Manuscript En02-B@1 · En02-B-v1.tex · lemma · ca28dee4

Lemma: Intertwining factorization

An explicit batch subsystem decoder · lines 832–865 · proof included or linked

En02-B@1#lem:factor, En02-B@1#eq:factor

For a fixed normalized vector $χ \in H_{F} \otimes H_{E}$ ,

(U_{A} \otimes I_{E}) V ∣ ψ ⟩ = ∣ ψ ⟩_{2} \otimes ∣ χ ⟩_{F E}

for every logical vector $ψ$ . Moreover,

Tr_{E} ∣ χ ⟩ ⟨ χ ∣ = \frac{I _{F}}{4} .

Proof. Write $W = (U_{A} \otimes I_{E}) V$ . Lemma lem:conjugation gives $X_{2} W = W X$ and $Z_{2} W = W Z$ . The $Z$ identity makes $W ∣ 0 ⟩ = ∣ 0 ⟩ ∣ χ_{0} ⟩$ and $W ∣ 1 ⟩ = ∣ 1 ⟩ ∣ χ_{1} ⟩$ . The $X$ identity gives $χ_{0} = χ_{1} = χ$ . Isometry of $W$ gives normalization.

The encoded maximally mixed logical input is $P /2$ . Its reduction on $A$ is $I_{A} /8$ , by the weight-four stabilizer table. Equation (eq:factor) therefore gives

\frac{I _{2}}{2} \otimes Tr_{E} ∣ χ ⟩ ⟨ χ ∣ = \frac{I _{8}}{8},

proving the syndrome statement.

Displayed formulas in this passage

En02-B@1#eq:factor · line 835
$(U_{A} \otimes I_{E}) V ∣ ψ ⟩ = ∣ ψ ⟩_{2} \otimes ∣ χ ⟩_{F E}$
unlabeled · line 843
$Tr_{E} ∣ χ ⟩ ⟨ χ ∣ = \frac{I _{F}}{4} .$
unlabeled · line 861
$\frac{I _{2}}{2} \otimes Tr_{E} ∣ χ ⟩ ⟨ χ ∣ = \frac{I _{8}}{8},$

Lemma: Intertwining factorization

Displayed formulas in this passage

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