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

Theorem: Finite-window unitary limit

Transient entanglement and controlled comparisons · lines 803–849 · proof included or linked

Em04-A@2#thm:largechi, Em04-A@2#eq:Sunit, Em04-A@2#eq:unitbound

Fix $N, S_{0}, h, r$ , and let

S_{u} (τ) = e^{- i τ h} S_{0} e^{i τ h}, ρ_{u} = V S_{u} V^{†} .

For finite $χ > 0$ ,

∥ S (τ) - S_{u} (τ) ∥_{1} d_{tr} (ρ (τ), ρ_{u} (τ)) \leq \frac{4 ( 1 - r ) τ}{χ}, \leq min {1, 2 τ / χ} .

All diagnostics above converge uniformly to their unitary counterparts on every fixed $[0, T]$ . In particular,

0 \leq τ \leq T sup ∣ C_{ij} - C_{ij}^{u} ∣ \leq min {1, 4 T / χ} .

Proof. Variation of constants with the unitary propagator $U$ gives

S (τ) - S_{u} (τ) = - κ \int_{0}^{τ} U_{τ - s} (S (s) - Δ S (s)) d s .

Both matrices in the integrand difference are density matrices, so its trace norm is at most two. This proves (eq:Sunit). Reconstruction gives

d_{tr} (ρ, ρ_{u}) \leq (1 - q) + q d_{tr} (S, S_{u}) \leq \frac{2 r τ}{χ} + \frac{2 ( 1 - r ) τ}{χ} .

For Hermitian $M$ and distinct orthonormal $u, v$ , $2∣ ⟨ u ∣ M ∣ v ⟩ ∣ \leq ∥ M ∥_{1}$ , by testing against a phase-aligned Hermitian off-diagonal operator of norm one. Together with (eq:C), this proves the concurrence bound. The remaining diagnostics are continuous functions of $q, S$ ; uniform continuity on their compact finite-dimensional state domain gives the stated uniform convergence.

Displayed formulas in this passage

Em04-A@2#eq:Sunit · line 812
$∥ S (τ) - S_{u} (τ) ∥_{1} d_{tr} (ρ (τ), ρ_{u} (τ)) \leq \frac{4 ( 1 - r ) τ}{χ}, \leq min {1, 2 τ / χ} .$
Em04-A@2#eq:unitbound · line 812
$∥ S (τ) - S_{u} (τ) ∥_{1} d_{tr} (ρ (τ), ρ_{u} (τ)) \leq \frac{4 ( 1 - r ) τ}{χ}, \leq min {1, 2 τ / χ} .$
unlabeled · line 806
$S_{u} (τ) = e^{- i τ h} S_{0} e^{i τ h}, ρ_{u} = V S_{u} V^{†} .$
unlabeled · line 820
$0 \leq τ \leq T sup ∣ C_{ij} - C_{ij}^{u} ∣ \leq min {1, 4 T / χ} .$
unlabeled · line 829
$S (τ) - S_{u} (τ) = - κ \int_{0}^{τ} U_{τ - s} (S (s) - Δ S (s)) d s .$
unlabeled · line 837
$d_{tr} (ρ, ρ_{u}) \leq (1 - q) + q d_{tr} (S, S_{u}) \leq \frac{2 r τ}{χ} + \frac{2 ( 1 - r ) τ}{χ} .$

Theorem: Finite-window unitary limit

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