Years 7–10 · Week 9 of 12
Entanglement and Bell Correlations
Learning goals
By the end, you can…
- Describe how H followed by CNOT prepares a Bell state from |00⟩.
- Explain why the Bell state cannot be assigned separate pure states for its two qubits.
- Interpret Bell tests as tests of broad classes of local hidden-variable models.
- Apply the no-signalling principle to entangled measurements.
What you already know
Connect to a familiar idea
Applying H to the first qubit of |00⟩ and then CNOT gives an ideal joint state with amplitudes for |00⟩ and |11⟩. Matching outcomes in one basis are a starting observation, not the whole Bell-test argument.
- Two-qubit states use four computational-basis labels.
- H creates equal-magnitude amplitudes and CNOT has control and target roles.
- Correlation does not by itself prove entanglement.
Opening story
Start with something familiar
Two distant teams each receive one qubit from the same source. On every trial, each team independently chooses a measurement setting and records one outcome. Only later do they compare settings and outcomes. The pattern of correlations across many setting pairs can violate a Bell inequality.
Plain-English explanation
Build the idea carefully
Preparing and recognising a Bell state
The ideal H–CNOT circuit prepares the Bell state (|00⟩ + |11⟩)/√2 from |00⟩. This Bell state is entangled because no choice of one pure state for the first qubit and another for the second can reproduce the joint state.
Bell tests and no signalling
When both qubits are measured in the computational basis, the ideal outcomes 00 and 11 each have probability one half, while 01 and 10 have probability zero. Bell tests use correlations across multiple independently chosen measurement settings. Experimental violations of Bell inequalities rule out broad classes of local hidden-variable explanations, subject to the test's assumptions and controls. Entanglement does not allow controllable faster-than-light signalling: each local result is not chosen by the distant user, and the correlation is established by comparing classical records.
Try the model
Bell-pair settings and correlation lab
Prepare the Bell pair, choose measurement settings independently, run seeded trials and compare the correlation chart with a local toy model.
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 educational simulation of predicted statistics; it is not a loophole-free laboratory Bell test.
Text alternative for this interactive
Bell correlation data table Select a setting pair and read seeded counts, correlation values and a plain-language comparison with the local toy-model bound.
Expected observation: Computational-basis measurements give only 00 and 11 ideally, while correlations across several settings follow a pattern that the supplied local toy model cannot match.
Guided activity
Correlation is not communication
- Verify the ideal 00/11 distribution in the computational basis.
- Run several setting pairs and record correlations rather than raw matches alone.
- Check whether one station can select the other station's individual result.
- Write a claim about what the combined data support and what they do not enable.
Evidence to collect: A multi-setting correlation table and an explanation that local outcomes cannot be controlled to transmit a message.
Glossary
Words to know
- Bell state
- One of four standard maximally entangled two-qubit states.
- entanglement
- A joint state that is not separable; a separable state is a product state or a probabilistic mixture of product states.
- measurement setting
- The chosen basis or observable used for a measurement.
- Bell inequality
- A constraint satisfied by a specified class of local hidden-variable models.
- local hidden-variable model
- A model using local properties to predetermine or probabilistically govern outcomes.
- correlation
- A quantified statistical relationship between paired outcomes.
- no-signalling
- The principle that choices at one site cannot change usable local statistics at a distant site.
Short recap
Keep these ideas
- H followed by CNOT can prepare an entangled Bell state.
- Bell tests compare correlations across several measurement settings, not just matching outcomes.
- Bell-inequality violations constrain local hidden-variable explanations but do not enable faster-than-light communication.
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, interpret a supplied CHSH-style correlation expression without deriving it. Compare an ideal quantum prediction with the bound for the specified local hidden-variable model and discuss sampling uncertainty.
Try this
Calculate four provided correlation values from count tables, insert them into the supplied CHSH combination and state only the conclusion supported under the model assumptions.
Adult support Teacher and parent notes
Discuss
- Do not present Bell tests as settling every philosophical interpretation of quantum mechanics.
- Separate no-signalling from the presence of strong joint correlations.
Answer guidance
Strong answers mention several settings, Bell-inequality constraints and the need for classical comparison; avoid 'instant force' language.
Offline activity
Analyse a teacher-provided, clearly labelled simulated Bell dataset; do not claim a classroom card game reproduces entanglement.
Safety
No special hazards; this lesson uses simulated or supplied data 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.
- 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.