Manuscript En02-D@4 · En02-D-v4.tex · section · e19ec871

Recovery SDPs and numerical verification

Recovery SDPs and numerical verification · lines 1190–1250

Use output-first unnormalized Choi matrices

J (Φ) = i, j \sum Φ (∣ i ⟩ ⟨ j ∣) \otimes ∣ i ⟩ ⟨ j ∣ .

For a decoder $R = J (D)$ ,

R ⪰ 0, Tr_{O} R = I_{B}, D_{R} (A) = Tr_{B} [R (I_{O} \otimes A^{T})] .

For logical dimension $d$ , let

C_{N} = \frac{1}{d ^{2}} i, j \sum ∣ j ⟩ ⟨ i ∣_{O} \otimes N (∣ i ⟩ ⟨ j ∣)^{T} .

Entanglement fidelity is $tr (C_{N} R)$ . Its recovery SDP and dual are

R ⪰ 0, Tr_{O} R = I_{B} max tr (C_{N} R), Y = Y^{†}, I_{O} \otimes Y ⪰ C_{N} min tr Y .

Strict feasibility holds on both sides. This is fixed-noise fidelity recovery Fletcher. A verified dual value $U$ implies

ε_{IT} \geq max {0, 1 - U},

by a normalized Bell-projector test. It is not diamond optimization.

For a difference $Δ$ of CPTP channels, the SDP

max {tr (J (Δ) W) : 0 ⪯ W ⪯ I_{O} \otimes ρ, ρ ⪰ 0, tr ρ = 1}

has dual

min {λ : Z ⪰ 0, Z ⪰ J (Δ), Tr_{O} Z ⪯ λ I_{M}} .

Their common value is $∥ Δ ∥_{⋄} /2$ WatrousSDP.

Unrestricted diamond recovery is therefore

minimize subject to λ R ⪰ 0, Tr_{O} R = I_{B}, Z ⪰ 0, Z ⪰ J (D_{R} N - id), Tr_{O} Z ⪯ λ I_{M} .

This affine finite-dimensional SDP is not a bounded-memory strategy optimization.

Displayed formulas in this passage

unlabeled · line 1194
$J (Φ) = i, j \sum Φ (∣ i ⟩ ⟨ j ∣) \otimes ∣ i ⟩ ⟨ j ∣ .$
unlabeled · line 1198
$R ⪰ 0, Tr_{O} R = I_{B}, D_{R} (A) = Tr_{B} [R (I_{O} \otimes A^{T})] .$
unlabeled · line 1204
$C_{N} = \frac{1}{d ^{2}} i, j \sum ∣ j ⟩ ⟨ i ∣_{O} \otimes N (∣ i ⟩ ⟨ j ∣)^{T} .$
unlabeled · line 1211
$R ⪰ 0, Tr_{O} R = I_{B} max tr (C_{N} R), Y = Y^{†}, I_{O} \otimes Y ⪰ C_{N} min tr Y .$
unlabeled · line 1219
$ε_{IT} \geq max {0, 1 - U},$
unlabeled · line 1226
$max {tr (J (Δ) W) : 0 ⪯ W ⪯ I_{O} \otimes ρ, ρ ⪰ 0, tr ρ = 1}$
unlabeled · line 1231
$min {λ : Z ⪰ 0, Z ⪰ J (Δ), Tr_{O} Z ⪯ λ I_{M}} .$
unlabeled · line 1240
$minimize subject to λ R ⪰ 0, Tr_{O} R = I_{B}, Z ⪰ 0, Z ⪰ J (D_{R} N - id), Tr_{O} Z ⪯ λ I_{M} .$