Manuscript En08-B@1 · En08-B.tex · proposition · 23cac52a

Proposition: A small nonlocal perturbation can signal

Implementation errors and their boundaries › Small error is not exact causal containment · lines 1180–1212 · proof included or linked

En08-B@1#prop:near-local

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For every $0 < ϵ \leq 1$ , there is a channel within half-diamond distance at most $ϵ$ of the identity that signals across an arbitrarily large spatial separation.

Proof. Place two $D$ -dimensional registers $A, B$ at the chosen separation and define

C = (1 - ϵ) id_{A B} + ϵ Ad_{SWAP_{A B}} .

It is a channel, and

d_{⋄} (C, id_{A B}) \leq ϵ,

because the half-diamond distance between any two channels is at most one.

Prepare $B$ in $∣ 0 ⟩$ . Inputs $∣ 0 ⟩$ and $∣ 1 ⟩$ at $A$ produce respective $B$ -outputs

∣ 0 ⟩ ⟨ 0 ∣, (1 - ϵ) ∣ 0 ⟩ ⟨ 0 ∣ + ϵ ∣ 1 ⟩ ⟨ 1 ∣ .

Their trace distance is $ϵ > 0$ . The separation can exceed $a_{j}$ , making this channel inadmissible as one tick of the frozen model.

Displayed formulas in this passage

unlabeled · line 1190
$C = (1 - ϵ) id_{A B} + ϵ Ad_{SWAP_{A B}} .$
unlabeled · line 1196
$d_{⋄} (C, id_{A B}) \leq ϵ,$
unlabeled · line 1205
$∣ 0 ⟩ ⟨ 0 ∣, (1 - ϵ) ∣ 0 ⟩ ⟨ 0 ∣ + ϵ ∣ 1 ⟩ ⟨ 1 ∣ .$

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section: Small error is not exact causal containment