Manuscript Oai282-A@1 · Oai282-A.tex · theorem · 5b6134ec

Theorem: Exact full-code recovery

Stage 3: modular null cuts and recovery are different interfaces › The independent complete-channel criterion · lines 1249–1298 · proof included or linked

Oai282-A@1#thm:recovery

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There exists a deterministic CPTP map $D : B (R) \to B (C)$ with $D N = id_{C}$ if and only if

C (X) = Tr (X) τ_{B}

for a fixed state $τ_{B}$ . If $r = rank τ_{B}$ , then

dim R \geq d r .

Recovery preserves every independent external reference.

Proof. Dilate an exact decoder by an isometry $U : R \to C \otimes G$ . Then $Z = (U \otimes I_{B}) W$ dilates identity. Purity of the decoded basis outputs gives

Z ∣ i ⟩ = ∣ i ⟩ ∣ χ_{i} ⟩ .

Exact recovery of off-diagonal matrix units gives $⟨ χ_{j}, χ_{i} ⟩ = 1$ ; all normalized $χ_{i}$ coincide. Tracing $C G$ proves constancy of $C$ .

Conversely, diagonalize

τ_{B} = a = 1 \sum^{r} λ_{a} ∣ a ⟩ ⟨ a ∣, λ_{a} > 0.

Write

W ∣ i ⟩ = a \sum λ_{a} ∣ u_{ia} ⟩_{R} ∣ a ⟩_{B} .

The complementary matrix-unit equations imply

⟨ u_{j b}, u_{ia} ⟩ = δ_{ij} δ_{ab} .

Thus $R$ contains $d r$ orthonormal vectors. Define the isometry $A : C \otimes C^{r} \to R$ by $A (∣ i ⟩ ∣ a ⟩) = u_{ia}$ . For any fixed code state $ω_{C}$ ,

D (Y) = Tr_{C^{r}} (A^{†} Y A) + Tr ((I - A A^{†}) Y) ω_{C}

is CPTP and recovers every matrix unit. Equality of linear maps extends after tensoring with every reference.

Displayed formulas in this passage

unlabeled · line 1253
$C (X) = Tr (X) τ_{B}$
unlabeled · line 1258
$dim R \geq d r .$
unlabeled · line 1268
$Z ∣ i ⟩ = ∣ i ⟩ ∣ χ_{i} ⟩ .$
unlabeled · line 1276
$τ_{B} = a = 1 \sum^{r} λ_{a} ∣ a ⟩ ⟨ a ∣, λ_{a} > 0.$
unlabeled · line 1280
$W ∣ i ⟩ = a \sum λ_{a} ∣ u_{ia} ⟩_{R} ∣ a ⟩_{B} .$
unlabeled · line 1284
$⟨ u_{j b}, u_{ia} ⟩ = δ_{ij} δ_{ab} .$
unlabeled · line 1291
$D (Y) = Tr_{C^{r}} (A^{†} Y A) + Tr ((I - A A^{†}) Y) ω_{C}$

Theorem: Exact full-code recovery

Displayed formulas in this passage

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