Project: A Simple Bell Test

The Bell test is one of the most striking experiments in physics: it shows that two entangled qubits cannot be described as two independent classical bits. In this project you will build the simplest entangled state — a Bell pair — and read off the correlation between the two qubits by sampling the measurement outcomes.

The Bell state $|\Phi^+\rangle$

Start both qubits in $|00\rangle$. Apply a Hadamard to the first qubit:

Now apply a CNOT with qubit 0 as the control and qubit 1 as the target. The CNOT flips the target whenever the control is $|1\rangle$:

This is one of the four Bell states, the maximally entangled two-qubit states. It cannot be factored as a product of two single-qubit states — that is the definition of entanglement.

What sampling reveals

Measuring both qubits in the computational basis collapses the state to either $|00\rangle$ or $|11\rangle$, each with probability $\tfrac{1}{2}$ (by the Born rule applied to the amplitudes $\tfrac{1}{\sqrt{2}}$). The outcomes $|01\rangle$ and $|10\rangle$ never appear. This perfect correlation — the two bits always agree — is the signature of entanglement and is what a real Bell test verifies.

Estimating the correlation

In practice a Bell test works by running the circuit many times (many shots), recording the pairs of outcomes, and computing a correlation coefficient. For $|\Phi^+\rangle$ measured in the $Z$-basis, every shot gives matching bits, so the correlation between qubit 0 and qubit 1 is $+1$. The simulator gives you the exact probabilities, which correspond to the infinite-shot limit.

Try it

Build the circuit that prepares $|\Phi^+\rangle$ and measures both qubits. Look at the Probabilities tab: you should see exactly two equal bars for outcomes 00 and 11, and nothing for 01 or 10.

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