Entanglement Generation and Information Spreading in Weakly Monitored Quantum Circuits

We study the non-equilibrium dynamics in (1+1)D weakly monitored quantum circuits, focusing on the entanglement generation and information spreading. Due to the non-local nature of projective measurements, entanglement dynamics in monitored circuits is "faster" than the unitary ones in several ways. Specifically, we find that a pair of well-separated regions can build up nontrivial entanglement in a timescale l^2/3­­­, sub-linear in their distance l; and initially local information can spread super-ballistically as t. By viewing the dynamics as a dynamical error correcting code, we find the code distance grows sub-linearly as t^1/2 until saturation; and the input state's local information contained in size l is lost at a timescale l­­². Some notions we developed to quantify information dynamics apply to more general monitored quantum processes and are of their own interest.