Manuscript Hawk04-B@2 · Hawk04-B.tex · proposition · ba3b0713

Proposition: Classical assistance is a different access task

Residual readout, local forgetting, and the apparatus › The explicitly added residual programs · lines 1075–1105 · proof included or linked

Hawk04-B@2#prop:assistance

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Suppose residual readout produces the hybrid channel

N_{R K} (X) = x \sum ∣ x ⟩ ⟨ x ∣ \otimes N_{x} (X), x \sum N_{x} = N .

A deterministic decoder on its classical interface has the form

D_{R K} (N_{R K} (X)) = x \sum D_{x} (N_{x} (X)),

where each $D_{x}$ is CPTP on $R$ . Moreover,

ε_{rec} (N_{R K}) \leq ε_{rec} (N) .

The inequality can be strict.

Proof. A channel on a direct sum of classical blocks acts by a CPTP channel on each block and adds the outputs. A decoder can ignore $K$ , so every radiation-only strategy is available with $R K$ access. Example ex:assistance proves strictness.

Displayed formulas in this passage

unlabeled · line 1078
$N_{R K} (X) = x \sum ∣ x ⟩ ⟨ x ∣ \otimes N_{x} (X), x \sum N_{x} = N .$
unlabeled · line 1086
$D_{R K} (N_{R K} (X)) = x \sum D_{x} (N_{x} (X)),$
unlabeled · line 1093
$ε_{rec} (N_{R K}) \leq ε_{rec} (N) .$

Proposition: Classical assistance is a different access task

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