Manuscript XVII-A@1 · XVII-A.tex · proposition · 711d7ee4

Proposition: Named conditional perturbation allowance

A conditional, reference-stable error budget · lines 773–822 · proof included or linked

XVII-A@1#prop:stability, XVII-A@1#eq:channel-perturbation, XVII-A@1#eq:complete-budget

Suppose a self-adjoint boundary matrix family $β (t)$ satisfies

t \in I_{B} sup ∥ β (t) - β (t) ∥_{\infty} \leq δ .

Define $Q_{B}, O_{B}, R_{B}$ by the same integral, functional calculus, and complete record prescription as before. Then

d_{⋄} (R_{B} \circ N_{B}, M_{O_{bulk}}) \leq min {1, \frac{∣ λ _{*} ∣ π}{4} δ} .

More generally, if an actual CPTP boundary interface $N$ obeys $d_{⋄} (N, N_{B}) \leq η$ , then

d_{⋄} (R_{B} \circ N, M_{O_{bulk}}) \leq min {1, η + \frac{∣ λ _{*} ∣ π}{4} δ} .

Proof. The interval has length $π$ , so

∥ Q_{B} - Q_{B} ∥_{\infty} \leq \frac{π}{2} δ .

Lemma lem:sine yields

∥ O_{B} - O_{B} ∥_{\infty} \leq \frac{∣ λ _{*} ∣}{2} ∥ Q_{B} - Q_{B} ∥_{\infty} \leq \frac{∣ λ _{*} ∣ π}{4} δ .

Lemma lem:binary-diamond, unitary precomposition, and Theorem thm:main prove (eq:channel-perturbation). CPTP postprocessing contracts diamond distance, so the triangle inequality gives (eq:complete-budget). Half-diamond distance between channels is at most one.

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XVII-A@1#eq:channel-perturbation · line 785
$d_{⋄} (R_{B} \circ N_{B}, M_{O_{bulk}}) \leq min {1, \frac{∣ λ _{*} ∣ π}{4} δ} .$
XVII-A@1#eq:complete-budget · line 794
$d_{⋄} (R_{B} \circ N, M_{O_{bulk}}) \leq min {1, η + \frac{∣ λ _{*} ∣ π}{4} δ} .$
unlabeled · line 777
$t \in I_{B} sup ∥ β (t) - β (t) ∥_{\infty} \leq δ .$
unlabeled · line 805
$∥ Q_{B} - Q_{B} ∥_{\infty} \leq \frac{π}{2} δ .$
unlabeled · line 810
$∥ O_{B} - O_{B} ∥_{\infty} \leq \frac{∣ λ _{*} ∣}{2} ∥ Q_{B} - Q_{B} ∥_{\infty} \leq \frac{∣ λ _{*} ∣ π}{4} δ .$

Proposition: Named conditional perturbation allowance

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