Manuscript Oai265-A@2 · Oai265-A.tex · lemma · c0a34a45

Lemma: Optimal deterministic recovery from a constant channel

Stage 5: complete channels and the entropy separation › The sharp replacement-channel benchmark · lines 1371–1405 · proof included or linked

Oai265-A@2#lem:constant

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For $d \geq 2$ ,

σ in f d_{⋄} (id_{d}, R_{σ}) = 1 - d^{- 2} .

The maximally mixed replacement attains the infimum. Consequently every constant accessible channel has optimal recovery error $1 - d^{- 2}$ .

Proof. On a maximally entangled input, the identity output is $∣ Ω_{d} ⟩ ⟨ Ω_{d} ∣$ , while a replacement output is $I_{d} / d \otimes σ$ . Their overlap against the ideal Bell projector is $d^{- 2}$ , independently of $σ$ . The effect test therefore gives the lower bound.

For the upper bound, let $(W_{ab})_{a, b = 0}^{d - 1}$ be the shift-phase Weyl unitaries. Their twirl is

R_{I / d} = \frac{1}{d ^{2}} a, b \sum Ad_{W_{ab}} .

Hence

id - R_{I / d} = (1 - d^{- 2}) id - \frac{1}{d ^{2}} (a, b) \neq = (0, 0) \sum Ad_{W_{ab}} .

The diamond norm is at most $2 (1 - d^{- 2})$ . Division by two gives equality. A decoder after a constant channel is still constant and can prepare $I / d$ , proving the final assertion.

Displayed formulas in this passage

unlabeled · line 1374
$σ in f d_{⋄} (id_{d}, R_{σ}) = 1 - d^{- 2} .$
unlabeled · line 1391
$R_{I / d} = \frac{1}{d ^{2}} a, b \sum Ad_{W_{ab}} .$
unlabeled · line 1395
$id - R_{I / d} = (1 - d^{- 2}) id - \frac{1}{d ^{2}} (a, b) \neq = (0, 0) \sum Ad_{W_{ab}} .$

Lemma: Optimal deterministic recovery from a constant channel

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