  We construct an exact map between a tight-binding model on any bipartite
lattice in presence of dephasing noise and a Hubbard model with imaginary
interaction strength. In one dimension, the exact many-body Liouvillian
spectrum can be obtained by application of the Bethe ansatz method. We find
that both the non-equilibrium steady state and the leading decay modes
describing the relaxation at late times are related to the eta-pairing symmetry
of the Hubbard model. We show that there is a remarkable relation between the
time-evolution of an arbitrary k-point correlation function in the dissipative
system and k-particle states of the corresponding Hubbard model.
