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

Corollary: Survival approximation and diagnostic error

Validated dynamical residual interfaces › Accuracy without a positivity assumption · lines 1530–1563 · proof included or linked

Em04-A@2#cor:residualdiagnostics

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Under Theorem thm:residual, suppose $0 \leq q \leq 1$ , $∣ q - q ∣ \leq ζ$ , and reconstruct with $q$ . Then, without a positivity assumption,

∥ ρ - ρ ∥_{1} \leq 2 ζ + q E .

If $S$ is a density matrix, this is a trace-distance certificate after division by two. With exact $q$ , the physical reconstructed approximation obeys

∣ C_{ij} - C_{ij} ∣ \leq q E, ∣ Tr ρ^{2} - Tr ρ^{2} ∣ \leq 2 q E .

Proof. Write

ρ - ρ = (q - q) (V S V^{†} - ∣Ω ⟩ ⟨ Ω∣) + q V (S - S) V^{†} .

The first parenthesis has trace norm two. For concurrence, use the Hermitian off-diagonal inequality from Theorem thm:largechi. For purity,

∣ Tr (ρ^{2} - ρ^{2}) ∣ = ∣ Tr ((ρ - ρ) (ρ + ρ)) ∣ \leq 2 ∥ ρ - ρ ∥_{1} .

Displayed formulas in this passage

unlabeled · line 1535
$∥ ρ - ρ ∥_{1} \leq 2 ζ + q E .$
unlabeled · line 1541
$∣ C_{ij} - C_{ij} ∣ \leq q E, ∣ Tr ρ^{2} - Tr ρ^{2} ∣ \leq 2 q E .$
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$ρ - ρ = (q - q) (V S V^{†} - ∣Ω ⟩ ⟨ Ω∣) + q V (S - S) V^{†} .$
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$∣ Tr (ρ^{2} - ρ^{2}) ∣ = ∣ Tr ((ρ - ρ) (ρ + ρ)) ∣ \leq 2 ∥ ρ - ρ ∥_{1} .$

Nearby in this section

theorem: Theorem: Cellwise a posteriori residual bound