  Quantum many-body dynamics generically results in increasing entanglement
that eventually leads to thermalization of local observables. This makes the
exact description of the dynamics complex despite the apparent simplicity of
(high-temperature) thermal states. For accurate but approximate simulations one
needs a way to keep track of essential (quantum) information while discarding
inessential one. To this end, we first introduce the concept of the information
lattice, which supplements the physical spatial lattice with an additional
dimension and where a local Hamiltonian gives rise to well defined locally
conserved von Neumann information current. This provides a convenient and
insightful way of capturing the flow, through time and space, of information
during quantum time evolution, and gives a distinct signature of when local
degrees of freedom decouple from long-range entanglement. As an example, we
describe such decoupling of local degrees of freedom for the mixed field
transverse Ising model. Building on this, we secondly construct algorithms to
time-evolve sets of local density matrices without any reference to a global
state. With the notion of information currents, we can motivate algorithms
based on the intuition that information for statistical reasons flow from small
to large scales. Using this guiding principle, we construct an algorithm that,
at worst, shows two-digit convergence in time-evolutions up to very late times
for diffusion process governed by the mixed field transverse Ising Hamiltonian.
While we focus on dynamics in 1D with nearest-neighbor Hamiltonians, the
algorithms do not essentially rely on these assumptions and can in principle be
generalized to higher dimensions and more complicated Hamiltonians.
