Manuscript XXII-D@1 · XXII-D.tex · proposition · 17889d18

Proposition: A moving arrival jump

Failure tests › F3: moving arrivals obstruct uniform comparison across the event · lines 1392–1417 · proof included or linked

XXII-D@1#prop:jump

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Let $τ_{n} > τ$ and $τ_{n} \to τ$ . On a common message-plus-flag interface define

J_{τ} (t) (X) = {Tr (X) σ_{0} \otimes ∣0 ⟩ ⟨ 0∣, X \otimes ∣1 ⟩ ⟨ 1∣, t < τ, t \geq τ .

Then endpoint arrival times converge and the post-arrival transport channels are identical, but

t sup d_{⋄} (J_{τ_{n}} (t), J_{τ} (t)) = 1 for every n .

Proof. For $τ \leq t < τ_{n}$ , the flags of the two outputs are orthogonal. Their trace distance is one for every state input. The half-diamond distance is therefore one. The channel distance cannot exceed one.

Displayed formulas in this passage

unlabeled · line 1396
$J_{τ} (t) (X) = {Tr (X) σ_{0} \otimes ∣0 ⟩ ⟨ 0∣, X \otimes ∣1 ⟩ ⟨ 1∣, t < τ, t \geq τ .$
unlabeled · line 1405
$t sup d_{⋄} (J_{τ_{n}} (t), J_{τ} (t)) = 1 for every n .$

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section: F3: moving arrivals obstruct uniform comparison across the event