Manuscript I@2 · I.tex · proposition · 85631fa7

Proposition: Concrete record equation

Typed records and experiment interfaces › The record transformation · lines 1020–1052 · proof included or linked

I@2#prop:concrete-record

In $EC_{Born}$ ,

Obs_{B} [ψ, t_{S, u}] = Obs_{B} [ψ, defer_{S} (t_{S, u})] .

Proof. Write

ψ = κ \sum ∣ κ ⟩_{S} ∣ ξ_{κ} ⟩_{B} .

For a path $h$ of $u$ , let $K_{h}$ be its path map. The source's joint weight for $h, κ$ is $∥ K_{h} ξ_{κ} ∥^{2}$ .

Allocation and copying in the target produce

κ \sum ∣ κ ⟩_{S} ∣ ξ_{κ} ⟩_{B} ∣ κ ⟩_{R} .

After path $h$ , the unnormalized vector is

κ \sum ∣ κ ⟩_{S} (K_{h} ξ_{κ}) ∣ κ ⟩_{R} .

Record readout has the same joint weight $∥ K_{h} ξ_{κ} ∥^{2}$ . Sum histories with the same reported continuation output.

Displayed formulas in this passage

unlabeled · line 1023
$Obs_{B} [ψ, t_{S, u}] = Obs_{B} [ψ, defer_{S} (t_{S, u})] .$
unlabeled · line 1032
$ψ = κ \sum ∣ κ ⟩_{S} ∣ ξ_{κ} ⟩_{B} .$
unlabeled · line 1040
$κ \sum ∣ κ ⟩_{S} ∣ ξ_{κ} ⟩_{B} ∣ κ ⟩_{R} .$
unlabeled · line 1045
$κ \sum ∣ κ ⟩_{S} (K_{h} ξ_{κ}) ∣ κ ⟩_{R} .$