Manuscript En01-A@1 · En01-A.tex · proposition · 558b972b

Proposition: Discarded-copy dilation

Recorded circuits and the two statistical models › Pinching, anchoring, and dilation · lines 979–999 · proof included or linked

En01-A@1#prop:deeper

Choose one anchor in each class $j \in J$ . Let $C$ copy those anchors into a fresh register $R^{'}$ . Then

Δ_{J} ρ = tr_{R^{'}} (C ρ C^{†}) .

Consequently, the dephased model is the coherent Born model of an extended tower when its last copies are inaccessible.

Proof. For one anchor,

∣ b ⟩ ⟨ b^{'} ∣ \otimes ∣0 ⟩ ⟨ 0∣ ⟼ ∣ b ⟩ ⟨ b^{'} ∣ \otimes ∣ b ⟩ ⟨ b^{'} ∣.

Tracing the copy yields the factor $δ_{b b^{'}}$ , hence pinching. Repeat for each anchor and apply Lemma lem:anchor.

Displayed formulas in this passage

unlabeled · line 983
$Δ_{J} ρ = tr_{R^{'}} (C ρ C^{†}) .$
unlabeled · line 992
$∣ b ⟩ ⟨ b^{'} ∣ \otimes ∣0 ⟩ ⟨ 0∣ ⟼ ∣ b ⟩ ⟨ b^{'} ∣ \otimes ∣ b ⟩ ⟨ b^{'} ∣.$