Entanglement Purification for Quantum Repeaters

Raw-Pair Fidelity

Initial Werner fidelity F₀ 0.80

Purification rounds R 3

Protocol Control

Ready. 2^R = 8 raw Bell pairs will be consumed to attempt one distilled pair.

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Round

0 / 3

Fidelity F

80.0%

Round success p

—

Pairs alive

How it works

Photon loss and decoherence leave each elementary link in a repeater chain holding a noisy Werner state — a Bell pair mixed with white noise at fidelity F = ⟨Φ⁺|ρ|Φ⁺⟩. Entanglement swapping can extend distance, but it also multiplies noise: low-fidelity links must be purified before they're swapped together.

The BBPSSW recurrence protocol (Bennett, Brassard, Popescu, Schumacher, Smolin, Wootters, 1996) consumes two pairs at once: each station applies a local bilateral CNOT between its two qubits, then measures the target pair in the computational basis. Alice and Bob compare outcomes over a classical channel — if they agree, the source pair survives with boosted fidelity; if they disagree, both pairs are discarded.

p_succ(F) = F² + (2/3)F(1-F) + (5/9)(1-F)²

F' = [ F² + (1/9)(1-F)² ] / p_succ(F)

F₀ slider — fidelity of the raw pairs arriving from the elementary links.
Rounds R — nested purification depth; consumes 2^R raw pairs per surviving distilled pair.
Step round — pairs branches two-by-two, runs the CNOT-and-measure flash, then keeps or discards each result.
F = 1/2 is the fixed point: below it F' < F and purification makes things worse — real repeater links are engineered to clear this floor before distillation is attempted. Above 1/2, F converges toward 1 but the surviving fraction shrinks with every round, which is why real quantum-repeater budgets treat purification depth as an explicit resource cost, not a free upgrade.

ALICEBOB

High fidelity (→1.0) Low fidelity (→0.25) CNOT + measurement flash

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