Many-body localization (MBL) is a result of the balance between interference-based Anderson localization and many-body interactions in an ultra-high dimensional Fock space. It is usually expected that dissipation is blurring interference and destroying that balance so that the asymptotic state of a system with an MBL Hamiltonian does not bear localization signatures. We demonstrate, within the framework of the Lindblad formalism, that the system can be brought into a steady state with non-vanishing MBL signatures. We use a set of dissipative operators acting on pairs of connected sites (or spins), and show that the difference between ergodic and MBL Hamiltonians is encoded in the imbalance, entanglement entropy, and level spacing characteristics of the density operator. An MBL system which is exposed to the combined impact of local dephasing and pairwise dissipation evinces localization signatures hitherto absent in the dephasing-outshaped steady state.
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