Entanglement Purification

Entanglement Purification

Entanglement purification (also called entanglement distillation) takes multiple copies of a noisy Bell pair and produces a single higher-fidelity pair. It is essential for quantum repeaters: each segment distributes noisy pairs, and purification converts them into pairs clean enough for teleportation or QKD.

The Problem: Noise Degrades Fidelity

After distribution over a noisy channel, a Bell pair may have Werner state fidelity F < 1. Two uses of such a pair for QKD or teleportation accumulate errors. Purification allows trading quantity (two noisy pairs) for quality (one better pair).

Requirement: input fidelity F > 0.5 (any entanglement at all). With F > 0.5, one round of the BBPSSW protocol can boost fidelity significantly.

The BBPSSW Purification Protocol

The Bennett-Brassard-Popescu-Schumacher-Smolin-Wootters (BBPSSW) protocol uses two Werner pairs with fidelity F to produce one pair with higher fidelity F':

Step 1: Local CNOT operations Alice applies CNOT (control=qubit from pair 1, target=qubit from pair 2) on her side. Bob applies CNOT (control=qubit from pair 1, target=qubit from pair 2) on his side.

Step 2: Measurement Both Alice and Bob measure their "target" qubits (from pair 2) in the Z basis.

For $F=0.8$: $P_{\text{success}}\approx 0.77$, $F^{\prime }\approx 0.84$, a modest improvement.

Iterative Purification

One round of BBPSSW improves fidelity but uses 2 pairs to produce 1. Multiple rounds can approach $F\to 1$, but at exponential resource cost: to achieve $F>0.95$ from $F_0=0.7$ requires approximately 8 rounds, using $2^8=256$ initial pairs.

In practical repeater designs, purification is performed 1-2 times per link, with the trade-off between fidelity improvement and resource consumption carefully optimized.