Hardy Uncertainty Principle and unique continuation properties of covariant Schrodinger flows

We prove a logarithmic convexity result for exponentially weighted $L^2$-norms of solutions to electromagnetic Schr"odinger equation, without needing to assume smallness of the magnetic potential. As a consequence, we can prove a unique continuation result in the style of the Hardy uncertainty principle, which generalizes the analogous theorems which have been recently proved by Escauriaza, Kenig, Ponce and Vega.