Many-body Entanglement and Tensor Networks: an Introduction

Quantum systems, such as lattices of quantum spins, are very hard to study because the associated many-body Hilbert space is simply huge: its dimension grows exponentially with the size of the lattice. As a result, not even the most powerful supercomputer on Earth is large enough to deal with the many-body state of, say, 60 quantum spins. In recent years, however, progress in our understanding of quantum entanglement has revealed that only a small region of this huge Hilbert space is actually relevant to the study of quantum many-body systems. Using tensor networks, we have then been able to efficiently describe some of the states in this small, physically relevant region of the many-body Hilbert space. In this talk, directed to non-experts, I will (i) review our current understanding of many-body entanglement, (ii) introduce tensor networks as an efficient description of many-body states, and (iii) mention some of its current applications, which nowadays includes quantum gravity, statistical physics and machine learning.