Manuscript En03-A@1 · En03-A.tex · proposition · 5398614e

Proposition: Diagnostic decomposition

Scale, records, and residual instruments › Transport and coherence diagnostics · lines 2422–2449 · proof included or linked

En03-A@1#prop:decomposition, En03-A@1#eq:transport-coherence, En03-A@1#eq:transport-coherence-terms, En03-A@1#eq:transport-coherence-bound

For every state,

Tr (ρ D_{w}) = a_{w} (ρ) + b_{w} (ρ),

where

a_{w} (ρ) b_{w} (ρ) = z \sum [Tr (E_{w} T (P_{z} ρ P_{z})) - 1_{{q (z) = w}} Tr (P_{z} ρ)], = z \neq = z^{'} \sum Tr (E_{w} T (P_{z} ρ P_{z^{'}})) .

Both sums are real, and

d_{TV} (p_{T}, p_{f}) \leq \frac{1}{2} w \sum (∣ a_{w} (ρ)∣ + ∣ b_{w} (ρ)∣) .

Proof. Insert $ρ = \sum_{z, z^{'}} P_{z} ρ P_{z^{'}}$ . Off-diagonal terms occur in conjugate pairs. Apply the triangle inequality.

Displayed formulas in this passage

En03-A@1#eq:transport-coherence · line 2425
$Tr (ρ D_{w}) = a_{w} (ρ) + b_{w} (ρ),$
En03-A@1#eq:transport-coherence-terms · line 2430
$a_{w} (ρ) b_{w} (ρ) = z \sum [Tr (E_{w} T (P_{z} ρ P_{z})) - 1_{{q (z) = w}} Tr (P_{z} ρ)], = z \neq = z^{'} \sum Tr (E_{w} T (P_{z} ρ P_{z^{'}})) .$
En03-A@1#eq:transport-coherence-bound · line 2438
$d_{TV} (p_{T}, p_{f}) \leq \frac{1}{2} w \sum (∣ a_{w} (ρ)∣ + ∣ b_{w} (ρ)∣) .$