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

Lemma: Constant-output recovery benchmark

Quantitative channel certificates › Normalization checks · lines 732–755 · proof included or linked

Hawk04-A@1#lem:replacement

For every code density operator $σ$ ,

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

Thus a constant accessible channel has optimal recovery error $a_{d}$ .

Proof. For the Bell input, a replacement output is $(I_{Q} / d) \otimes σ$ , whose overlap with $∣ Φ_{d} ⟩$ is $d^{- 2}$ . The Bell-projector test proves the lower bound.

The Weyl twirl represents

T_{τ_{C}} = d^{- 2} k \sum Ad_{U_{k}} .

Separating the identity term and using that unitary channels have diamond norm one gives the matching upper bound. A decoder after a constant channel remains constant.

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unlabeled · line 735
$d_{⋄} (id_{C}, T_{σ}) \geq a_{d}, d_{⋄} (id_{C}, T_{τ_{C}}) = a_{d} .$
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$T_{τ_{C}} = d^{- 2} k \sum Ad_{U_{k}} .$

Lemma: Constant-output recovery benchmark

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