Manuscript En07-B@1 · En07-B.tex · section · 45dff4a1

Two necessary debugging corrections

Scope, provenance, and the corrected question › Two necessary debugging corrections · lines 157–199

En07-B@1#conv:normalization, En07-B@1#conv:direction

Every operational error in this paper is a half-diamond distance:

d_{⋄} (Φ, Ψ) := \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .

Consequently a lower bound $a$ for $d_{⋄}$ is a lower bound $2 a$ for the unhalved diamond norm.

Thus the record-only bound is $1 - d^{- 1}$ in half-diamond distance, and $2 (1 - d^{- 1})$ in unhalved diamond norm. The featureless-output bound is normalized in the same way.

For an isometry $V : C \to R \otimes B$ ,

B -complement nearly constant R -output nearly constant ⟹ ⟹ small achievable recovery error from R, large unavoidable recovery error from R .

The first implication uses approximate information–disturbance conversion. The second is the featureless-output obstruction proved in Proposition prop:featureless.

In particular, the Stage D sentence in the proposed programme cannot apply the featureless-radiation lower bound to a nearly constant endpoint complement. The repaired endpoint estimate is Theorem thm:endpoint.

Displayed formulas in this passage

unlabeled · line 163
$d_{⋄} (Φ, Ψ) := \frac{1}{2} ∥ Φ - Ψ ∥_{⋄} .$
unlabeled · line 178
$B -complement nearly constant R -output nearly constant ⟹ ⟹ small achievable recovery error from R, large unavoidable recovery error from R .$