In this paper we work with the approximation of unitary groups of operators of the form
e−itH where
H∈L(H) is the Hamiltonian of a given quantum dynamical system modeled in the discretizable Hilbert space
H=H(G), to perform such approximations we implement some techniques from operator theory that we name particular projection methods by compatibility with quantum theory conventions. Once particular representations are defined we study the interelation between some of them properties with the original operators that they mimic. In the end some estimates for numerical implementation are presented to verify the theoretical discussion.
In this paper, we study the possibility of approximating some bi-continuous semigroup on a bi-admissible locally convex space
X using Chernoff iterations
(Ft)t≥0 of some class of operator-valued functions
(Ft)t≥0 . We obtain results that do not require a direct verification of the complicated condition of the Chernoff theorem, which consists in the fact that for some
λ>0 the subspace
(λI−L)D is bi-dense in the ambient space. However, we have to abandon the fact that the limit bi-continuous semigroup is defined on the
X and restrict ourselves to some bi-closed subspace that is constructed from the system
(Ft)t≥0 based on vectors with certain properties connected with the behavior of
(Ft)t≥0 . The formulation of the problem of approximation of a semigroup for bi-continuous semigroups is motivated by the dynamic systems existing in quantum mechanics, the evolution of which is described in terms of bi-continuous semigroups, but not strongly continuous ones. Here we can, for example, include the example of the evolution of a quantum system under a continuous measurement process, described by the Schrödinger–Belavkin equation. The study of approximations of such dynamics can be carried out by the proposed method.
We investigate convergence properties of discrete-time semigroup quantum dynamics, including asymptotic stability, probability and speed of convergence to pure states and subspaces. These properties are of interest in both the analysis of uncontrolled evolutions and the engineering of controlled dynamics for quantum information processing. Our results include two Hilbert space decompositions that allow for deciding the stability of the subspace of interest and for estimating of the speed of convergence, as well as a formula to obtain the limit probability distribution for a set of orthogonal invariant subspaces.
Abstract. A recent characterisation of Fock-adapted contraction operator stochastic cocycles on a Hilbert space, in terms of their associated semigroups, yields a general principle for the construction of such cocycles by approximation of their stochastic generators. This leads to new existence results for quantum stochastic differential equations. We also give necessary and sufficient conditions for a cocycle to satisfy such an equation. In this paper we study the functional equation V0 = I, Vr+t = Vrσr(Vt) for all r, t ≥ 0. for a family of contractions on h ⊗ F adapted to the Fock operator filtration. Here F is the symmetric Fock space over L 2 (R+; k), h and k are fixed but arbitrary