Manuscript Hawk01-B@1 · Hawk01-B.tex · proposition · cfa976f1

Proposition: Lipschitz error propagation

Regulators, implementation errors, and certificates · lines 1831–1860 · proof included or linked

Hawk01-B@1#prop:robustness, Hawk01-B@1#eq:recLipschitz, Hawk01-B@1#eq:leakLipschitz, Hawk01-B@1#eq:telescoping

For channels on common interfaces,

ϵ_{rec} (N) - ϵ_{rec} (N) δ_{leak} (C) - δ_{leak} (C) \leq D_{⋄} (N, N), \leq D_{⋄} (C, C) .

For composable CPTP maps,

D_{⋄} (Φ_{n} \dots Φ_{1}, Φ_{n} \dots Φ_{1}) \leq k = 1 \sum^{n} D_{⋄} (Φ_{k}, Φ_{k}) .

Proof. For any decoder, apply the triangle inequality and contractivity under decoder composition, then optimize and exchange the channels. For leakage use the same triangle inequality before optimizing the constant state. Expand the sequential difference as a telescoping sum and use contractivity of the diamond norm under CPTP pre- and post-composition.

Displayed formulas in this passage

Hawk01-B@1#eq:recLipschitz · line 1834
$ϵ_{rec} (N) - ϵ_{rec} (N) δ_{leak} (C) - δ_{leak} (C) \leq D_{⋄} (N, N), \leq D_{⋄} (C, C) .$
Hawk01-B@1#eq:leakLipschitz · line 1834
$ϵ_{rec} (N) - ϵ_{rec} (N) δ_{leak} (C) - δ_{leak} (C) \leq D_{⋄} (N, N), \leq D_{⋄} (C, C) .$
Hawk01-B@1#eq:telescoping · line 1843
$D_{⋄} (Φ_{n} \dots Φ_{1}, Φ_{n} \dots Φ_{1}) \leq k = 1 \sum^{n} D_{⋄} (Φ_{k}, Φ_{k}) .$

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section: Regulators, implementation errors, and certificates