Quantum-Dynamical Semigroups and the Church of the Larger Hilbert Space

TL;DR

This paper explores the structure of quantum-dynamical semigroups through Stinespring dilations, revealing that non-unitary evolutions require infinite-dimensional environments and unbounded Hamiltonians, and characterizes their generators.

Contribution

It demonstrates that non-unitary quantum-dynamical semigroups necessarily involve infinite-dimensional environments and unbounded Hamiltonians, providing a new characterization of their generators.

Findings

Non-unitary semigroups require infinite-dimensional environment Hilbert spaces.

Bounded Hamiltonians lead to dissipative parts of quantum-dynamical semigroups.

Characterization of generators of quantum-dynamical semigroups via Stinespring dilations.

Abstract

In this work we investigate Stinespring dilations of quantum-dynamical semigroups, which are known to exist by means of a constructive proof given by Davies in the early 70s. We show that if the semigroup describes an open system, that is, if it does not consist of only unitary channels, then the evolution of the dilated closed system has to be generated by an unbounded Hamiltonian; subsequently the environment has to correspond to an infinite-dimensional Hilbert space, regardless of the original system. Moreover, we prove that the second derivative of Stinespring dilations with a bounded total Hamiltonian yields the dissipative part of some quantum-dynamical semigroup -- and vice versa. In particular this characterizes the generators of quantum-dynamical semigroups via Stinespring dilations.