Manuscript Oai280-C@1 · Oai280-C.tex · proposition · 7d44b8c0

Proposition: Fixed-spectator trap in the connected model

Approximate-recovery examples on the declared models › Spectator-dependent error · lines 1770–1794 · proof included or linked

Oai280-C@1#prop:spectatortrap

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Let

C_{Z} = I_{L} \otimes ∣ 0 ⟩ ⟨ 0 ∣_{S} + Z_{L} \otimes ∣ 1 ⟩ ⟨ 1 ∣_{S}

act before $W$ . The printed left subsystem decoder then implements

Λ (ρ) = Tr_{S} (C_{Z} ρ C_{Z}^{†}) .

It is exact for every logical state with spectator $∣ 0 ⟩ ⟨ 0 ∣$ , but

d_{⋄} (Λ, Tr_{S}) = 1.

Proof. The channel identity follows from Theorem thm:patch. The $s = 0$ statement is immediate. For input $∣ + ⟩ ⟨ + ∣_{L} \otimes ∣ 1 ⟩ ⟨ 1 ∣_{S}$ , the actual and target outputs are $∣ - ⟩ ⟨ - ∣$ and $∣ + ⟩ ⟨ + ∣$ , which are orthogonal. The channel distance is therefore at least one, hence equals one.

Displayed formulas in this passage

unlabeled · line 1773
$C_{Z} = I_{L} \otimes ∣ 0 ⟩ ⟨ 0 ∣_{S} + Z_{L} \otimes ∣ 1 ⟩ ⟨ 1 ∣_{S}$
unlabeled · line 1779
$Λ (ρ) = Tr_{S} (C_{Z} ρ C_{Z}^{†}) .$
unlabeled · line 1783
$d_{⋄} (Λ, Tr_{S}) = 1.$

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section: Spectator-dependent error