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

Theorem: Cellwise a posteriori residual bound

Validated dynamical residual interfaces › Accuracy without a positivity assumption · lines 1471–1528 · proof included or linked

Em04-A@2#thm:residual, Em04-A@2#eq:residual, Em04-A@2#eq:rawreconstruction

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Fix the scaled generator

A (M) = - i [h, M] - κ (M - Δ M), κ = 2 (1 - r) / χ .

Let $0 = t_{0} < t_{1} < \dots < t_{K} = T$ , and let $S : [0, T] \to M_{N} (C)$ be continuous, Hermitian, and piecewise continuously differentiable. No positivity or trace condition is required. Define on each cell

R (τ) = S^{'} (τ) - A (S (τ)) .

Suppose validated bounds give

S (0) - S_{0}_{1} \leq E_{0}, τ \in [t_{k}, t_{k + 1}] sup ∥ R (τ) ∥_{1} \leq B_{k} .

For $τ \in [t_{j}, t_{j + 1}]$ ,

S (τ) - S (τ)_{1} \leq E (τ) := E_{0} + k < j \sum (t_{k + 1} - t_{k}) B_{k} + (τ - t_{j}) B_{j} .

For exact survival $q$ , the Hermitian reconstruction

ρ = (1 - q) ∣Ω ⟩ ⟨ Ω∣ + q V S V^{†}

satisfies

∥ ρ - ρ ∥_{1} \leq q E (τ) .

It is a density matrix only if $S$ is positive and has trace one.

Proof. The error $e = S - S$ satisfies $e^{'} = A (e) + R$ on each cell. Continuity allows variation of constants over the whole interval:

e (τ) = e^{τ A} e (0) + \int_{0}^{τ} e^{(τ - s) A} R (s) d s .

The propagator is CPTP and contracts trace norm on Hermitian matrices. Integrating the cell bounds proves (eq:residual). The reconstruction difference is $q V e V^{†}$ , whose trace norm is $q ∥ e ∥_{1}$ , proving (eq:rawreconstruction).

Displayed formulas in this passage

Em04-A@2#eq:residual · line 1494
$S (τ) - S (τ)_{1} \leq E (τ) := E_{0} + k < j \sum (t_{k + 1} - t_{k}) B_{k} + (τ - t_{j}) B_{j} .$
Em04-A@2#eq:rawreconstruction · line 1508
$∥ ρ - ρ ∥_{1} \leq q E (τ) .$
unlabeled · line 1474
$A (M) = - i [h, M] - κ (M - Δ M), κ = 2 (1 - r) / χ .$
unlabeled · line 1484
$R (τ) = S^{'} (τ) - A (S (τ)) .$
unlabeled · line 1489
$S (0) - S_{0}_{1} \leq E_{0}, τ \in [t_{k}, t_{k + 1}] sup ∥ R (τ) ∥_{1} \leq B_{k} .$
unlabeled · line 1503
$ρ = (1 - q) ∣Ω ⟩ ⟨ Ω∣ + q V S V^{†}$
unlabeled · line 1520
$e (τ) = e^{τ A} e (0) + \int_{0}^{τ} e^{(τ - s) A} R (s) d s .$

Nearby in this section

corollary: Corollary: Survival approximation and diagnostic error