Robust edge states and non-Abelian excitations are the trademark of topological states of matter, with promising applications such as ‘topologically protected’ quantum memory and computing. So far, topological phases have been exclusively discussed in a Hamiltonian context. Here we show that such phases and the associated topological protection and phenomena also emerge in open quantum systems with engineered dissipation. The specific system studied here is a quantum wire of spinless atomic fermions in an optical lattice coupled to a bath. The key feature of the dissipative dynamics described by a Lindblad master equation is the existence of Majorana edge modes, representing a non-local decoherence-free subspace. The isolation of the edge states is enforced by a dissipative gap in the p-wave paired bulk of the wire. We describe dissipative non-Abelian braiding operations within the Majorana subspace, and illustrate the insensitivity to imperfections. Topological protection is granted by a non-trivial winding number of the system density matrix.
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