Manuscript X-I-ST@1 · X-I-ST.tex · proposition · 9745fd40

Proposition: Complete numerical-report comparison

Complete destructive instruments and their composition › Numerical report encoding and an exact error formula · lines 1833–1860 · proof included or linked

X-I-ST@1#prop:reporterror, X-I-ST@1#eq:reporterror

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The maps $I_{v}$ are CPTP. On the same report interface,

d_{⋄} (I_{v}, I_{v ~}) = \frac{1}{2} j max ∣ v_{j} - v ~_{j} ∣, d_{⋄} (Λ, Γ) = \frac{1}{2} ∥ Λ - Γ ∥_{⋄} .

The comparison includes every untouched reference.

Proof. The Kraus maps are $p_{y} (v_{j}) ∣ j, y, w_{j} ⟩ \otimes D_{j}$ . Their completeness follows from $\sum_{y} p_{y} (v_{j}) = 1$ .

After the clock pinching, the difference has orthogonal $(j, y)$ -blocks $y (v_{j} - v ~_{j}) Z_{j j} /2$ . Pinching is trace-norm contractive, giving the diamond upper bound $max_{j} ∣ v_{j} - v ~_{j} ∣$ . A basis input at a maximizing $j$ attains it. Divide by two.

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X-I-ST@1#eq:reporterror · line 1837
$d_{⋄} (I_{v}, I_{v ~}) = \frac{1}{2} j max ∣ v_{j} - v ~_{j} ∣, d_{⋄} (Λ, Γ) = \frac{1}{2} ∥ Λ - Γ ∥_{⋄} .$

Proposition: Complete numerical-report comparison

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