  Interacting quantum many-body systems are usually expected to thermalise, in
the sense that the evolution of local expectation values approach a stationary
value resembling a thermal ensemble. This intuition is notably contradicted in
systems exhibiting many-body localisation, a phenomenon receiving significant
recent attention. One of its most intriguing features is that, in stark
contrast to the non-interacting case, entanglement of states grows without
limit over time, albeit slowly. In this work, we establish a novel link between
quantum information theory and notions of condensed matter, capturing the
phenomenon in the Heisenberg picture. We show that the existence of local
constants of motion, often taken as the defining property of many-body
localisation, together with a generic spectrum, is sufficient to rigorously
prove information propagation: These systems can be used to send a signal over
arbitrary distances, in that the impact of a local perturbation can be detected
arbitrarily far away. We perform a detailed perturbation analysis of
quasi-local constants of motion and also show that they indeed can be used to
construct efficient spectral tensor networks, as recently suggested. Our
results provide a detailed and model-independent picture of information
propagation in many-body localised systems.
