Dynamical maps beyond Markovian regime

Quantum dynamical maps provide suitable mathematical representation of quantum evolutions. It is the very notion of complete positivity which provides a proper mathematical representation of quantum evolution and gives rise to the powerful generalization of unitary evolution of closed Hamiltonian systems. A prominent example of quantum evolution of an open system is a Markovian semigroup. In what follows, we analyze both the semigroups of positive and completely positive maps. In the latter case the dynamics is governed by the celebrated Gorini-Kossakowski-Lindblad-Sudarshan (GKLS) Master Equation. Markovian semigroups, however, provide only an approximate description of general quantum evolution. The main topic of our analysis are dynamical maps beyond this regime. Non-Markovian quantum evolution attracted a lot of attention in recent years and there is a vast literature dedicated to it. In this report we analyze quantum dynamics governed by time-local generators and/or non-local memory kernels. A special attention is devoted to the concept of {\em divisibility} which is often used as a definition of Markovianity. In particular, the concept of so called CP-divisibility (in contrast to P-divisibility) is widely accepted as a proper definition of quantum Markovianity. We discuss a number of important physical implications of divisibility. We also briefly discuss the notion of Markovianity beyond the dynamical map, that is, when one has an access to the evolution of `system + environment'. The entire exposition is concentrated more on the general concepts and intricate connections between them than on studying particular systems. We illustrate the analyzed concepts by paradigmatic models of open quantum systems like the amplitude damping and phase damping models.