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

Lemma: Coherent recording followed by restriction

Restricted destructive deferred readout · lines 981–1004 · proof included or linked

VII-E@1#lem:recording-restriction, VII-E@1#eq:copy-trace

For every state on $S \otimes B$ ,

Tr_{S} (V ρ V^{†}) = z \sum ∣ z ⟩ ⟨ z ∣_{A} \otimes M_{z} ρ M_{z}^{†} .

Proof. Write

ρ = z, z^{'} \sum ∣ z ⟩ ⟨ z^{'} ∣_{S} \otimes ρ_{z z^{'}} .

Then

V ρ V^{†} = z, z^{'} \sum ∣ z ⟩ ⟨ z^{'} ∣_{S} \otimes ∣ z ⟩ ⟨ z^{'} ∣_{A} \otimes ρ_{z z^{'}} .

Tracing out $S$ removes precisely the terms with $z \neq = z^{'}$ .

Displayed formulas in this passage

VII-E@1#eq:copy-trace · line 984
$Tr_{S} (V ρ V^{†}) = z \sum ∣ z ⟩ ⟨ z ∣_{A} \otimes M_{z} ρ M_{z}^{†} .$
unlabeled · line 993
$ρ = z, z^{'} \sum ∣ z ⟩ ⟨ z^{'} ∣_{S} \otimes ρ_{z z^{'}} .$
unlabeled · line 997
$V ρ V^{†} = z, z^{'} \sum ∣ z ⟩ ⟨ z^{'} ∣_{S} \otimes ∣ z ⟩ ⟨ z^{'} ∣_{A} \otimes ρ_{z z^{'}} .$