We prove new a priori estimates for the resolvent of a class of minimal Quantum Dynamical Semigroup (QDS).These estimates simplify the proof of the unital property of QDS and suggest a continuity condition for time-dependent in- finitesimal generators to ensure existence and conservativity of the Markov master evolution equation. \S 1. INTRODUCTION.The theory of the Markov
mas ter equation has been intensively studied dur- ing the recent years [1][2][3][4][5][6][7][8][9][10].Important applications of this theory were contributed to qu.antum chemistry [3] and quantum optics [4-6].Numerical Monte-Carlo and Runge-Kutta algorithms for these equations are discussed in [7-8].One of efficient analytical tools in the theory of master equation is the interaction representation technique [6-7], where the Markov
m..aster equations with time-dependent coeffi- cients arise in a natural way.To study Markov
mas ter equations with unbounded time-dependent coefficients describing evolution of observables from the algebra
B(H) of bounded operators or states from the algebra
T(H) of operators with finite trace, we introduce the continuity conditions for the generator.We recall that one parameter contraction semigroup
Pt(⋅) acting in
B(H) is called a Quantum Dynamical Semigroup (QDS) if it is completely positive, normal, conservative, and ultraweak continuous
[2,11] .Normal and ultraweak continuity properties mean respectively that l.u.b.
Pt(xn)=Pt(1.u.b.xn) and
Tr{ρ(Pt(B)− B)}arrow0 as
tarrowO for any
ρ∈T(H) and
B∈B(H) .In the Heisenberg rep- resentation, the conservative (or unital) property means the conservation of the unit operator
I in the algebra
B(H) of observables:
Pt(I)≡I∀t≥0 ; in the Schr\"odinger representation, it means the conservation of the trace of an initial state
ρ∈T(H) during the evolution:
TrTt(β)≡bρt≥0 , where
Tt :
T(H)arrowT(H) and
Pt :
B(H)arrowB(H) are dual semigroups:
Tr{τt(ρ)B}=Tr{ρPt(B)} .The semigroup
Tt=Pt1 is called the predual of the semigroup
Pt .The dagger "\dagger '' is used to denote predual operators and maps.The map.
P(⋅) with the predual
P↑(⋅) is referred to as a completely positive
(CP) if
∑i,jTr{P(Bi∗Bj)σjσi}∗≥0 or equivalently
∑i,jTr{B∗iBjPt(σjσi)∗}≥0 (1.1)
We derive sufficient conditions for the memory-kernel governing nonlocal master equation which guarantee a legitimate (completely positive and trace-preserving) dynamical map. It turns out that these conditions provide natural parametrizations of the dynamical map being a generalization of the Markovian semigroup. This parametrization is defined by the so-called legitimate pair---monotonic quantum operation and completely positive map---and it is shown that such a class of maps covers almost all known examples from the Markovian semigroup, the semi-Markov evolution, up to collision models and their generalization.
Abstract We derive Heisenberg equations for arbitrary high order moments of creation and annihilation operators in the case of the quantum master equation with a multimode generator which is quadratic in creation and annihilation operators and obtain their solutions. Based on them we also derive similar equations for the case of the quantum master equation, which occur after averaging the dynamics with a quadratic generator with respect to the classical Poisson process. This allows us to show that dynamics of arbitrary finite-order moments of creation and annihilation operators is fully defined by finite number of linear differential equations in this case.
In this chapter, we define and explore basic properties of a quantum Markov semigroup { I t , t ≥ 0} of linear maps on the C* -algebra or von Neumann algebra A that characterizes the quantum system. The quantum Markov semigroup (QMS) plays a key role in describing quantum Markov processes that are to be explored in the subsequent chapters. The concept of QMS extends the semigroup of probability transition operators { T t , t ≥ 0} for a classical Markov process to its noncommutative counterpart. Suppose the classical Markov process { X t , t ≥ 0} is defined on the complete filtered classical probability space (Ω, Ƒ, P ; {Ƒ t , t ≥ 0}) and with values in a measurable space (핏, B (X)). Recall that for each t ≥ 0, the probability transition operator T t : L ∞ (X, B (X)) → L ∞ (X, B (X)) by ( T t f )( X s ) = E x [ f ( X s + t ) | Ƒ s ] ∀ s , t ≥ 0. A semigroup of linear maps on the C* -algebra or von Neumann algebra A is said to be a quantum dynamical semigroup (QDS) if (i) I 0 = J (the identity operator on A ); (ii) I t I s = I t + s for all t , s ≥ 0; (iii) I t is completely positive for each t ≥ 0; and (iv) I t is σ weakly continuous on A , i.e., a ↦ tr ( ρI t ( a )) is continuous from A to ℂ for each ρ ∈ S ( A ) (the space of quantum states) and for each t ≥ 0. If in addition I t ( I ) = I , (respectively, I t ( I ) ≤ I ) for all t ≥ 0, then the QDS is said to be a quantum Makrov semigroup (QMS) (respectively, quantum sub-Markov semigroup).
In recent years, digraph induced generators of quantum dynamical semigroups have been introduced and studied, particularly in the context of unique relaxation and invariance. We define the class of pair block diagonal generators, which allows for additional interaction coefficients but preserves the main structural properties. Namely, when the basis of the underlying Hilbert space is given by the eigenbasis of the Hamiltonian (for example the generic semigroups), then the action of the semigroup leaves invariant the diagonal and off-diagonal matrix spaces. In this case, we explicitly compute all invariant states of the semigroup. In order to define this class we provide a characterization of when the Gorini- Kossakowski-Sudarshan-Lindblad (GKSL) equation defines a proper generator when arbitrary Lindblad operators are allowed (in particular, they do not need to be trace- less as demanded by the GKSL Theorem). Moreover, we consider the converse con- struction to show that every generator naturally gives rise to a digraph, and that under certain assumptions the properties of this digraph can be exploited to gain knowledge of both the number and the structure of the invariant states of the corre- sponding semigroup. We also consider more general constructions on the von Neumann algebra of all bounded linear operators on a Hilbert space, perhaps infinite dimensional. In partic- ular, we prove that for every semigroup of Schwarz maps on such an algebra which has a subinvariant faithful normal state there exists an associated semigroup of con- tractions on the space of Hilbert-Schmidt operators of the Hilbert space. Moreover, we show that if the original semigroup is weak∗ continuous then the associated semigroup is strongly continuous. We introduce the notion of the extended generator of a semigroup on the bounded operators of a Hilbert space with respect to an or- thonormal basis of the Hilbert space. We describe this form of the generator of a quantum Markov semigroup on the von Neumann algebra of all bounded linear op- erators on a Hilbert space which has an invariant faithful normal state under the assumption that the generator of the associated semigroup has compact resolvent, or under the assumption that the minimal unitary dilation of the associated semigroup of contractions is compact.
The concept of complete positivity fixes in considerable detail the mathematical structure of quantum Markovian master equations which describe the combined reversible-irreversible dynamics of open systems for the case of time-independent Hamilton operators. For many applications such as, for instance, problems of time-resolved spectroscopy, similar results would be of great value but involve time-varying external fields with a corresponding time-dependent Hamiltonian. For this reason, we propose a suitable extension of the Kossakowski generator to the time-dependent case by maintaining complete positivity and using arguments suggested by experimental facts. As an example, generalized Bloch-like equations can be obtained with three different relaxation times and a nonlinear dependence upon field strength.