Manuscript VII-E@1 · VII-E.tex · lemma · b713cf3c

Lemma: Restriction commutes with an untouched spectator

Spectator-extension invariance · lines 834–854 · proof included or linked

VII-E@1#lem:partial-trace, VII-E@1#eq:trace-channel

For a channel $Φ : L (H_{Γ}) \to L (H_{Γ^{'}})$ ,

Tr_{N} ((Φ \otimes id_{N}) (ρ)) = Φ (Tr_{N} ρ) .

Proof. For every $A \in L (H_{Γ^{'}})$ ,

Tr (A Tr_{N} [(Φ \otimes id_{N}) (ρ)]) = Tr ((Φ^{*} (A) \otimes I_{N}) ρ) = Tr (Φ^{*} (A) Tr_{N} ρ) = Tr (A Φ (Tr_{N} ρ)) .

Equality against every output observable proves the result.

Displayed formulas in this passage

VII-E@1#eq:trace-channel · line 838
$Tr_{N} ((Φ \otimes id_{N}) (ρ)) = Φ (Tr_{N} ρ) .$
unlabeled · line 847
$Tr (A Tr_{N} [(Φ \otimes id_{N}) (ρ)]) = Tr ((Φ^{*} (A) \otimes I_{N}) ρ) = Tr (Φ^{*} (A) Tr_{N} ρ) = Tr (A Φ (Tr_{N} ρ)) .$