Quantum evolution with a large number of negative decoherence rates

We analyze the evolution of open quantum systems governed by time-local master equations beyond the Markovian semigroup. Non-Markovian effects are usually attributed to the negativity of some decoherence rates in the time-dependent generator. For the qubit dynamics, it is well known that there can be one permanently negative rate in the so-called eternally non-Markovian evolution. We show that for qudits one can have (d − 1)² out of the total d ² − 1 rates that are always negative, and the evolution is still physically legitimate—that is, represented by a completely positive, trace-preserving dynamical map. (paper)