TY - BOOK

T1 - Unbounded generators of dynamical semigroups

AU - Siemon, Inken

PY - 2024

Y1 - 2024

N2 - The time evolution of a closed quantum system was described quite early in the history of quantum physics. These dynamics are reversible, and the time evolution is implemented by a continuous unitary group, which is in turn generated by a selfadjoint Hamiltonian operator. So, we have a complete mathematical characterization of all such evolutions. For open quantum systems the time evolution is given by dynamical semigroups. In the case of uniform continuity the generator of the dynamical semigroup is a bounded operator in the famous GKLS-form that has been found by V. Gorini, A. Kossakowski, G. Sudarshan and, independently, G. Lindblad. But the problem of characterizing also the merely strongly continuous dynamical semigroups or, equivalently, their unbounded generators, is open. In the first part of this thesis we introduce a standard formfor the generator of quantum dynamical semigroups that is an unbounded version of the GKLS-form. The basis of the standard form are so-called no-event semigroups, describing an evolution of a quantum system, that maps pure states to multiples of pure states, and completely positive perturbations of their generator that correspond to jumps in this evolution, like absorption by a measurement device. We will give examples of standard semigroups, which appear to be probability preserving to first order (i.e., when looking only at the generator on the finite-rank part of its domain) but not for finite times. Additionally, we construct examples of generators not of standard form by modifying the previous examples. In the second part we relate the notion of standardness toW. Arveson’s classification of endomorphism semigroups. He divided them into three classes, Type I, Type II and Type III. We show that a conservative dynamical semigroup is standard if and only if the minimal dilation of its adjoint is of Type I. The key feature is the set of ketbras in the domain of the no-event generator and whether it is a core for the standard generator. With this knowledge, we suggest to extend this classification to (not necessarily conservative) semigroups that are standard or can be constructed as a series of completely positive perturbations of a no-event semigroup. By construction these are either of Type I or Type II.

AB - The time evolution of a closed quantum system was described quite early in the history of quantum physics. These dynamics are reversible, and the time evolution is implemented by a continuous unitary group, which is in turn generated by a selfadjoint Hamiltonian operator. So, we have a complete mathematical characterization of all such evolutions. For open quantum systems the time evolution is given by dynamical semigroups. In the case of uniform continuity the generator of the dynamical semigroup is a bounded operator in the famous GKLS-form that has been found by V. Gorini, A. Kossakowski, G. Sudarshan and, independently, G. Lindblad. But the problem of characterizing also the merely strongly continuous dynamical semigroups or, equivalently, their unbounded generators, is open. In the first part of this thesis we introduce a standard formfor the generator of quantum dynamical semigroups that is an unbounded version of the GKLS-form. The basis of the standard form are so-called no-event semigroups, describing an evolution of a quantum system, that maps pure states to multiples of pure states, and completely positive perturbations of their generator that correspond to jumps in this evolution, like absorption by a measurement device. We will give examples of standard semigroups, which appear to be probability preserving to first order (i.e., when looking only at the generator on the finite-rank part of its domain) but not for finite times. Additionally, we construct examples of generators not of standard form by modifying the previous examples. In the second part we relate the notion of standardness toW. Arveson’s classification of endomorphism semigroups. He divided them into three classes, Type I, Type II and Type III. We show that a conservative dynamical semigroup is standard if and only if the minimal dilation of its adjoint is of Type I. The key feature is the set of ketbras in the domain of the no-event generator and whether it is a core for the standard generator. With this knowledge, we suggest to extend this classification to (not necessarily conservative) semigroups that are standard or can be constructed as a series of completely positive perturbations of a no-event semigroup. By construction these are either of Type I or Type II.

U2 - 10.15488/15855

DO - 10.15488/15855

M3 - Doctoral thesis

CY - Hannover

ER -