Exponential Stability of Subspaces for Quantum Stochastic Master\n Equations

Abstract

We study the stability of quantum pure states and, more generally, subspaces\nfor stochastic dynamics that describe continuously--monitored systems. We show\nthat the target subspace is almost surely invariant if and only if it is\ninvariant for the average evolution, and that the same equivalence holds for\nthe global asymptotic stability. Moreover, we prove that a strict linear\nLyapunov function for the average evolution always exists, and latter can be\nused to derive sharp bounds on the Lyapunov exponents of the associated\nsemigroup. Nonetheless, we also show that taking into account the measurements\ncan lead to an improved bound on stability rate for the stochastic,\nnon-averaged dynamics. We discuss explicit examples where the almost sure\nstability rate can be made arbitrary large while the average one stays\nconstant.\n