Adiabaticity in open quantum systems

We provide a rigorous generalization of the quantum adiabatic theorem for open systems described by a Markovian master equation with time-dependent Liouvillian $\mathcal{L}(t)$. We focus on the finite system case relevant for adiabatic quantum computing and quantum annealing. Adiabaticity is defined in terms of closeness to the instantaneous steady state. While the general result is conceptually similar to the closed system case, there are important differences. Namely, a system initialized in the zero-eigenvalue eigenspace of $\mathcal{L}(t)$ will remain in this eigenspace with a deviation that is inversely proportional to the total evolution time $T$. In the case of a finite number of level crossings the scaling becomes $T^{-\eta}$ with an exponent $\eta$ that we relate to the rate of the gap closing. For master equations that describe relaxation to thermal equilibrium, we show that the evolution time $T$ should be long compared to the corresponding minimum inverse gap squared of $\mathcal{L}(t)$. Our results are illustrated with several examples.