Manuscript En03-A@1 · En03-A.tex · lemma · 5398614e

Lemma: Recording followed by restriction

Restricted destructive deferred readout › Measurement identities · lines 1079–1099 · proof included or linked

En03-A@1#lem:copytrace, En03-A@1#eq:copytrace

For every $ρ \in D (H_{S B})$ ,

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

Proof. Write $ρ = \sum_{z, z^{'}} ∣ z ⟩ ⟨ z^{'} ∣_{S} \otimes ρ_{z z^{'}}$ . Recording gives

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

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

Displayed formulas in this passage

En03-A@1#eq:copytrace · line 1082
$Tr_{S} (V ρ V^{†}) = z \sum ∣ z ⟩ ⟨ z ∣_{A} \otimes M_{z} ρ M_{z}^{†} .$
unlabeled · line 1093
$V ρ V^{†} = z, z^{'} \sum ∣ z ⟩ ⟨ z^{'} ∣_{S} \otimes ∣ z ⟩ ⟨ z^{'} ∣_{A} \otimes ρ_{z z^{'}} .$