Entanglement Purification

Distributing entanglement over a real link produces noisy Bell pairs with fidelity $F < 1$ . Swapping makes the noise worse, so a quantum repeater must occasionally run the noise backwards. Entanglement purification (also called distillation) trades quantity for quality: it consumes several low-fidelity pairs to produce a smaller number of higher-fidelity ones, using only local operations and classical communication (LOCC).

The trade you are making

You start with two noisy copies of $|\Phi^+\rangle$ and end with — at best — one copy whose fidelity is higher than either input. Purification can never reach $F = 1$ in a single round, and it always discards some pairs, but iterating the protocol drives the fidelity arbitrarily close to 1 as long as the input fidelity exceeds a threshold ( $F > 1/2$ for the standard protocols).

The bilateral CNOT

The heart of the BBPSSW / Deutsch et al. protocols is a bilateral CNOT. Label the two pairs:

a kept pair — Alice's $q_0$ with Bob's $q_1$ ;
a sacrificial pair — Alice's $q_2$ with Bob's $q_3$ .

Each party applies a CNOT on their own side, from the kept qubit to the sacrificial qubit:

\text{Alice: } \mathrm{CNOT}_{q_0 \to q_2}, \qquad \text{Bob: } \mathrm{CNOT}_{q_1 \to q_3}.

No gate crosses the link — this is genuinely local on each node, which is why purification is implementable with only classical coordination.

The parity check

After the bilateral CNOT, Alice and Bob each measure their sacrificial qubit ( $q_2$ and $q_3$ ) in the computational basis and compare the bits over the classical channel:

Outcomes agree ( $q_2 = q_3$ ): the round succeeds. The kept pair is retained, and on noisy inputs its fidelity has increased.
Outcomes disagree ( $q_2 \neq q_3$ ): an error was detected. The kept pair is discarded.

For ideal input pairs the agreement check passes deterministically, and afterwards the kept pair is still a perfect Bell pair: the populated outcomes have $q_2 = q_3$ always, and the kept qubits $q_0, q_1$ remain perfectly correlated, split equally between the $00$ and $11$ branches.

Why thresholds exist

Each successful round maps the input fidelity $F$ to a new fidelity $F'$ that is larger than $F$ only when $F$ is above the protocol's fixed point. Below that threshold, the local gates inject more noise than the parity check removes, and iterating makes things worse. This is the same threshold phenomenon that governs fault-tolerant error correction — and it is why repeaters must keep each link's raw fidelity above the line before purification can help.

Try it

Run one bilateral-CNOT round on two ideal Bell pairs and measure. The grader checks the purification signature: the sacrificial qubits $q_2, q_3$ always agree (no probability on disagreeing outcomes), and the kept pair stays a clean Bell pair — half its weight on $00\!\ldots$ and half on $11\!\ldots$ in the kept qubits.

Run your code to see the quantum state.

In the Probabilities tab, every populated bar has matching $q_2$ and $q_3$ bits — the heralding condition that tells Alice and Bob to keep the other pair.