What type of dynamics arise in E_0-dilations of commuting quantum Markov process?

TL;DR

This paper investigates the dynamics of E_0-dilations of commuting quantum Markov processes, showing how their structure depends on whether the original semigroup is an automorphism or not, with implications for their classification.

Contribution

It proves that non-automorphic CP_0-semigroups have E_0-dilations cocycle conjugate to their minimal *-endomorphic dilations, and classifies the resulting E_0-semigroups.

Findings

If $$ is not an automorphism, $$'s dilation is cocycle conjugate to its minimal *-endomorphic dilation.

If $$ is an automorphism, its dilation is also an automorphism.

For non-automorphic $$ with bounded generator, the dilation is a type I E_0-semigroup.

Abstract

Let H be a separable Hilbert space. Given two strongly commuting CP_0-semigroups $\phi$ and $\theta$ on B(H), there is a Hilbert space K containing H and two (strongly) commuting E_0-semigroups $\alpha$ and $\beta$ such that $\phi_s \circ \theta_t (P_H A P_H) = P_H \alpha_s \circ \beta_t (A) P_H$ for all s,t and all A in B(K). In this note we prove that if $\phi$ is not an automorphism semigroup then $\alpha$ is cocycle conjugate to the minimal *-endomorphic dilation of $\phi$, and that if $\phi$ is an automorphism semigroup then $\alpha$ is also an automorphism semigroup. In particular, we conclude that if $\phi$ is not an automorphism semigroup and has a bounded generator (in particular, if H is finite dimensional) then $\alpha$ is a type I E_0-semigroup.