Manuscript Hawk04-A@1 · Hawk04-A.tex · section · 526092ba

A finite residual optimization

What a positive finite-time certificate must contain › A finite residual optimization · lines 1667–1697

For the finite complementary Choi state $J$ , a trace-norm certificate can be obtained from

minimize subject to \frac{1}{2} Tr (X_{+} + X_{-}), J - \frac{I _{Q}}{d} \otimes σ = X_{+} - X_{-}, X_{\pm} ⪰ 0, σ ⪰ 0, Tr σ = 1.

Every rigorously feasible tuple gives an upper bound on $e$ . The objective is a Choi trace distance, not the exact diamond norm.

A decoder search uses an unnormalized decoder Choi operator satisfying

J_{D} ⪰ 0, Tr_{C} J_{D} = I_{R} .

Composition with the known finite channel is linear. A feasible decoder and a validated bound on its decoded Choi distance supply (eq:direct-certificate).

An exploratory floating-point optimum is not a proof of positivity, trace preservation, or the claimed error. No numerical execution is asserted here.

Displayed formulas in this passage

unlabeled · line 1671
$minimize subject to \frac{1}{2} Tr (X_{+} + X_{-}), J - \frac{I _{Q}}{d} \otimes σ = X_{+} - X_{-}, X_{\pm} ⪰ 0, σ ⪰ 0, Tr σ = 1.$
unlabeled · line 1685
$J_{D} ⪰ 0, Tr_{C} J_{D} = I_{R} .$