  The operator space entanglement entropy, or simply 'operator entanglement'
(OE), is an indicator of the complexity of quantum operators and of their
approximability by Matrix Product Operators (MPO). We study the OE of the
density matrix of 1D many-body models undergoing dissipative evolution. It is
expected that, after an initial linear growth reminiscent of unitary quench
dynamics, the OE should be suppressed by dissipative processes as the system
evolves to a simple stationary state. Surprisingly, we find that this scenario
breaks down for one of the most fundamental dissipative mechanisms: dephasing.
Under dephasing, after the initial 'rise and fall' the OE can rise again,
increasing logarithmically at long times. Using a combination of MPO
simulations for chains of infinite length and analytical arguments valid for
strong dephasing, we demonstrate that this growth is inherent to a $U(1)$
conservation law. We argue that in an XXZ spin-model and a Bose-Hubbard model
the OE grows universally as $\frac{1}{4} \log_2 t$ at long times, and as
$\frac{1}{2} \log_2 t$ for a Fermi-Hubbard model. We trace this behavior back
to anomalous classical diffusion processes.
