Geometry, robustness, and emerging unitarity in dissipation-projected dynamics

Quantum information can be encoded in the set of steady states (SSS) of a driven-dissipative system. Nonsteady states are separated by a large dissipative gap that adiabatically decouples them while the dynamics inside the SSS is governed by an effective, dissipation-projected, Hamiltonian. The latter results from the interplay between a weak driving and the fast relaxation process that continuously projects the system back to the SSS. This amounts to a different type of environment-induced quantum Zeno effect. We prove that the dissipation-projected dynamics is of geometric nature and that it is robust against different types of Hamiltonian and dissipative perturbations. Remarkably, in some cases an effective unitary dynamics can emerge out of purely dissipative interactions.