Manuscript Hawk07-A@1 · Hawk07-A.tex · theorem · f9c0f5da

Theorem: Exact whole-code recovery

Complete restoration and the complement criterion · lines 475–541 · proof included or linked

Hawk07-A@1#thm:exact, Hawk07-A@1#eq:normalform, Hawk07-A@1#eq:exactdecoder

For $V : L \to R B$ ,

DN = id_{L} for some CPTP D ⟺ C = T_{τ_{B}} for some state τ_{B} .

If $s = rank τ_{B}$ , then

dim R \geq d s .

There are an isometry $T : L \otimes C^{s} \to R$ and a state $σ$ such that

N (X) = T (X \otimes σ) T^{†} .

An exact decoder on the entire accessible space is

D (Y) = Tr_{C^{s}} (T^{†} Y T) + Tr ((I - T T^{†}) Y) ω_{L} .

Proof. Suppose a decoder is exact and dilate it by $W : R \to L G$ . The isometry $(W \otimes I_{B}) V$ dilates the identity. Its Choi operator has rank one, so every Kraus operator is proportional to identity. Therefore

(W \otimes I_{B}) V ∣ ψ ⟩ = ∣ ψ ⟩_{L} ∣ χ ⟩_{GB}

with fixed $χ$ . Tracing $L G$ proves complement constancy.

Conversely, diagonalize

τ_{B} = α = 1 \sum^{s} λ_{α} ∣ α ⟩ ⟨ α ∣, λ_{α} > 0,

and write

V ∣ i ⟩ = α \sum λ_{α} ∣ u_{i α} ⟩_{R} ∣ α ⟩_{B} .

The identities $C (∣ i ⟩ ⟨ j ∣) = δ_{ij} τ_{B}$ give

⟨ u_{j β} ∣ u_{i α} ⟩ = δ_{ij} δ_{α β} .

These $d s$ orthonormal vectors define $T$ . Equations (eq:normalform) and (eq:exactdecoder) follow. The decoder is CP, its two trace contributions sum to $Tr Y$ , and it recovers every matrix unit.

Displayed formulas in this passage

Hawk07-A@1#eq:normalform · line 491
$N (X) = T (X \otimes σ) T^{†} .$
Hawk07-A@1#eq:exactdecoder · line 496
$D (Y) = Tr_{C^{s}} (T^{†} Y T) + Tr ((I - T T^{†}) Y) ω_{L} .$
unlabeled · line 478
$DN = id_{L} for some CPTP D ⟺ C = T_{τ_{B}} for some state τ_{B} .$
unlabeled · line 486
$dim R \geq d s .$
unlabeled · line 511
$(W \otimes I_{B}) V ∣ ψ ⟩ = ∣ ψ ⟩_{L} ∣ χ ⟩_{GB}$
unlabeled · line 519
$τ_{B} = α = 1 \sum^{s} λ_{α} ∣ α ⟩ ⟨ α ∣, λ_{α} > 0,$
unlabeled · line 524
$V ∣ i ⟩ = α \sum λ_{α} ∣ u_{i α} ⟩_{R} ∣ α ⟩_{B} .$
unlabeled · line 532
$⟨ u_{j β} ∣ u_{i α} ⟩ = δ_{ij} δ_{α β} .$

Theorem: Exact whole-code recovery

Displayed formulas in this passage

Nearby in this section