The index of a quantum dynamical semigroup

TL;DR

This paper introduces a numerical index for quantum dynamical semigroups of completely positive maps, generalizing existing concepts and establishing conditions under which the index aligns with that of their minimal dilations to E_0-semigroups.

Contribution

It defines a new index for quantum dynamical semigroups and proves its equivalence to the E_0-semigroup index under minimal dilation.

Findings

The index generalizes the E_0-semigroup index.

The index of a unital semigroup matches that of its minimal dilation.

Provides a framework for classifying quantum dynamical semigroups.

Abstract

A numerical index is introduced for semigroups of completely positive maps of $\Cal B(H)$ which generalizes the index of E_0-semigroups. It is shown that the index of a unital completely positive semigroup agrees with the index of its dilation to an E_0-semigroup, provided that the dilation is minimal.