  We introduce a quantum generalization of classical kinetic Ising models,
described by a certain class of quantum many body master equations. Similarly
to kinetic Ising models with detailed balance that are equivalent to certain
Hamiltonian systems, our models reduce to a set of Hamiltonian systems
determining the dynamics of the elements of the many body density matrix. The
ground states of these Hamiltonians are well described by matrix product, or
pair entangled projected states. We discuss critical properties of such
Hamiltonians, as well as entanglement properties of their low energy states.
