Manuscript VII-G@1 · VII-G.tex · section · 54f561e8

History instruments

Observation boundaries and stochastic histories › History instruments · lines 582–605

VII-G@1#eq:path-block, VII-G@1#eq:history-channel

Allow a finite sequence of channels with branches $Ψ^{(t)}_{h ∣ g}$ , $t = 1, \dots, T$ . All tables are fixed; the current label already contains the controller memory.

For a history $γ = (g_{0}, \dots, g_{n})$ , define

ρ_{γ} = Ψ^{(n)}_{g_{n} ∣ g_{n - 1}} \circ \dots \circ Ψ^{(1)}_{g_{1} ∣ g_{0}} (ρ_{g_{0}}) .

Its probability is $p_{γ} = Tr ρ_{γ}$ . The complete history-retaining output is

H_{n} (ρ) = γ \in G^{n + 1} ⨁ ρ_{γ} .

Forbidden paths have zero blocks.

Displayed formulas in this passage

VII-G@1#eq:path-block · line 590
$ρ_{γ} = Ψ^{(n)}_{g_{n} ∣ g_{n - 1}} \circ \dots \circ Ψ^{(1)}_{g_{1} ∣ g_{0}} (ρ_{g_{0}}) .$
VII-G@1#eq:history-channel · line 599
$H_{n} (ρ) = γ \in G^{n + 1} ⨁ ρ_{γ} .$

Nearby in this section

lemma: Lemma: Completeness of history retention
theorem: Theorem: History-sensitive observation principle
section: History instruments