Manuscript X-E-ST@1 · X-E-ST.tex · proposition · 44614221

Proposition: Complete-report stability

Finite supplied evaluations and the Enter instrument › An explicit finite-report error · lines 2363–2393 · proof included or linked

X-E-ST@1#prop:report, X-E-ST@1#eq:report

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The map $I_{v}$ is CPTP, and, with half-diamond distance $‡ (Λ, Γ) = \frac{1}{2} ∥Λ - Γ ∥_{⋄}$ ,

‡ (I_{v}, I_{v}) = \frac{1}{2} k max ∣ v_{k} - v_{k} ∣.

The formula includes all untouched references.

Proof. Use Kraus maps $p_{y} (v_{k}) ∣ k, y ⟩_{R} \otimes D_{k}$ . For any input operator, including a reference, the difference has orthogonal blocks

\frac{y ( v _{k} - v _{k} )}{2} X_{k k} .

The output trace norm is bounded by

k max ∣ v_{k} - v_{k} ∣ k \sum ∥ X_{k k} ∥_{1} \leq k max ∣ v_{k} - v_{k} ∣ ∥ X ∥_{1} .

The last inequality is contraction of pinching, an average of unitary conjugations. A clock-basis input at a maximizing $k$ saturates the bound.

Displayed formulas in this passage

X-E-ST@1#eq:report · line 2368
$‡ (I_{v}, I_{v}) = \frac{1}{2} k max ∣ v_{k} - v_{k} ∣.$
unlabeled · line 2381
$\frac{y ( v _{k} - v _{k} )}{2} X_{k k} .$
unlabeled · line 2385
$k max ∣ v_{k} - v_{k} ∣ k \sum ∥ X_{k k} ∥_{1} \leq k max ∣ v_{k} - v_{k} ∣ ∥ X ∥_{1} .$

Proposition: Complete-report stability

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