Manuscript Hawk01-A@1 · Hawk01-A.tex · proposition · 11979653

Proposition: Continuity with the correct normalization

Exact delayed-query scoring and classical nulls · lines 754–785 · proof included or linked

Hawk01-A@1#prop:scorecontinuity, Hawk01-A@1#eq:scorecontinuity, Hawk01-A@1#eq:scorediamond

For two output ensembles on a common space,

∣ Q_{opt} ({ρ_{x}}) - Q_{opt} ({ρ_{x}}) ∣ \leq \frac{1}{4} x \sum D_{tr} (ρ_{x}, ρ_{x}) .

In particular, for regulated channels,

Q_{opt} (N) - Q_{opt} (N) \leq D_{⋄} (N, N) .

Proof. For a fixed decoder, put

F_{x} = \frac{1}{2} j = 1 \sum^{2} M_{x_{j} ∣ j} .

Then $0 \leq F_{x} \leq I$ , and its score is $\frac{1}{4} \sum_{x} Tr (F_{x} ρ_{x})$ . The variational characterization of trace distance bounds the score difference by the right-hand side of (eq:scorecontinuity). Taking suprema in both directions proves the assertion. A half-diamond bound controls the trace distance for every input state, proving (eq:scorediamond).

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Hawk01-A@1#eq:scorecontinuity · line 757
$∣ Q_{opt} ({ρ_{x}}) - Q_{opt} ({ρ_{x}}) ∣ \leq \frac{1}{4} x \sum D_{tr} (ρ_{x}, ρ_{x}) .$
Hawk01-A@1#eq:scorediamond · line 765
$Q_{opt} (N) - Q_{opt} (N) \leq D_{⋄} (N, N) .$
unlabeled · line 774
$F_{x} = \frac{1}{2} j = 1 \sum^{2} M_{x_{j} ∣ j} .$

Proposition: Continuity with the correct normalization

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