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Dissipative preparation of a pure steady state usually involves a commutativeaction of a coherent and a dissipative dynamics on the target state. Namely,the target pure state is an eigenstate of both the coherent and dissipativeparts of the dynamics. We show that working in the Zeno regime, i.e. forinfinitely large dissipative coupling, one can generate a pure state by a noncommutative action, in the above sense, of the coherent and dissipativedynamics. A corresponding Zeno regime pureness criterion is derived. Weillustrate the approach, looking at both its theoretical and applicativeaspects, in the example case of an open $XXZ$ spin-$1/2$ chain, driven out ofequilibrium by boundary reservoirs targeting different spin orientations. Usingour criterion, we find two families of pure nonequilibrium steady states, inthe Zeno regime, and calculate the dissipative strengths effectively needed togenerate steady states which are almost indistinguishable from the target purestates.

Obtaining pure steady states in nonequilibrium quantum systems with strong dissipative couplings