The ‘operator entanglement’ is an indicator of the complexity of quantum operators, and of their approximability by Matrix Product Operators (MPO) in 1D quantum systems. I will discuss the ‘entanglement barrier’: the fact that the OE of a reduced density matrix initially grows linearly as entanglement builds up between the local degrees of freedom, then reaches a maximum, and ultimately decays to a small finite value as the reduced density matrix converges to a simple stationary state through standard thermalization mechanisms.This ‘entanglement barrier’ can be obtained in various models, and it has recently been found in experimental data in the trapped ion setup of [Brydges et al., Science 364, 260 (2019)], by performing a new analysis of the data using the randomized measurement toolbox. I will also briefly discuss the influence of chaotic vs. integrable dynamics, as well as dissipation, on the shape of the ‘entanglement barrier’.