"In this talk we consider the continuous monitoring of quantum systems and elucidate the role of the measurement back action in quantum dynamics.

Specifically, we consider the limits to the speed of evolution under continuous quantum measurements. The pace of evolution of physical systems is fundamentally constrained by quantum speed limits (QSL), which have found broad applications in quantum science and technology. It is shown that there are trajectories for which standard QSL are violated. We determine the dispersion of the speed of evolution and characterize the full statistics of single trajectories.

In a second part of the talk, we focus on spontaneous symmetry breaking (SSB), that is responsible for structure formation in scenarios ranging from condensed matter to cosmology. SSB is broadly understood in terms of perturbations to the Hamiltonian governing the dynamics or to the state of the system. Under continuous quantum measurements, the acquisition of information during the measurement process induces a measurement back-action that we identify as the crucial ingredient for SSB. An observer can thus tailor the topology of the vacuum manifold and the pattern of symmetry breaking by monitoring different observables."