Manuscript III-B@1 · III-B-v1.tex · proposition · d2c98f62

Proposition: Sampled stationarity and conditioning

Finite Enter experiments and the clock instrument › A finite conditional-evolution lemma · lines 1622–1650 · proof included or linked

III-B@1#prop:stationarity

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(S_{C} \otimes U) ρ (S_{C}^{†} \otimes U^{†}) = ρ,

then

σ_{k + 1} = U σ_{k} U^{†}, p_{k + 1} = p_{k} .

Thus supported normalized conditional states obey the same sampled unitary law. The cyclic condition also requires $U^{4} σ_{0} U^{† 4} = σ_{0}$ .

If $U^{4} = I$ , an inhabited class is supplied by the isometry

W ∣ ψ ⟩ = \frac{1}{2} k = 0 \sum^{3} ∣ τ_{k} ⟩ \otimes U^{k} ∣ ψ ⟩,

including arbitrary input references.

Proof. Take the $(k + 1, k + 1)$ clock block of the stationarity equation. Trace gives equality of probabilities. For the example, $W^{†} W = I$ , and $(S_{C} \otimes U) W = W$ . These are isometry identities and hence preserve references.

Displayed formulas in this passage

unlabeled · line 1625
$(S_{C} \otimes U) ρ (S_{C}^{†} \otimes U^{†}) = ρ,$
unlabeled · line 1629
$σ_{k + 1} = U σ_{k} U^{†}, p_{k + 1} = p_{k} .$
unlabeled · line 1637
$W ∣ ψ ⟩ = \frac{1}{2} k = 0 \sum^{3} ∣ τ_{k} ⟩ \otimes U^{k} ∣ ψ ⟩,$

Proposition: Sampled stationarity and conditioning

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