Quantum Markov semigroup

In quantum mechanics, a quantum Markov semigroup describes the dynamics in a Markovian open quantum system. The axiomatic definition of the prototype of quantum Markov semigroups was first introduced by A. M. Kossakowski in 1972, and then developed by V. Gorini, A. M. Kossakowski, E. C. G. Sudarshan and Göran Lindblad in 1976.

Motivation
An ideal quantum system is not realistic because it should be completely isolated while, in practice, it is influenced by the coupling to an environment, which typically has a large number of degrees of freedom (for example an atom interacting with the surrounding radiation field). A complete microscopic description of the degrees of freedom of the environment is typically too complicated. Hence, one looks for simpler descriptions of the dynamics of the open system. In principle, one should investigate the unitary dynamics of the total system, i.e. the system and the environment, to obtain information about the reduced system of interest by averaging the appropriate observables over the degrees of freedom of the environment. To model the dissipative effects due to the interaction with the environment, the Schrödinger equation is replaced by a suitable master equation, such as a Lindblad equation or a stochastic Schrödinger equation in which the infinite degrees of freedom of the environment are "synthesized" as a few quantum noises. Mathematically, time evolution in a Markovian open quantum system is no longer described by means of one-parameter groups of unitary maps, but one needs to introduce quantum Markov semigroups.

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 $$ \mathcal{A} $$ be a von Neumann algebra acting on Hilbert space $$ \mathcal{H} $$, a quantum dynamical semigroup on $$ \mathcal{A} $$ is a collection of bounded operators on $$ \mathcal{A} $$, denoted by $$ \mathcal{T} := \left( \mathcal{T}_t \right)_{t \ge 0} $$, with the following properties: Under the condition of complete positivity, the operators $$ \mathcal{T}_t $$ are $$ \sigma $$-weakly continuous if and only if $$ \mathcal{T}_t $$ are normal. Recall that, letting $$ \mathcal{A}_+ $$ denote the convex cone of positive elements in $$ \mathcal{A} $$, a positive operator $$ T : \mathcal{A} \rightarrow \mathcal{A} $$ is said to be normal if for every increasing net $$ \left( x_\alpha \right)_\alpha $$ in $$ \mathcal{A}_+ $$ with least upper bound $$ x $$ in $$ \mathcal{A}_+ $$ one has
 * 1) $$ \mathcal{T}_0 \left( a \right) = a $$, $$ \forall a \in \mathcal{A} $$,
 * 2) $$ \mathcal{T}_{t + s} \left( a \right) = \mathcal{T}_t \left( \mathcal{T}_s \left( a \right) \right) $$, $$ \forall s, t \ge 0 $$, $$ \forall a \in \mathcal{A} $$,
 * 3) $$ \mathcal{T}_t $$ is completely positive for all $$ t \ge 0 $$,
 * 4) $$ \mathcal{T}_t $$ is a $$ \sigma $$-weakly continuous operator in $$ \mathcal{A} $$ for all $$ t \ge 0 $$,
 * 5) For all $$ a \in \mathcal{A} $$, the map $$ t \mapsto \mathcal{T}_t \left( a \right) $$ is continuous with respect to the $$ \sigma $$-weak topology on $$ \mathcal{A} $$.
 * $$ \lim_{\alpha} \langle u, (T x_\alpha) u \rangle = \sup_{\alpha} \langle u, (T x_\alpha) u \rangle = \langle u, (T x) u \rangle $$

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

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

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

The Condition ($$) 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.

Infinitesimal generator of QDS
The infinitesimal generator of a quantum dynamical semigroup $$ \mathcal{T} $$ is the operator $$ \mathcal{L} $$ with domain $$ \operatorname{Dom} (\mathcal{L}) $$, where
 * $$ \operatorname{Dom} \left( \mathcal{L} \right) := \left\{ a \in \mathcal{A} ~\left\vert~ \lim_{t \rightarrow 0} \frac{\mathcal{T}_t(a) - a}{t} = b \text{ in } \sigma\text{-weak topology} \right. \right\} $$

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

Characterization of generators of uniformly continuous QMSs
If the quantum Markov semigroup $$ \mathcal{T} $$ is uniformly continuous in addition, which means $$ \lim_{t \rightarrow 0^+} \left\| \mathcal{T}_t - \mathcal{T}_0 \right\| = 0 $$, then
 * the infinitesimal generator $$ \mathcal{L} $$ will be a bounded operator on von Neumann algebra $$ \mathcal{A} $$ with domain $$ \mathrm{Dom} (\mathcal{L}) = \mathcal{A} $$,
 * the map $$ t \mapsto \mathcal{T}_t a $$ will automatically be continuous for every $$ a \in \mathcal{A} $$,
 * the infinitesimal generator $$ \mathcal{L} $$ will be also $$ \sigma $$-weakly continuous.

Under such assumption, the infinitesimal generator $$ \mathcal{L} $$ has the characterization
 * $$ \mathcal{L} \left( a \right) = i \left[ H, a \right] + \sum_{j} \left( V_j^\dagger a V_j - \frac{1}{2} \left\{ V_j^\dagger V_j, a \right\} \right) $$

where $$ a \in \mathcal{A} $$, $$ V_j \in \mathcal{B} (\mathcal{H}) $$, $$ \sum_{j} V_j^\dagger V_j \in \mathcal{B} (\mathcal{H}) $$, and $$ H \in \mathcal{B} (\mathcal{H}) $$ is self-adjoint. Moreover, above $$ \left[ \cdot, \cdot \right] $$ denotes the commutator, and $$ \left\{ \cdot, \cdot \right\} $$ the anti-commutator.