Entanglement Purification — Cascade & Ensemble (2D)

Raw-Pair Fidelity

Initial Werner fidelity F₀ 0.80

Purification rounds R 3

Protocol Control

Ready. 2^R = 8 raw Bell pairs feed the tree; 400 independent chains run alongside it as a statistical ensemble.

Single-Chain Readout

Round

0 / 3

Fidelity F

80.0%

Round success p

—

Pairs alive

Ensemble Readout (N chains)

Empirical survival

100.0%

Theory ∏p(round)

100.0%

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⁺⟩. The BBPSSW recurrence protocol (Bennett, Brassard, Popescu, Schumacher, Smolin, Wootters, 1996) consumes two pairs at once: a bilateral CNOT plus a Z-basis measurement either boosts the surviving pair's fidelity or discards both.

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

F' = [ F² + (1/9)(1-F)² ] / p_succ(F)

This 2D view runs the same recursion two ways at once, computed independently from scratch: the top panel draws one literal binary merge tree — a dendrogram — where each node is a Bell pair and each merge is a CNOT-and-measure gate symbol; the bottom panel runs a Monte-Carlo ensemble of hundreds of independent chains through the identical recursion and plots, round by round, what fraction of chains still have a surviving pair. The dashed curve is the exact theoretical prediction ∏ p_succ(Fᵢ) — with a large enough ensemble the bars should hug it.

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 in the tree, and every ensemble chain runs the same R rounds.
Step round — advances both panels together by one round: the tree merges pairs two-by-two, and every still-alive ensemble chain draws its own independent Bernoulli trial at that round's p_succ.
F = 1/2 is the fixed point: below it F' < F and purification makes things worse. Above 1/2, F converges toward 1 but the surviving fraction shrinks with every round — the ensemble panel makes that shrinkage visible as a statistic, not just a story about one lucky (or unlucky) tree.

MERGE TREE — one chainENSEMBLE SURVIVAL — N chains

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