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Open Systems & Information Dynamics

On a Certain Class of Semigroups of Operators

Paolo Aniello

Published Jun 1, 2011• 12 citations
Publisher

Abstract

We define an interesting class of semigroups of operators in Banach spaces, namely, the randomly generated semigroups. This class contains as a remarkable subclass a special type of quantum dynamical semigroups introduced in the early 1970s by Kossakowski. Each randomly generated semigroup is associated, in a natural way, with a pair formed by a representation or an antirepresentation of a locally compact group in a Banach space and by a convolution semigroup of probability measures on this group. Examples of randomly generated semigroups having important applications in physics are briefly illustrated.

DOI
10.1142/s123016121100008x

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We define an interesting class of semigroups of operators in Banach spaces, namely, the randomly generated semigroups. This class contains as a remarkable subclass a special type of quantum dynamical semigroups introduced in the early 1970s by Kossakowski. Each randomly generated semigroup is associated, in a natural way, with a pair formed by a representation or an antirepresentation of a locally compact group in a Banach space and by a convolution semigroup of probability measures on this group. Examples of randomly generated semigroups having important applications in physics are briefly illustrated.
Semigroups of operators are known to play an important role in theoretical physics. In particular, quantum dynamical semigroups are fundamental in the theory of open quantum systems. We will describe a class of semigroups of operators which has interesting applications, for instance, in quantum information science. Each of these semigroups of operators is generated, in a suitable way, by a representation (or an antirepresentation) of a group in a Banach space and by a convolution semigroup of probability measures on that group. Some significant examples—including a remarkable type of quantum dynamical semigroups introduced by Kossakowski in the pioneering times of the theory of open quantum systems—and their mutual relations will be discussed.
We study the twirling semigroups of (super) operators, namely certain quantum dynamical semigroups that are associated, in a natural way, with the pairs formed by a projective representation of a locally compact group and a convolution semigroup of probability measures on this group. The link connecting this class of semigroups of operators with (classical) Brownian motion is clarified. It turns out that every twirling semigroup associated with a finite-dimensional representation is a random unitary semigroup, and, conversely, every random unitary semigroup arises as a twirling semigroup. Using standard tools of the theory of convolution semigroups of measures and of convex analysis, we provide a complete characterization of the infinitesimal generator of a twirling semigroup associated with a finite-dimensional unitary representation of a Lie group.
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.
Quantum mechanics can be formulated in terms of phase-space functions, according to Wigner's approach. A generalization of this approach consists in replacing the density operators of the standard formulation with suitable functions, the so-called generalized Wigner functions or (group-covariant) tomograms, obtained by means of group-theoretical methods. A typical problem arising in this context is to express the evolution of a quantum system in terms of tomograms. In the case of a (suitable) open quantum system, the dynamics can be described by means of a quantum dynamical semigroup 'in disguise', namely, by a semigroup of operators acting on tomograms rather than on density operators. We focus on a special class of quantum dynamical semigroups, the twirling semigroups, that have interesting applications, e.g., in quantum information science. The 'disguised counterparts' of the twirling semigroups, i.e., the corresponding semigroups acting on tomograms, form a class of semigroups of operators that we call tomographic semigroups . We show that the twirling semigroups and the tomographic semigroups can be encompassed in a unique theoretical framework, a class of semigroups of operators including also the probability semigroups of classical probability theory, so achieving a deeper insight into both the mathematical and the physical aspects of the problem.
This paper is devoted to the aspects of derivation of dynamics equation of quantum system under a stochastic dynamics. We study the conditions, under which a sequence of random variations of wave function can approximate a random diffusion process in a Hilbert space. A random variation $|\psi_0\rangle \mapsto G_{t_N}\ldots G_{t_1}|\psi_0\rangle =|\psi_{t_N}\rangle $ is associated with a transform of distribution of vector $|\psi\rangle $, as well as with the variation of its characteristic functional φ(v)=Eexp⁡(iRe⁡⟨v∣ψ⟩).\varphi(v)=\mathbb{E}\exp(i\operatorname{Re}\langle v|\psi\rangle ).φ(v)=Eexp(iRe⟨v∣ψ⟩). For a continuous random walk we study the approximation of a Markov semigroup by the Markov operators of discrete random walk. We pay a special attention to the cases, when the derivative of random operator F′(0)F'(0)F′(0) is an unbounded operator. However, we restrict the consideration to the case when the Markov operators of random walks with the operators GtG_tGt​ mutually commute. The characteristic functional is transformed by the Markov operator of adjoint process, and in contrast to the dynamics of wave function, it has a deterministic nature that allows us to rely on the developed theory of semigroups in Banach spaces. The most illustrative examples are the process of continuous measurement, that is, the measurement of trajectories of some observable, and the random control.