Manuscript IX-C-ST@1 · IX-C-ST.tex · section · 892f50c4

Finite evaluation records and a normed comparison

Shared finite clock, certificates, and regular transport › Finite evaluation records and a normed comparison · lines 768–801

IX-C-ST@1#eq:tablechannel

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A finite catalogue entry records:

(σ, ℏ -order, c -identifier, f, g, f^{'} -identifiers, ρ, μ, coefficient type, error enclosure) .

Only finitely many entries are installed in one experiment. For an arbitrary real $c$ , a finite entry may store its symbolic specification or a declared finite approximation. There is no finite-bit exact encoding of all arbitrary real numbers or all smooth functions.

Let $i$ be a classical catalogue index and let $P$ be an arbitrary finite payload. Pinch the catalogue in its designated basis and write

X_{k i} = (⟨ τ_{k} ∣ ⟨ i ∣ \otimes I_{P}) X (∣ τ_{k} ⟩ ∣ i ⟩ \otimes I_{P}) .

For installed values $v_{k i} \in [- 1, 1]$ , set

p_{y} (k, i) = \frac{1 + y v _{k i}}{2}, y = \pm 1,

and let $r (k, i)$ be the fixed report word. Define

I_{v} (X) = k, i, y \sum ∣ k, i, y, r (k, i)⟩ ⟨ k, i, y, r (k, i) ∣_{W} \otimes p_{y} (k, i) X_{k i} .

Displayed formulas in this passage

IX-C-ST@1#eq:tablechannel · line 795
$I_{v} (X) = k, i, y \sum ∣ k, i, y, r (k, i)⟩ ⟨ k, i, y, r (k, i) ∣_{W} \otimes p_{y} (k, i) X_{k i} .$
unlabeled · line 771
$(σ, ℏ -order, c -identifier, f, g, f^{'} -identifiers, ρ, μ, coefficient type, error enclosure) .$
unlabeled · line 785
$X_{k i} = (⟨ τ_{k} ∣ ⟨ i ∣ \otimes I_{P}) X (∣ τ_{k} ⟩ ∣ i ⟩ \otimes I_{P}) .$
unlabeled · line 791
$p_{y} (k, i) = \frac{1 + y v _{k i}}{2}, y = \pm 1,$

Finite evaluation records and a normed comparison

Displayed formulas in this passage

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