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Quantum dynamical semigroup

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In quantum mechanics, and especially the theory of open quantum systems, a quantum dynamical semigroup is family of quantum dynamical maps ϕ t on the space of density matrices indexed by a single time parameter t ≥ 0 that obey the semigroup property

ϕ s ( ϕ t ( ρ ) ) = ϕ t + s ( ρ ) , t , s ≥ 0.

A quantum dynamical semigroup is generated by a Lindblad superoperator. The generator can be obtained by

L ( ρ ) = l i m Δ t → 0 ϕ Δ t ( ρ ) − ϕ 0 ( ρ ) Δ t

which, by the linearity of ϕ t , is a linear superoperator. The semigroup can be recovered as

ϕ t + s ( ρ ) = e L s ϕ t ( ρ ) .

References

Quantum dynamical semigroup Wikipedia
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