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

Lemma: Patch channel identity

One connected two-cell encoding › Triple patch decoders · lines 759–781 · proof included or linked

Oai280-C@1#lem:patchidentity

The matrix $U_{K}$ is unitary and

Tr_{b} (F ρ F^{†}) = U_{K} (ρ \otimes I_{h} /2) U_{K}^{†}

for every operator $ρ$ on $L \otimes e$ . Equivalently,

F = (U_{K} \otimes I_{b}) (I_{L e} \otimes ∣ Φ_{2} ⟩_{hb}) .

Proof. Equation (eq:UK) is the balanced reshaping $8 T$ from Proposition prop:perfect, hence is unitary. The second formula follows directly from $F_{a_{K}, b; l, e} = 2 V_{a_{K}, b, e; l}$ . Tracing $b$ proves the channel identity.

Displayed formulas in this passage

unlabeled · line 762
$Tr_{b} (F ρ F^{†}) = U_{K} (ρ \otimes I_{h} /2) U_{K}^{†}$
unlabeled · line 769
$F = (U_{K} \otimes I_{b}) (I_{L e} \otimes ∣ Φ_{2} ⟩_{hb}) .$

Nearby in this section

section: Triple patch decoders
theorem: Theorem: Connected subsystem and full-bulk recovery
section: Triple patch decoders