The aim of this course is to provide the basics on the necessary tools to understand quantum many-body systems from a quantum information perspective. We will review the theory of quantum entanglement and its quantification, its application in the study of quantum phase transitions, and its implications for quantum matter through the tensor network representation of quantum states. As a practical application, we will explain the basics of the Density Matrix Renormalization Group algorithm, the most successful numerical tool in the study of the low-energy properties of 1d quantum lattice systems, as well as other methods for 1d systems. We will also provide some basics on advanced tensor network structures such as PEPS, MERA, and their connections to current research topics.

• Entanglement and its quantification

• Entanglement in quantum many-body systems

• Tensor Network Theory

• 1d Matrix Product States (MPS)

• Density Matrix Renormalization Group (DMRG)

• Time-Evolving Block Decimation (TEBD)

• Basics and properties

• Algorithms

• Fermions

• Basics and properties

• Algorithms

• Symmetries

• Entanglement Hamiltonians

• Branching MERA

• AdS/CFT and holography