Manuscript Psy02-B@2 · Psy02-B-v2.tex · theorem · cfd06e2e

Theorem: Complete state and matched coherence witness

Track A1: carrier, matched branches, and exact dynamics › Separate memoryless loss–dephasing model · lines 411–451 · proof included or linked

Psy02-B@2#thm:pilot, Psy02-B@2#eq:witness

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The generator (eq:generator) defines a CPTP semigroup. For $ρ_{+}$ ,

ρ_{R} (t) = (1 - q) ∣ Ω ⟩ ⟨ Ω ∣ + \frac{q}{2} (1 c e^{i ω t} c e^{- i ω t} 1)_{{0, 1}}, q = e^{- κ t}, c = e^{- (γ + R) t} .

The comparison input $Δ ρ_{+}$ has the same populations and zero code coherence. The complete-state witness is

D_{R} (t) = \frac{1}{2} \norm Λ_{R} (t) (ρ_{+} - Δ ρ_{+})_{1} = \frac{1}{2} e^{- (κ + γ + R) t} .

With a common recorded preparation failure, $D_{0} = p /2$ and $D_{R} (t) = D_{0} e^{- (κ + γ + R) t}$ .

Proof. The finite generator has GKSL form GKS,Lindblad. Its code block satisfies

B ˙ = - \frac{i ω}{2} [Z, B] - κ B + \frac{γ + R}{2} (Z B Z - B) .

Trace gives $q = e^{- κ t}$ ; the off-diagonal equation gives $c$ . The omitted trace appears in $Ω$ . The coherent-minus-dephased output has eigenvalues $\pm q c /2$ . Multiplication by the common preparation-success probability gives the last claim.

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Psy02-B@2#eq:witness · line 428
$D_{R} (t) = \frac{1}{2} \norm Λ_{R} (t) (ρ_{+} - Δ ρ_{+})_{1} = \frac{1}{2} e^{- (κ + γ + R) t} .$
unlabeled · line 415
$ρ_{R} (t) = (1 - q) ∣ Ω ⟩ ⟨ Ω ∣ + \frac{q}{2} (1 c e^{i ω t} c e^{- i ω t} 1)_{{0, 1}}, q = e^{- κ t}, c = e^{- (γ + R) t} .$
unlabeled · line 441
$B ˙ = - \frac{i ω}{2} [Z, B] - κ B + \frac{γ + R}{2} (Z B Z - B) .$

Theorem: Complete state and matched coherence witness

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