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

Corollary: Mixed-state extension

Stability and represented measurements › Mixed states and finite adaptivity · lines 2072–2095 · proof included or linked

I@2#cor:mixed

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In a model of $T$ , evaluation through a purification of a laboratory density operator $ρ$ gives

Pr (κ) = tr (P^{S}_{κ} ρ), ρ^{'}_{κ} = \frac{tr _{S} ( P ^{S} _{κ} ρ P ^{S} _{κ} )}{tr ( P ^{S} _{κ} ρ )}

on positive branches. Both expressions are independent of the purification.

Proof. Apply the probability and update theorems to a purification. Trace its inaccessible purifying factors from the conditioned residual. The formulas depend only on $ρ$ . Purifying names can be returned as inert spectators in the total pure configuration; no term-level discard is needed.

Displayed formulas in this passage

unlabeled · line 2076
$Pr (κ) = tr (P^{S}_{κ} ρ), ρ^{'}_{κ} = \frac{tr _{S} ( P ^{S} _{κ} ρ P ^{S} _{κ} )}{tr ( P ^{S} _{κ} ρ )}$

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corollary: Corollary: Finite adaptive execution