Manuscript Oai211-A@1 · Oai211-A.tex · section · 3f3e4f63

An actual serial half-diamond certificate

An actual serial half-diamond certificate · lines 843–866

Oai211-A@1#eq:weaksteps

A meaningful serial theorem requires a composition law. We now supply one explicitly rather than infer it from the map.

Let $s (k, i, A)$ be a fixed nonnegative label with

0 \leq s (k, i, A) \leq C_{s} k, μ = E_{sp} s < \infty, S (v) = path (v) \sum s .

Define the completely depolarizing channel

T_{d} (X) = Tr (X) I_{C} / d

and the designed local channel

D_{N, s} = e^{- s / N} id_{C} + (1 - e^{- s / N}) T_{d} .

This $N^{- 1/2}$ weakness is part of the model definition. It is not deduced from a geometric scaling exponent.

Displayed formulas in this passage

Oai211-A@1#eq:weaksteps · line 859
$D_{N, s} = e^{- s / N} id_{C} + (1 - e^{- s / N}) T_{d} .$
unlabeled · line 849
$0 \leq s (k, i, A) \leq C_{s} k, μ = E_{sp} s < \infty, S (v) = path (v) \sum s .$
unlabeled · line 855
$T_{d} (X) = Tr (X) I_{C} / d$