Entanglement and entropy rates in open quantum systems

We study a recent conjecture about the behavior of the quantum relative entropy compared to the relative entropy of entanglement in open bipartite systems. The conjecture states that, under a dissipative time-evolution, the positive rate of change of the relative entropy will always be larger than that of the relative entropy of entanglement. After explicitly solving a two-qubit master equation of Lindblad type with separable and entangled stationary states, we show that the conjecture can be violated for initial states with an entangled asymptotic state, while it appears to be confirmed when the asymptotic states are separable.