Manuscript VII-E-ST@1 · VII-E-ST.tex · proposition · d28d9c5d

Proposition: Finite parameter compression

Finite clock, residual audit, and encoding limits › Finite compression and implementation bounds · lines 2078–2117 · proof included or linked

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

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Fix the finite routing and geometric label dictionary. For active source instruments with $0 \leq η, η \leq η_{*} < 1$ , put

L_{*} = \frac{1}{2 2 ( 1 - η _{*} )} .

Their complete one-step channels satisfy

d_{⋄} (F, F) \leq min {1, 2 (L_{*} ∣ η - η ∣ + h max ∣ ϑ_{h} - ϑ_{h} ∣)} .

Thus finite numerical approximations to the matrix calibrations converge in complete channel norm on this fixed finite model. No uniform Lipschitz assertion at $η = 1$ is made.

Proof. The derivatives of the square-root diagonal entries of $M_{a}$ are bounded by $L_{*}$ . Also

e^{- i ϑQ} - e^{- i ϑQ}_{\infty} \leq ∣ ϑ - ϑ ∣.

For contraction operators $K, K$ ,

Ad_{K} - Ad_{K}_{⋄} \leq 2 K - K_{\infty},

by expanding the difference into two terms, including after amplification. Sum over the two recorded branches and divide by two.

Displayed formulas in this passage

VII-E-ST@1#eq:compression · line 2087
$d_{⋄} (F, F) \leq min {1, 2 (L_{*} ∣ η - η ∣ + h max ∣ ϑ_{h} - ϑ_{h} ∣)} .$
unlabeled · line 2083
$L_{*} = \frac{1}{2 2 ( 1 - η _{*} )} .$
unlabeled · line 2105
$e^{- i ϑQ} - e^{- i ϑQ}_{\infty} \leq ∣ ϑ - ϑ ∣.$
unlabeled · line 2110
$Ad_{K} - Ad_{K}_{⋄} \leq 2 K - K_{\infty},$

Nearby in this section

section: Finite compression and implementation bounds