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). In this talk I will present the study of OE of the density matrix of a 1D spin chain undergoing dissipative evolution. While it is expected that, after an initial linear growth the OE should be supressed by dissipative processes as the system evolves to a simple stationary state, 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. Through a combination of MPO simulations for chains of infinite length and analytical arguments valid for strong dephasing, I show that the growth is inherent to a U(1) conservation law. I argue that in an XXZ model the OE grows universally as 1/4log_2 t at long times, and trace this behavior back to an anomalous classical diffusion process. 

* Guillermo Preisser, University of Strasbourg, France