Manuscript I-B@1 · I-B-v1.tex · corollary · 5357688f

Corollary: Complete clock-readout and transport identity

One finite complete-channel null transport protocol › Composition with the clock instrument · lines 1316–1341 · proof included or linked

I-B@1#cor:clock-transport, I-B@1#eq:composed-channel

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For every input state on $C \otimes A_{0} \otimes E$ , clock readout at the sender followed by the route has complete output

k \sum ∣ k ⟩ ⟨ k ∣_{R_{sender}} \otimes (ι \otimes id_{E}) (σ_{A_{0} E, k}) \otimes i = 0 ⨂^{m - 1} σ_{A_{i}},

where

σ_{A_{0} E, k} = (D_{k} \otimes I_{E}) ρ (D_{k}^{†} \otimes I_{E}) .

The clock factor $C$ is absent.

Proof. Apply Theorem thm:transport to each subnormalized positive block of Theorem thm:instrument, and sum the blocks. Linearity gives (eq:composed-channel).

Displayed formulas in this passage

I-B@1#eq:composed-channel · line 1320
$k \sum ∣ k ⟩ ⟨ k ∣_{R_{sender}} \otimes (ι \otimes id_{E}) (σ_{A_{0} E, k}) \otimes i = 0 ⨂^{m - 1} σ_{A_{i}},$
unlabeled · line 1329
$σ_{A_{0} E, k} = (D_{k} \otimes I_{E}) ρ (D_{k}^{†} \otimes I_{E}) .$

Corollary: Complete clock-readout and transport identity

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