Manuscript VII-A-ST@1 · VII-A-ST.tex · proposition · c39a67a3

Proposition: Fixed-table approximation

Finite encoding and the destructive-clock instrument › A controlled finite-compression limit · lines 1776–1817 · proof included or linked

VII-A-ST@1#prop:compressionlimit, VII-A-ST@1#eq:compression

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Suppose finite diagonal tables $A_{s}^{(n)}$ satisfy

ϵ_{n} = s max A_{s}^{(n)} - A_{s} ⟶ 0.

Use a common $Λ$ such that every effect considered has spectrum in $[1/4, 3/4]$ . The corresponding complete instruments satisfy

d_{⋄} (I_{n}, I) \leq min {1, 4 ϵ_{n} /Λ} ⟶ 0, d_{⋄} = \frac{1}{2} ∥ \cdot ∥_{⋄} .

Smooth convergence of the classical inputs on common compact dependency neighborhoods gives such entrywise table convergence.

Proof. For diagonal positive matrices with spectra in $[1/4, 3/4]$ , the scalar square-root identity gives

M_{s, y}^{(n)} - M_{s, y} \leq \frac{ϵ _{n}}{2Λ} .

For contraction Kraus operators,

Ad_{L} - Ad_{L}_{⋄} \leq 2 L - L,

by inserting and subtracting $L Z L^{†}$ , also after a reference extension. Sum over the eight recorded branches and divide by two to obtain (eq:compression).

The classical continuity assertion follows from Lemma lem:Cauchy, finite differentiation, multiplication, and compact integration in the explicit coefficient formulas.

Displayed formulas in this passage

VII-A-ST@1#eq:compression · line 1786
$d_{⋄} (I_{n}, I) \leq min {1, 4 ϵ_{n} /Λ} ⟶ 0, d_{⋄} = \frac{1}{2} ∥ \cdot ∥_{⋄} .$
unlabeled · line 1779
$ϵ_{n} = s max A_{s}^{(n)} - A_{s} ⟶ 0.$
unlabeled · line 1800
$M_{s, y}^{(n)} - M_{s, y} \leq \frac{ϵ _{n}}{2Λ} .$
unlabeled · line 1805
$Ad_{L} - Ad_{L}_{⋄} \leq 2 L - L,$

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section: A controlled finite-compression limit