  When a quantum system initialized in a product state is subjected to either
coherent or incoherent dynamics, the entropy of any of its connected partitions
generically increases as a function of time, signalling the inevitable
spreading of (quantum) information throughout the system. Here, we show that,
in the presence of continuous symmetries and under ubiquitous experimental
conditions, symmetry-resolved information spreading is inhibited due to the
competition of coherent and incoherent dynamics: in given quantum number
sectors, entropy decreases as a function of time, signalling dynamical
purification. Such dynamical purification bridges between two distinct short
and intermediate time regimes, characterized by a log-volume and log-area
entropy law, respectively. It is generic to symmetric quantum evolution, and as
such occurs for different partition geometry and topology, and classes of
(local) Liouville dynamics. We then develop a protocol to measure
symmetry-resolved entropies and negativities in synthetic quantum systems based
on the random unitary toolbox, and demonstrate the generality of dynamical
purification using experimental data from trapped ion experiments [Brydges et
al., Science 364, 260 (2019)]. Our work shows that symmetry plays a key role as
a magnifying glass to characterize many-body dynamics in open quantum systems,
and, in particular, in noisy-intermediate scale quantum devices.
