Dynamical maps beyond Markovian regime

Quantum dynamical maps provide suitable mathematical representation of quantum evolutions. When representing states by density operators, the evident requirements for any map are positivity and trace-preservation. However, these properties not consistent with mechanics composite systems. It is very notion complete which provides a proper evolution gives rise to powerful generalization unitary closed Hamiltonian A prominent example an open system Markovian semigroup. In what follows, we analyze both semigroups positive completely maps. latter case dynamics governed celebrated Gorini–Kossakowski–Lindblad–Sudarshan (GKLS) Master Equation. semigroups, however, only approximate description general evolution. The main topic our analysis beyond this regime. Non-Markovian attracted lot attention in recent years there vast literature dedicated it. report time-local generators and/or non-local memory kernels. special devoted concept divisibility often used as definition Markovianity. particular, so called CP-divisibility (in contrast P-divisibility) widely accepted We discuss number important physical implications divisibility. also briefly Markovianity map, that is, when one has access ‘system + environment’. entire exposition concentrated more on concepts intricate connections between them than studying particular illustrate analyzed paradigmatic models systems like amplitude damping phase models.