Manuscript Em04-A@2 · Em04-A-v2.tex · corollary · f5b28abc

Corollary: Purity, coherence, and vacuum convergence

Exact diagnostic and entanglement identities › Purity and computational-basis coherence · lines 571–605 · proof included or linked

Em04-A@2#cor:diagnostics, Em04-A@2#eq:purity, Em04-A@2#eq:coherence, Em04-A@2#eq:Spurity

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For (eq:reduction),

Tr ρ^{2} C_{ℓ_{1}}^{norm} (ρ) \frac{d}{d t} Tr S^{2} = (1 - q)^{2} + q^{2} Tr S^{2}, = \frac{q \sum _{i \neq = j} ∣ S _{ij} ∣}{2 ^{N} - 1} \leq \frac{N - 1}{2 ^{N} - 1} q, = - 4 γ_{d} ∥ S - Δ S ∥_{F}^{2} \leq 0.

For $γ_{r} > 0$ ,

d_{tr} (ρ (t), ∣Ω ⟩ ⟨ Ω∣) = q (t), Tr ρ (t)^{2} ⟶ 1.

Proof. The vacuum and excitation blocks are orthogonal, proving (eq:purity). All full-state off-diagonal entries belong to $q S$ . Positivity and Cauchy–Schwarz give

i \neq = j \sum ∣ S_{ij} ∣ \leq (i \sum S_{ii})^{2} - 1 \leq N - 1.

Differentiating conditional purity eliminates the commutator, and $Tr (S (S - Δ S)) = ∥ S - Δ S ∥_{F}^{2}$ . The difference from the vacuum has blocks $- q$ and $q S$ , so its trace norm is $2 q$ . The limiting purity follows from (eq:purity).

Displayed formulas in this passage

Em04-A@2#eq:purity · line 574
$Tr ρ^{2} C_{ℓ_{1}}^{norm} (ρ) \frac{d}{d t} Tr S^{2} = (1 - q)^{2} + q^{2} Tr S^{2}, = \frac{q \sum _{i \neq = j} ∣ S _{ij} ∣}{2 ^{N} - 1} \leq \frac{N - 1}{2 ^{N} - 1} q, = - 4 γ_{d} ∥ S - Δ S ∥_{F}^{2} \leq 0.$
Em04-A@2#eq:coherence · line 574
$Tr ρ^{2} C_{ℓ_{1}}^{norm} (ρ) \frac{d}{d t} Tr S^{2} = (1 - q)^{2} + q^{2} Tr S^{2}, = \frac{q \sum _{i \neq = j} ∣ S _{ij} ∣}{2 ^{N} - 1} \leq \frac{N - 1}{2 ^{N} - 1} q, = - 4 γ_{d} ∥ S - Δ S ∥_{F}^{2} \leq 0.$
Em04-A@2#eq:Spurity · line 574
$Tr ρ^{2} C_{ℓ_{1}}^{norm} (ρ) \frac{d}{d t} Tr S^{2} = (1 - q)^{2} + q^{2} Tr S^{2}, = \frac{q \sum _{i \neq = j} ∣ S _{ij} ∣}{2 ^{N} - 1} \leq \frac{N - 1}{2 ^{N} - 1} q, = - 4 γ_{d} ∥ S - Δ S ∥_{F}^{2} \leq 0.$
unlabeled · line 586
$d_{tr} (ρ (t), ∣Ω ⟩ ⟨ Ω∣) = q (t), Tr ρ (t)^{2} ⟶ 1.$
unlabeled · line 596
$i \neq = j \sum ∣ S_{ij} ∣ \leq (i \sum S_{ii})^{2} - 1 \leq N - 1.$

Corollary: Purity, coherence, and vacuum convergence

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