Divisibility and Information Flow Notions of Quantum Markovianity for Noninvertible Dynamical Maps

TL;DR

This paper explores the relationship between CP-divisibility and information backflow in quantum dynamical maps, especially focusing on noninvertible cases, revealing conditions under which these notions are equivalent and implications for quantum Markovianity.

Contribution

It extends the understanding of quantum Markovianity by analyzing noninvertible maps, showing when lack of information backflow implies CP-divisibility, and clarifying the structure of time-local generators.

Findings

Lack of information backflow implies CP-divisibility for a wide class of non-invertible maps.

For non-invertible maps, positivity of dissipation rates is not necessary for CP-divisibility.

In certain cases, the CP propagator becomes trace-preserving, linking backflow and divisibility.

Abstract

We analyze the relation between CP-divisibility and the lack of information backflow for an arbitrary -- not necessarily invertible -- dynamical map. It is well known that CP-divisibility always implies lack of information backflow. Moreover, these two notions are equivalent for invertible maps. In this letter it is shown that for a map which is not invertible the lack of information backflow always implies the existence of completely positive (CP) propagator which, however, needs not be trace-preserving. Interestingly, for a {\em wide class of image non-increasing dynamical maps} this propagator becomes trace-preserving as well and hence the lack of information backflow implies CP-divisibility. This result sheds new light into the structure of the time-local generators giving rise to CP-divisible evolutions. We show that if the map is not invertible then positivity of dissipation/decoherence rates is no longer necessary for {CP-}divisibility.