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Quantum information can be encoded in the set of steady-states (SSS) of adriven-dissipative system. Non steady-states are separated by a largedissipative gap that adiabatically decouples them way while the dynamics insidethe SSS is governed by an effective, dissipation-projected, Hamiltonian. Thelatter results from a highly non-trivial interplay between a weak driving withthe fast relaxation process that continuously projects the system back to theSSS. This amounts to a novel type of environment-induced quantum Zeno effect.We prove that the dissipation-projected dynamics is of geometric nature andthat it is robust against different types of hamiltonian and dissipativeperturbations. Remarkably, in some cases an effective unitary dynamics canemerge out of purely dissipative interactions.

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