Low rank approximability and entropy area law for PDEs

We want to identify and describe the mathematical structure of sparsity in high-dimensional problems. Systems that depend on a large number of variables are known to suffer from the curse of dimensionality: their complexity generally grows exponentially in the number of variables. Nevertheless, decades of research have shown that in many cases such systems can be accurately approximated with polynomial complexity.Perhaps the most studied phenomena in this context are entangled quantum mechanical systems obeying are laws.In such case, the information content scales much slower than the size of the system. In this talk we will discuss the links between entropy area laws and low-rank approximation. We will see how a PDE operator with local (NNI) structure admits eigenfunctions with favorable approximation properties.This will lead to an area law for the system states described by the eigenfunctions.