A Priori Estimates for Quantum Dynamical Semigroups

J. C. García, Roberto B. Quezada

Alexander Mikhailovich Chebotarev, Julio C. García, Roberto B. Quezada

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 $\mathrm{m}\mathrm{a}s$ 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 $\mathrm{m}.$.aster equations with time-dependent coeffi- cients arise in a natural way.To study Markov $\mathrm{m}\mathrm{a}s$ ter equations with unbounded time-dependent coefficients describing evolution of observables from the algebra $B(\mathcal{H})$ of bounded operators or states from the algebra $\mathcal{T}(\mathcal{H})$ of operators with finite trace, we introduce the continuity conditions for the generator.We recall that one parameter contraction semigroup $P_{t}(\cdot)$ acting in $B(\mathcal{H})$ 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. $P_{t}(xn)=P_{t}(1.\mathrm{u}.\mathrm{b}.x_{n})$ and $\mathrm{T}\mathrm{r}\{\rho(P_{t}(B)-$ $B)\}arrow 0$ as $tarrow \mathrm{O}$ for any $\rho\in \mathcal{T}(\mathcal{H})$ and $B\in B(\mathcal{H})$ .In the Heisenberg rep- resentation, the conservative (or unital) property means the conservation of the unit operator $I$ in the algebra $B(\mathcal{H})$ of observables: $P_{t}(I)\equiv I\forall t\geq 0$ ; in the Schr\"odinger representation, it means the conservation of the trace of an initial state $\rho\in \mathcal{T}(\mathcal{H})$ during the evolution: $\mathrm{T}\mathrm{r}T_{t}(\beta)\equiv \mathrm{b}\rho t\geq 0$ , where $T_{t}$ : $\mathcal{T}(\mathcal{H})arrow \mathcal{T}(\mathcal{H})$ and $P_{t}$ : $B(\mathcal{H})arrow B(\mathcal{H})$ are dual semigroups: $\mathrm{T}\mathrm{r}\{\tau_{t}(\rho)B\}=\mathrm{T}\mathrm{r}\{\rho P_{t}(B)\}$ .The semigroup $T_{t}=P_{t}^{1}$ is called the predual of the semigroup $P_{t}$ .The dagger "\dagger '' is used to denote predual operators and maps.The map. $P(\cdot)$ with the predual $P^{\uparrow}(\cdot)$ is referred to as a completely positive $(CP)$ if $\sum_{i,j}\mathrm{T}\mathrm{r}\{P(B_{i}^{*}B_{j})\sigma j\sigma_{i}\}*\geq 0$ or equivalently $\sum_{i,j}\mathrm{T}\mathrm{r}\{B^{*}iBjP^{\mathrm{t}}(\sigma_{j}\sigma_{i})*\}\geq 0$ (1.1)

Wawrzyniec Guz

E. B. Davies

Lorenzo Bertini, Alberto De Sole, Gustavo Posta +1 more

Abstract We analyze a class of quantum Feller semigroups describing the evolution of interacting quantum lattice systems, specified either as generic qudits or as fermions. The corresponding generators, which include both conservative and dissipative evolutions, are given by the superposition of local generators in the Lindblad form. The associated infinite volume dynamics can be obtained as the strong limit of the finite volume dynamics. By regarding the interacting evolution as a perturbation of a non-interacting dissipative dynamics, we obtain a quantitative criterion that yields the uniqueness of the stationary state together with the exponential convergence of local observables. The analysis is based on suitable a priori bounds on the resolvent equation which yield quantitive estimates on the evolution of local observables.

Andrzej Posiewnik