Manuscript X-D-ST@1 · X-D-ST.tex · proposition · e0cf8fdb

Proposition: Reference-stable finite-table convergence

Finite Enter encoding of supplied limiting evaluations › A specific complete-channel limit for the table · lines 2265–2307 · proof included or linked

X-D-ST@1#prop:table, X-D-ST@1#eq:table-error

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The channel $I_{v}$ is CPTP, and

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

Consequently the explicitly encoded ramp and limiting tables obey

d_{⋄} (I_{v_{ε}}, I_{v_{0}}) \leq \frac{M}{2 B _{tab}} ε .

All statements include untouched references.

Proof. A complete Kraus family is

p_{y} (v_{k}) ∣ k, y ⟩_{R} \otimes D_{k} .

For an arbitrary input operator with a reference included, the difference channel has orthogonal blocks

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

Its trace norm is at most

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

The final inequality is trace-norm contraction of pinching, an average of unitary conjugations. A clock-basis input at a maximizing index saturates the estimate. Divide by two. Equation (eq:evaluation-rate) gives the rate.

Displayed formulas in this passage

X-D-ST@1#eq:table-error · line 2268
$d_{⋄} (I_{v}, I_{v}) = \frac{1}{2} k max ∣ v_{k} - v_{k} ∣, d_{⋄} (Λ, Γ) = \frac{1}{2} ∥Λ - Γ ∥_{⋄} .$
unlabeled · line 2278
$d_{⋄} (I_{v_{ε}}, I_{v_{0}}) \leq \frac{M}{2 B _{tab}} ε .$
unlabeled · line 2287
$p_{y} (v_{k}) ∣ k, y ⟩_{R} \otimes D_{k} .$
unlabeled · line 2293
$\frac{y ( v _{k} - v _{k} )}{2} X_{k k} .$
unlabeled · line 2297
$k max ∣ v_{k} - v_{k} ∣ k \sum ∥ X_{k k} ∥_{1} \leq k max ∣ v_{k} - v_{k} ∣ ∥ X ∥_{1} .$

Proposition: Reference-stable finite-table convergence

Displayed formulas in this passage

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