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

Intrinsic observation and histories

Contract G: finite completely positive hybrid evolution › Intrinsic observation and histories · lines 1272–1289

VII-E-ST@1#eq:history

This is the direct-sum specialization of a general adjoint criterion, not a new general channel theorem.

For a finite sequence of fixed tables and a history $γ = (g_{0}, \dots, g_{n})$ , define

X_{γ} = Ψ^{(n)}_{g_{n} ∣ g_{n - 1}} \dots Ψ^{(1)}_{g_{1} ∣ g_{0}} (X_{g_{0}}), H_{n} (X) = γ ⨁ X_{γ} .

The history map is a channel: each path map is completely positive, and summing the last successor, then the preceding ones, proves trace preservation.

Displayed formulas in this passage

VII-E-ST@1#eq:history · line 1279
$X_{γ} = Ψ^{(n)}_{g_{n} ∣ g_{n - 1}} \dots Ψ^{(1)}_{g_{1} ∣ g_{0}} (X_{g_{0}}), H_{n} (X) = γ ⨁ X_{γ} .$

Nearby in this section

theorem: Theorem: Hybrid observation criterion
theorem: Theorem: History-sensitive observation
section: Intrinsic observation and histories
example: Example: An aggregate continuation can hide branch failure
proposition: Proposition: Minimal backward boundary
section: Intrinsic observation and histories
theorem: Theorem: Spectator consistency for Contract G