Manuscript Hawk02-D@1 · Hawk02-D.tex · lemma · e6e4b72f

Lemma: Replacement-channel benchmark

Channels, complete residuals, and reference-stable estimates · lines 294–341 · proof included or linked

Hawk02-D@1#lem:replacement

Let

a_{d} = 1 - d^{- 2}, τ_{d} = I_{C} / d .

Then

d_{⋄} (id_{C}, T_{τ_{d}}) = a_{d}, σ in f d_{⋄} (id_{C}, T_{σ}) = a_{d} .

Consequently, the optimal recovery error of any constant code-to-output channel is $a_{d}$ .

Proof. Let

∣ Φ_{d} ⟩ = d^{- 1/2} j = 1 \sum^{d} ∣ j ⟩ ∣ j ⟩ .

For every $σ$ ,

⟨ Φ_{d} ∣ (I_{C} / d) \otimes σ ∣ Φ_{d} ⟩ = d^{- 2} .

Testing the Bell projector gives the lower bound $a_{d}$ .

For the upper bound, let $U_{α}$ , $α = 1, \dots, d^{2}$ , be a Weyl unitary basis with $U_{1} = I$ . Its twirl is

T_{τ_{d}} = d^{- 2} α = 1 \sum^{d^{2}} Ad_{U_{α}} .

Thus

id_{C} - T_{τ_{d}} = a_{d} id_{C} - d^{- 2} α \neq = 1 \sum Ad_{U_{α}} .

Every unitary channel has diamond norm one. The triangle inequality gives half-diamond distance at most $a_{d}$ .

A decoder after a constant channel remains constant. Choosing its output to be $τ_{d}$ attains the bound.

Displayed formulas in this passage

unlabeled · line 297
$a_{d} = 1 - d^{- 2}, τ_{d} = I_{C} / d .$
unlabeled · line 303
$d_{⋄} (id_{C}, T_{τ_{d}}) = a_{d}, σ in f d_{⋄} (id_{C}, T_{σ}) = a_{d} .$
unlabeled · line 314
$∣ Φ_{d} ⟩ = d^{- 1/2} j = 1 \sum^{d} ∣ j ⟩ ∣ j ⟩ .$
unlabeled · line 318
$⟨ Φ_{d} ∣ (I_{C} / d) \otimes σ ∣ Φ_{d} ⟩ = d^{- 2} .$
unlabeled · line 327
$T_{τ_{d}} = d^{- 2} α = 1 \sum^{d^{2}} Ad_{U_{α}} .$
unlabeled · line 332
$id_{C} - T_{τ_{d}} = a_{d} id_{C} - d^{- 2} α \neq = 1 \sum Ad_{U_{α}} .$