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

Theorem: History-sensitive observation principle

Observation boundaries and stochastic histories › History instruments · lines 621–658 · proof included or linked

VII-G@1#thm:histories, VII-G@1#eq:time-branch-inclusion

Suppose that, for each $t, g, h$ ,

Ψ^{(t) *}_{h ∣ g} (O_{t, h}) \subseteq O_{t - 1, g},

where each $O_{t, g} \subseteq L (H)$ contains $I$ . If two initial hybrid states agree on $⨁_{g} O_{0, g}$ , then:

all geometric history probabilities through time $T$ agree;
for every history ending at $g_{n}$ , the corresponding unnormalized blocks agree on $O_{n, g_{n}}$ ;
on every positive-probability history, the normalized conditional states agree on the same space.

Proof. For $B \in O_{n, g_{n}}$ , its history-conditioned expectation is

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

Successive applications of (eq:time-branch-inclusion) put the bracket in $O_{0, g_{0}}$ . Initial agreement therefore proves (ii). Taking $B = I$ proves (i). Equal positive probabilities permit division by the same scalar, proving (iii).

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VII-G@1#eq:time-branch-inclusion · line 624
$Ψ^{(t) *}_{h ∣ g} (O_{t, h}) \subseteq O_{t - 1, g},$
unlabeled · line 643
$Tr (B ρ_{γ}) = Tr ([Ψ^{(1) *}_{g_{1} ∣ g_{0}} \circ \dots \circ Ψ^{(n) *}_{g_{n} ∣ g_{n - 1}} (B)] ρ_{g_{0}}) .$

Theorem: History-sensitive observation principle

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