Entanglement and Bell states

Build a Bell pair

Start from $\lvert00\rangle$. Apply H to q0, then controlled-X with q0 as control:

$$\lvert00\rangle\xrightarrow{H_0} \frac{\lvert00\rangle+\lvert10\rangle}{\sqrt2} \xrightarrow{CX_{0,1}} \lvert\Phi^+\rangle=\frac{\lvert00\rangle+\lvert11\rangle}{\sqrt2}.$$

The Z-basis outcomes are 00 and 11, each with probability $1/2$. Each individual outcome is random, but the pair agrees.

Why it is not a product

Suppose $(a\lvert0\rangle+b\lvert1\rangle)\otimes(c\lvert0\rangle+d\lvert1\rangle)$ equalled this Bell state. We would need $ac$ and $bd$ nonzero, but $ad=bc=0$. Nonzero $ac$ and $bd$ require all four factors nonzero, contradicting those zeros. The state is not separable.

L04 · INTERACTIVE LABRuns on your device

Entanglement explorer

Prepare (|00⟩ + |11⟩)/√2 using H and CX. Compare this with independent superpositions in two measurement bases.

Make a prediction first

The Bell state agrees in Z. Will it still agree if both qubits are measured in X?

Experiment controls

X-basis measurements are implemented with an H gate before each Z-basis readout. Labels describe the readout bits.

Random seed

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Exact probability

|00⟩

50.0%

|01⟩

0.0%

|10⟩

0.0%

|11⟩

50.0%

Exact ideal-model predictions · run a measurement experiment to compare.

MODEL · Ideal pure-state vector, complex double precisionORDER · q0 is the leftmost bit

Take it further

Compare the Bell state with two independent |+⟩ states. Why are correlations in one basis alone insufficient to establish entanglement?

Prediction: The Bell state agrees in Z. Will it still agree if both qubits are measured in X? Challenge: Compare the Bell state with two independent |+⟩ states. Why are correlations in one basis alone insufficient to establish entanglement?

One basis is not enough evidence

A classical 50/50 mixture of 00 and 11 has the same Z-basis counts as this Bell state. Their X-basis behaviour differs: the ideal Bell state still has matching outcomes, while that classical mixture does not preserve those X correlations.

The lab compares Bell preparation with independent $\lvert+\rangle$ states. This distinguishes those two known ideal preparations; it is not a general experimental certification of an arbitrary unknown device. Such a claim needs carefully chosen observables and assumptions.

Correlation is not communication

Measuring one half gives a random local outcome. The other reader cannot use their own outcomes to learn which measurement was chosen remotely. Comparing correlations requires ordinary classical communication.

Challenge: predict the independent-state distributions in Z and X before switching the selector. Explain both using the product-state structure.

Reference: Microsoft Learn — Entanglement and correlations.