  We study the effects of dephasing noise on a prototypical many-body localized
system -- the XXZ spin 1/2 chain with a disordered magnetic field. At times
longer than the inverse dephasing strength the dynamics of the system is
described by a probabilistic Markov process on the space of diagonal density
matrices, while all off-diagonal elements of the density matrix decay to zero.
The generator of the Markovian process is a bond-disordered spin chain. The
scaling variable is identified, and independence of relaxation on the
interaction strength is demonstrated. We show that purity and von Neumann
entropy are extensive, showing no signatures of localization, while the
operator space entanglement entropy exhibits a logarithmic growth with time
until the final saturation corresponding to localization breakdown, suggesting
a many-body localized dynamics of the effective Markov process.
