Manuscript VII-G@1 · VII-G.tex · proposition · 54f561e8

Proposition: Quantitative approximation to the diagonal model

An explicit resource-budgeted feedback witness › An exact classical sector description · lines 1989–2025 · proof included or linked

VII-G@1#prop:classical-approx, VII-G@1#eq:classical-approximation

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Start at one label $g_{0} = (k_{0}, q_{0})$ . Let $Δ_{Z} ρ = P_{0} ρ P_{0} + P_{1} ρ P_{1}$ , and compare the complete history–matter outputs from $ρ$ and $Δ_{Z} ρ$ . After $n \leq B - k_{0}$ active steps,

d_{tr} (H_{n} (ρ), H_{n} (Δ_{Z} ρ)) = ∣ ρ_{01} ∣ (1 - η^{2})^{n /2} .

Their geometric history laws agree exactly. Forgetting histories can only decrease the distance in (eq:classical-approximation).

Proof. On a fixed history, the difference has only a pair of conjugate off-diagonal entries. Its trace distance contribution is the absolute value of one entry. The diagonal feedback unitaries change only its phase.

For either outcome,

⟨ 0∣ M_{a} ∣0 ⟩ ⟨ 1∣ M_{a} ∣1 ⟩ = \frac{1}{2} 1 - η^{2} .

Each length- $n$ history therefore contributes $∣ ρ_{01} ∣ (1 - η^{2} /2)^{n}$ . There are $2^{n}$ histories, so their direct-sum contributions give (eq:classical-approximation). Equal diagonals give equal history probabilities by (eq:classical-sector). Marginalization over histories is a channel.

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VII-G@1#eq:classical-approximation · line 1995
$d_{tr} (H_{n} (ρ), H_{n} (Δ_{Z} ρ)) = ∣ ρ_{01} ∣ (1 - η^{2})^{n /2} .$
unlabeled · line 2014
$⟨ 0∣ M_{a} ∣0 ⟩ ⟨ 1∣ M_{a} ∣1 ⟩ = \frac{1}{2} 1 - η^{2} .$

Proposition: Quantitative approximation to the diagonal model

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