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

Proposition: Uniform-channel incompatibility

Debugging the proposed imports and physical interpretation › Closeness to an information-losing semiclassical channel · lines 1738–1763 · proof included or linked

Hawk02-D@1#prop:semiclassical

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Let $N_{rep}$ be a constant semiclassical comparison channel. If

d_{⋄} (N, N_{rep}) \leq b,

then

ε_{rec} (N) \geq 1 - d^{- 2} - b .

Therefore a highly recoverable microscopic channel cannot also be uniformly close in diamond distance to that replacement channel.

Proof. For every decoder,

d_{⋄} (DN, id_{C}) \geq d_{⋄} (D N_{rep}, id_{C}) - b .

The decoded replacement channel is constant. Apply Lemma lem:replacement and take the decoder infimum.

Displayed formulas in this passage

unlabeled · line 1742
$d_{⋄} (N, N_{rep}) \leq b,$
unlabeled · line 1746
$ε_{rec} (N) \geq 1 - d^{- 2} - b .$
unlabeled · line 1756
$d_{⋄} (DN, id_{C}) \geq d_{⋄} (D N_{rep}, id_{C}) - b .$

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section: Closeness to an information-losing semiclassical channel