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

Lemma: Physical semigroup and trace-norm contraction

Physicality and exact reduction · lines 246–285 · proof included or linked

Em04-A@2#lem:physical, Em04-A@2#eq:contractivity

The finite-dimensional equation (eq:master) has a unique solution for all $t \geq 0$ . Its propagator is completely positive and trace preserving (CPTP). Every CPTP map $Φ$ satisfies

∥ Φ (M) ∥_{1} \leq ∥ M ∥_{1} for every Hermitian M .

Proof. Existence and uniqueness follow from $ρ (t) = e^{t L} ρ (0)$ . Enumerate the collapse operators as $L_{ℓ}$ , and put

K = - i H - \frac{1}{2} ℓ \sum L_{ℓ}^{†} L_{ℓ}, A (M) = K M + M K^{†}, J (M) = ℓ \sum L_{ℓ} M L_{ℓ}^{†} .

Then $L = A + J$ . The maps $e^{t A} (M) = e^{t K} M e^{t K^{†}}$ and $J$ are completely positive. The norm-convergent Dyson expansion for $e^{t L}$ consists of integrals of compositions of these maps. Complete positivity is preserved by these integrals, sums, and finite-dimensional limits. Cyclicity gives $Tr L (M) = 0$ , so the propagator preserves trace. This proves the needed sufficiency statement of the GKSL framework GKS,Lindblad,Kuramochi directly.

For contraction, write the Hermitian Jordan decomposition $M = M_{+} - M_{-}$ , $M_{\pm} \geq 0$ . Positivity and trace preservation give

∥ Φ (M) ∥_{1} \leq Tr Φ (M_{+}) + Tr Φ (M_{-}) = Tr M_{+} + Tr M_{-} = ∥ M ∥_{1} .

Displayed formulas in this passage

Em04-A@2#eq:contractivity · line 251
$∥ Φ (M) ∥_{1} \leq ∥ M ∥_{1} for every Hermitian M .$
unlabeled · line 262
$K = - i H - \frac{1}{2} ℓ \sum L_{ℓ}^{†} L_{ℓ}, A (M) = K M + M K^{†}, J (M) = ℓ \sum L_{ℓ} M L_{ℓ}^{†} .$
unlabeled · line 280
$∥ Φ (M) ∥_{1} \leq Tr Φ (M_{+}) + Tr Φ (M_{-}) = Tr M_{+} + Tr M_{-} = ∥ M ∥_{1} .$

Lemma: Physical semigroup and trace-norm contraction

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