Manuscript Oai365-A@1 · Oai365-A.tex · section · a3840e3d

Local errors may accumulate

Transport: obstructions, certificates, and an added-resource pilot › Local errors may accumulate · lines 1492–1515

For

R_{d} (X) = Tr (X) I_{d} / d, P_{p} = (1 - p) id + p R_{d},

one has

P_{p}^{L} = (1 - p)^{L} id + (1 - (1 - p)^{L}) R_{d} .

With $p = λ / L$ , each local error tends to zero, but testing a maximally entangled input gives

L \to \infty lim inf d_{⋄} (P_{λ / L}^{L}, id) \geq (1 - e^{- λ}) (1 - d^{- 2}) > 0.

This is the global-accumulation mechanism emphasized in En08-B CheungEn08B. Merely requiring $η_{n} \to 0$ is insufficient; the sufficient certificate above requires $H_{n} η_{n} \to 0$ .

Displayed formulas in this passage

unlabeled · line 1495
$R_{d} (X) = Tr (X) I_{d} / d, P_{p} = (1 - p) id + p R_{d},$
unlabeled · line 1500
$P_{p}^{L} = (1 - p)^{L} id + (1 - (1 - p)^{L}) R_{d} .$
unlabeled · line 1506
$L \to \infty lim inf d_{⋄} (P_{λ / L}^{L}, id) \geq (1 - e^{- λ}) (1 - d^{- 2}) > 0.$