Entanglement Distribution

The most basic service a quantum network provides is a shared Bell pair: one qubit held by node $A$ (Alice), the other by node $B$ (Bob), in the maximally entangled state

|\Phi^+\rangle_{AB} = \frac{|00\rangle + |11\rangle}{\sqrt{2}}.

Every higher-level protocol — teleportation, swapping, distributed gates — assumes this pair already exists. In this lesson you distribute one and measure the correlations that prove it is entangled.

Generating a Bell pair

A Bell pair is prepared with exactly two gates, regardless of how far apart the qubits will eventually sit:

|\Phi^+\rangle = \mathrm{CNOT}_{0\to 1}\,\big(H \otimes I\big)\,|00\rangle.

Reading it left to right:

Start in $|00\rangle$ .
A Hadamard on qubit 0 produces $\tfrac{1}{\sqrt 2}(|0\rangle + |1\rangle)\otimes|0\rangle$ .
A CNOT with qubit 0 as control entangles qubit 1, giving $\tfrac{1}{\sqrt 2}(|00\rangle + |11\rangle)$ .

In a real network the two qubits are created together at a source and one half is then transmitted to the partner node. In our simulator we model both halves in a single register; qubit 0 is Alice's and qubit 1 is Bob's.

What entanglement looks like in the data

A Bell pair is not the same as two qubits that are each independently in superposition. The signature is perfect correlation. Measuring both qubits in the computational ( $Z$ ) basis yields

P(00) = P(11) = \tfrac12, \qquad P(01) = P(10) = 0.

Alice's outcome is individually random — $50/50$ — yet it always matches Bob's. The same perfect correlation holds in the $X$ basis, which is what rules out any classical "shared coin" explanation and underpins entanglement-based quantum key distribution.

Fidelity and noise

Real links never deliver a perfect $|\Phi^+\rangle$ . Loss and decoherence degrade it to a mixed state $\rho$ , characterised by its fidelity

F = \langle \Phi^+ | \rho | \Phi^+ \rangle,

the overlap with the ideal pair. $F = 1$ is perfect; $F = 1/4$ is no better than random noise. Later lessons (purification, repeaters) are entirely about pushing $F$ back up after distribution has degraded it. Here we work in the idealised $F = 1$ regime so you can see the target those protocols aim for.

Try it

Distribute a Bell pair across the two nodes and measure both qubits. The grader checks the measurement distribution: only 00 and 11 should appear, each with probability $\tfrac12$ .

Run your code to see the quantum state.

After running, open the Probabilities tab: two bars of height $0.5$ at 00 and 11, zero everywhere else — the fingerprint of a perfectly distributed Bell pair.