Manuscript II-B@1 · II-B-v1.tex · section · bfd2eb3c

An inhabited finite preparation

The three dictionaries and calibration › An inhabited finite preparation · lines 1449–1482

For any probability vector $p = (p_{0}, \dots, p_{3})$ and fixed unitaries $U_{k}$ on $X$ , define an isometry

W_{p} ∣ ψ ⟩ = k = 0 \sum^{3} p_{k} ∣ τ_{k} ⟩_{C} \otimes U_{k} ∣ ψ ⟩_{X} .

Orthogonality of the clock basis gives $W_{p}^{†} W_{p} = I_{X}$ . It can be prepared by a fixed clock-state preparation and a clock-controlled unitary, all at the sender.

For any referenced input $ρ_{X E}$ ,

(M_{C} \otimes id_{E}) (W_{p} ρ_{X E} W_{p}^{†}) = k \sum p_{k} ∣ k ⟩ ⟨ k ∣_{R} \otimes (U_{k} \otimes I_{E}) ρ_{X E} (U_{k}^{†} \otimes I_{E}) .

Thus the conditional states and their probabilities are exact.

A default preparation is

p = (1/3, 1/3, 1/3, 0), U_{k} = I_{X} .

All three distinct contact rows then have positive probability.

Choosing $U_{k} = e^{- i H (t_{k} - t_{0})}$ would install sampled conditional evolution for a declared finite Hamiltonian $H$ . That is an engineered correlation, not a derivation of an external-time-free dynamics or of a Hamiltonian constraint. Page–Wootters conditioning is a useful comparison PageWootters,CheungEn09, not an additional premise.

Displayed formulas in this passage

unlabeled · line 1453
$W_{p} ∣ ψ ⟩ = k = 0 \sum^{3} p_{k} ∣ τ_{k} ⟩_{C} \otimes U_{k} ∣ ψ ⟩_{X} .$
unlabeled · line 1462
$(M_{C} \otimes id_{E}) (W_{p} ρ_{X E} W_{p}^{†}) = k \sum p_{k} ∣ k ⟩ ⟨ k ∣_{R} \otimes (U_{k} \otimes I_{E}) ρ_{X E} (U_{k}^{†} \otimes I_{E}) .$
unlabeled · line 1471
$p = (1/3, 1/3, 1/3, 0), U_{k} = I_{X} .$