Quantum dynamical semigroup (QDS)

In general, quantum dynamical semigroups can be defined on von Neumann algebras, so the dimensionality of the system could be infinite. Let ${\displaystyle {\mathcal {A}}}$ be a von Neumann algebra acting on Hilbert space ${\displaystyle {\mathcal {H}}}$, a quantum dynamical semigroup on ${\displaystyle {\mathcal {A}}}$ is a collection of bounded operators on ${\displaystyle {\mathcal {A}}}$, denoted by ${\displaystyle {\mathcal {T}}:=\left({\mathcal {T}}_{t}\right)_{t\geq 0}}$, with the following properties:^[5]

1. ${\displaystyle {\mathcal {T}}_{0}\left(a\right)=a}$, ${\displaystyle \forall a\in {\mathcal {A}}}$,
2. ${\displaystyle {\mathcal {T}}_{t+s}\left(a\right)={\mathcal {T}}_{t}\left({\mathcal {T}}_{s}\left(a\right)\right)}$, ${\displaystyle \forall s,t\geq 0}$, ${\displaystyle \forall a\in {\mathcal {A}}}$,
3. ${\displaystyle {\mathcal {T}}_{t}}$ is completely positive for all ${\displaystyle t\geq 0}$,
4. ${\displaystyle {\mathcal {T}}_{t}}$ is a ${\displaystyle \sigma }$-weakly continuous operator in ${\displaystyle {\mathcal {A}}}$ for all ${\displaystyle t\geq 0}$,
5. For all ${\displaystyle a\in {\mathcal {A}}}$, the map ${\displaystyle t\mapsto {\mathcal {T}}_{t}\left(a\right)}$ is continuous with respect to the ${\displaystyle \sigma }$-weak topology on ${\displaystyle {\mathcal {A}}}$.

It is worth mentioning that, under the condition of complete positivity, the operators ${\displaystyle {\mathcal {T}}_{t}}$ are ${\displaystyle \sigma }$-weakly continuous if and only if ${\displaystyle {\mathcal {T}}_{t}}$ are normal.^[5] Recall that, letting ${\displaystyle {\mathcal {A}}_{+}}$ denote the convex cone of positive elements in ${\displaystyle {\mathcal {A}}}$, a positive operator ${\displaystyle T:{\mathcal {A}}\rightarrow {\mathcal {A}}}$ is said to be normal if for every increasing net ${\displaystyle \left(x_{\alpha }\right)_{\alpha }}$ in ${\displaystyle {\mathcal {A}}_{+}}$ with least upper bound ${\displaystyle x}$ in ${\displaystyle {\mathcal {A}}_{+}}$ one has

${\displaystyle \lim _{\alpha }\langle u,(Tx_{\alpha })u\rangle =\sup _{\alpha }\langle u,(Tx_{\alpha })u\rangle =\langle u,(Tx)u\rangle }$

for each ${\displaystyle u}$ in a norm-dense linear sub-manifold of ${\displaystyle {\mathcal {H}}}$.

Quantum Markov semigroup (QMS)

A quantum dynamical semigroup ${\displaystyle {\mathcal {T}}}$ is said to be identity-preserving (or conservative, or Markovian) if

${\displaystyle {\mathcal {T}}_{t}\left({\boldsymbol {1}}\right)={\boldsymbol {1}},\quad \forall t\geq 0,}$

(1)

where ${\displaystyle {\boldsymbol {1}}\in {\mathcal {A}}}$ is the identity element. For simplicity, ${\displaystyle {\mathcal {T}}}$ is called quantum Markov semigroup. Notice that, the identity-preserving property and positivity of ${\displaystyle {\mathcal {T}}_{t}}$ imply ${\displaystyle \left\|{\mathcal {T}}_{t}\right\|=1}$ for all ${\displaystyle t\geq 0}$ and then ${\displaystyle {\mathcal {T}}}$ is a contraction semigroup.^[6]

The Condition (1) plays an important role not only in the proof of uniqueness and unitarity of solution of a Hudson–Parthasarathy quantum stochastic differential equation, but also in deducing regularity conditions for paths of classical Markov processes in view of operator theory.^[7]

Infinitesimal generator of QDS

The infinitesimal generator of a quantum dynamical semigroup ${\displaystyle {\mathcal {T}}}$ is the operator ${\displaystyle {\mathcal {L}}}$ with domain ${\displaystyle \operatorname {Dom} ({\mathcal {L}})}$, where

${\displaystyle \operatorname {Dom} \left({\mathcal {L}}\right):=\left\{a\in {\mathcal {A}}~\left\vert ~\lim _{t\rightarrow 0}{\frac {{\mathcal {T}}_{t}(a)-a}{t}}=b{\text{ in }}\sigma {\text{-weak topology}}\right.\right\}}$

and ${\displaystyle {\mathcal {L}}(a):=b}$.