |q₁q₀⟩, begin in |00⟩, apply H to q₀, then CNOT with control q₀ and target q₁. The intermediate state (|00⟩ + |01⟩)/√2 becomes |Φ+⟩ = (|00⟩ + |11⟩)/√2. The Bell state is non-separable: there are no normalised one-qubit states |a⟩ and |b⟩ whose tensor product equals |Φ+⟩. The pair must be described by one joint state even when its qubits are separated.
Bell correlations and the no-signalling boundary
A computational-basis measurement gives 00 or 11, each with probability 1/2 in the ideal circuit. Each qubit alone has a 50/50 outcome distribution, while the paired results are perfectly correlated. Bell tests use several measurement settings. Observed violations of a Bell inequality rule out broad classes of local hidden-variable models under the test's assumptions. They do not permit controllable faster-than-light messages: each local outcome remains random, and correlations are identified only after classical records are compared.
Try the model
Bell-state correlation laboratory
Build the Bell circuit, run repeated measurements in matched and different bases, and compare local outcome totals with joint correlations.
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: The setting activity uses ideal model probabilities. It does not claim to reproduce a loophole-free laboratory Bell test.
Text alternative for this interactive
Text alternative: the joint distribution is P(00)=1/2, P(11)=1/2, P(01)=P(10)=0; summing over the other qubit gives P(0)=P(1)=1/2 locally.
Expected observation: In the matched computational basis, the ideal Bell circuit produces only 00 and 11 with equal probabilities, while either local result alone remains approximately 50/50.
Guided activity
Joint versus local evidence
- Derive the state after H and after CNOT.
- Calculate all four computational-basis probabilities.
- Run 1000 ideal shots and create joint and marginal frequency tables.
- Explain why neither observer can choose their local random outcome to send a message.
Evidence to collect: A derivation of |Φ+⟩, a joint table containing only 00 and 11 in the ideal prediction, two 50/50 marginal tables and a no-signalling explanation.
Glossary
Words to know
- Bell state
- One of four standard maximally entangled two-qubit pure states.
- entanglement
- A property of a joint state that cannot be represented as independent subsystem states.
- correlation
- A statistical relationship between paired outcomes.
- marginal distribution
- The outcome distribution for one subsystem after summing over the other.
- Bell inequality
- A bound satisfied by a specified class of local hidden-variable models.
- local hidden-variable model
- A model with local influences and additional variables assigning outcome statistics.
- no-signalling
- The principle that local choices cannot be used to transmit information faster than light.
Short recap
Keep these ideas
- H followed by CNOT can prepare (|00⟩ + |11⟩)/√2.
- The Bell state is non-separable even though each local distribution is 50/50.
- Bell inequalities test broad classes of local hidden-variable explanations.
- Entanglement correlations do not provide faster-than-light messaging.
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
Starting from |00⟩, calculate (I⊗H)|00⟩ or the equivalent expression for the course's wire order, then apply the CNOT permutation. Derive the vector [1/√2,0,0,1/√2]ᵀ and all four computational-basis probabilities.
Try this
Attempt to factor the Bell vector as [a,b]ᵀ⊗[c,d]ᵀ. Use the zero middle amplitudes and non-zero outer amplitudes to show that no such normalised factors exist.
Adult support Teacher and parent notes
Discuss
- How can local randomness and perfect joint correlation both be present?
- What class of explanation does a Bell inequality test?
- Why must observers later compare records through a classical channel?
Answer guidance
Reject claims that Bell tests rule out every conceivable hidden-variable theory. Look for the qualified phrase 'local hidden-variable models' and a clear no-signalling explanation.
Offline activity
Give students paired synthetic datasets for several settings. They calculate local frequencies and correlations, then identify which claims require more than a single setting.
Safety
The activity uses generated ideal data. Label it clearly so it is not mistaken for results from a real Bell-test experiment.
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.
- Entanglement and correlations Microsoft Learn · official documentation · checked 2026-08-02
- Quantum information 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.