Manuscript En09-A@1 · En09-A.tex · section · eafec9de

Live and recorded interfaces

Typed hybrid evolution and the two notions of time › Live and recorded interfaces · lines 1150–1204

En09-A@1#eq:recorded-propagator

Before readout, use the live operator space

A_{-} = B (H_{C} \otimes H_{S}) .

After readout, use

A_{+} = {k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes X_{k} : X_{k} \in B (H_{S})} .

The space $A_{+}$ is a finite classical–quantum algebra. There is no surviving clock factor in it.

Choose a readout event with ordered germs

s_{*}^{-} < s_{*}^{+} .

They denote the interfaces immediately before and after the event. They may share a numerical scheduling coordinate with distinct event-order tags; no positive physical duration is inferred from this notation.

Let

U_{-} (s, u) = Ad_{e^{- i K (s - u) /ℏ}}

be the declared pre-event propagator. On a stationary preparation it leaves the preparation unchanged.

For the post-event interface, let $V_{k} (s, u)$ be any declared CPTP propagator satisfying

V_{k} (s, v) V_{k} (v, u) = V_{k} (s, u), V_{k} (u, u) = id .

Define

U_{+} (s, u) (k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes X_{k}) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes V_{k} (s, u) (X_{k}) .

The minimal construction chooses every $V_{k} = id$ . Time-dependent Hamiltonians or other admitted post-event dynamics are also allowed. No time-homogeneous semigroup is assumed.

Displayed formulas in this passage

En09-A@1#eq:recorded-propagator · line 1194
$U_{+} (s, u) (k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes X_{k}) = k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes V_{k} (s, u) (X_{k}) .$
unlabeled · line 1153
$A_{-} = B (H_{C} \otimes H_{S}) .$
unlabeled · line 1157
$A_{+} = {k \sum ∣ k ⟩ ⟨ k ∣_{R} \otimes X_{k} : X_{k} \in B (H_{S})} .$
unlabeled · line 1169
$s_{*}^{-} < s_{*}^{+} .$
unlabeled · line 1178
$U_{-} (s, u) = Ad_{e^{- i K (s - u) /ℏ}}$
unlabeled · line 1188
$V_{k} (s, v) V_{k} (v, u) = V_{k} (s, u), V_{k} (u, u) = id .$

Live and recorded interfaces

Displayed formulas in this passage

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