Manuscript V-B@1 · V-B-v1.tex · proposition · 3fadc1ad

Proposition: Covariance plus stationarity

Time, rods, histories, and conditional evolution · lines 1998–2024 · proof included or linked

V-B@1#prop:stationarity

Let finite generators $H_{C}, H_{X}$ satisfy

[H_{C} \otimes I + I \otimes H_{X}, ρ_{C X}] = 0, P_{k} = e^{- i H_{C} t_{k}} P_{0} e^{i H_{C} t_{k}} .

Then

σ_{k} = e^{- i H_{X} t_{k}} σ_{0} e^{i H_{X} t_{k}}, p_{k} = 1/ D .

The normalized conditional residuals obey the same sampled unitary law.

Proof. Stationarity implies

(e^{i H_{C} t} \otimes I) ρ (e^{- i H_{C} t} \otimes I) = (I \otimes e^{- i H_{X} t}) ρ (I \otimes e^{i H_{X} t}) .

Insert this identity in $Tr_{C} [(P_{k} \otimes I) ρ]$ . All probabilities coincide, and their sum is one.

Displayed formulas in this passage

unlabeled · line 2001
$[H_{C} \otimes I + I \otimes H_{X}, ρ_{C X}] = 0, P_{k} = e^{- i H_{C} t_{k}} P_{0} e^{i H_{C} t_{k}} .$
unlabeled · line 2007
$σ_{k} = e^{- i H_{X} t_{k}} σ_{0} e^{i H_{X} t_{k}}, p_{k} = 1/ D .$
unlabeled · line 2017
$(e^{i H_{C} t} \otimes I) ρ (e^{- i H_{C} t} \otimes I) = (I \otimes e^{- i H_{X} t}) ρ (I \otimes e^{i H_{X} t}) .$