The operator entanglement (OE) is a key quantifier of the complexity of a reduced density matrix. In out-of-equilibrium situations, e.g., after a quantum quench of a product state, it is expected to exhibit an entanglement barrier. The OE of a reduced density matrix initially grows linearly as entanglement builds up between the local degrees of freedom; it 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. Here, by performing a new data analysis of the published experimental results of Brydges et al. [Science 364, 260 (2019)], we obtain the first experimental estimation of the OE of a subsystem reduced density matrix in a quantum many-body system. We employ the randomized-measurements toolbox and we introduce and develop a new efficient method to postprocess experimental data in order to extract higher-order density-matrix functionals and access the OE. The OE thus obtained displays the expected barrier as long as the experimental system is large enough. For smaller systems, we observe a barrier with a double-peak structure, the origin of which can be interpreted in terms of pairs of quasiparticles being reflected at the boundary of the qubit chain. As U(1) symmetry plays a key role in our analysis, we introduce the notion of symmetry-resolved operator entanglement (SROE), in addition to the total OE. To gain further insights into the SROE, we provide a thorough theoretical analysis of this new quantity in chains of noninteracting fermions, which, in spite of their simplicity, capture most of the main features of OE and SROE. In particular, we uncover three main physical effects: the presence of a barrier in any charge sector, a time delay for the onset of the growth of SROE, and an effective equipartition between charge sectors.
