  We consider Markovian dynamics of a finitely dimensional open quantum system
featuring a weak unitary symmetry, i.e., when the action of a unitary symmetry
on the space of density matrices commutes with the master operator governing
the dynamics. We show how to encode the weak symmetry in quantum stochastic
dynamics of the system by constructing a weakly symmetric representation of the
master operator: a symmetric Hamiltonian, and jump operators connecting only
the symmetry eigenspaces with a fixed eigenvalue ratio. In turn, this
representation simplifies both the construction of the master operator as well
as quantum jump Monte Carlo simulations, where, for a symmetric initial state,
stochastic trajectories of the system state are supported within a single
symmetry eigenspace at a time, which is changed only by the action of an
asymmetric jump operator. Our results generalize directly to the case of
multiple Abelian weak symmetries.
