Replacing the linear Schröding equation for a quantum state with a non-linear equation for the local information

In this talk, I will show how one can time-evolve local information, i.e., the local density matrices of a system, in a closed way, without invoking the full state of the system.

I will present two semi-exact methods and one approximate method, where the linear Schröding equation for the full state is replaced with a non-linear equation for the local information. The first semi-exact method is for time-evolving states with little long-range entanglement, the other for imaginary time-evolution to get ground-states or low-temperature states.

These semi-exact methods have roughly the same efficiency as matrix-product-state (MPS) based approaches. Still, as opposed to MPS based approaches, they can straightforwardly be generalized to arbitrary dimensions.

The approximate method deals with quantum quenches starting from product states. In these situations, semi-exact methods will eventually always be limited by the exponential growth of entanglement. I will show that using the assumption that local information decouples from long-range entanglement, one avoids this exponential growth. The truncation value of this approach is to which scale one treats entanglement exactly. As I will show, in a few one-dimensional examples, there is remarkable convergence as this parameter is increased.