Manuscript Hawk07-B@1 · Hawk07-B.tex · section · b9785538

The decoder program

A specified receiver and two storage ledgers › The decoder program · lines 797–826

Hawk07-B@1#eq:sdp

Let $A_{*} = dim R_{*}$ . Use the unnormalized input-first Choi convention

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

A candidate decoder Choi matrix $C$ on $R_{*}^{'} \otimes L_{out}$ defines

D_{C} (Y) = Tr_{R_{*}^{'}} [(Y^{T} \otimes I_{L}) C] .

Let

J_{Δ} (C) = J (D_{C} N_{*}^{U} - id_{L}) .

Specify the receiver by

minimize subject to t /2 C \geq 0, Tr_{L} C = I_{R_{*}^{'}}, Z \geq J_{Δ} (C), Z \geq - J_{Δ} (C), Tr_{L} Z \leq t I_{L^{'}}, t \geq 0.

Among minimizing decoder matrices choose the one of least squared Frobenius norm, and denote its channel by $D_{*}$ .

Displayed formulas in this passage

Hawk07-B@1#eq:sdp · line 814
$minimize subject to t /2 C \geq 0, Tr_{L} C = I_{R_{*}^{'}}, Z \geq J_{Δ} (C), Z \geq - J_{Δ} (C), Tr_{L} Z \leq t I_{L^{'}}, t \geq 0.$
unlabeled · line 801
$J (Φ) = i, j \sum ∣ i ⟩ ⟨ j ∣ \otimes Φ (∣ i ⟩ ⟨ j ∣) .$
unlabeled · line 806
$D_{C} (Y) = Tr_{R_{*}^{'}} [(Y^{T} \otimes I_{L}) C] .$
unlabeled · line 810
$J_{Δ} (C) = J (D_{C} N_{*}^{U} - id_{L}) .$