The course will broadly lie at the interface of condensed matter physics, quantum information theory and high-energy physics. The aim is to study correlation structures in quantum many-body systems and understand their role in determining the physical properties of these systems. Owing to the complexity of many-body systems, exploring typical quantum information concepts in them will require us to invoke efficient approximation and renormalization techniques. This will lead us to introduce tensor networks and the multi-scale entanglement renormalization ansatz, which are standard workhorses in the modern literature. We will pay special attention to ground states and their entanglement properties, such as entanglement entropy area laws and how correlations decay over distance. Quantum correlations are also intertwined with the spreading of information and we shall examine this topic in the form of Lieb-Robinson bounds. A further topic we will investigate is how (gauge) symmetries affect the correlation structure and computation of entropies. While most of the discussion will focus on finite-dimensional many-body systems, we will proceed to studying some of these questions in quantum field theory towards the end of the course.

- Review of reduced density matrix, entanglement for pure and mixed states, entanglement (Renyi) entropies and mutual information.

- Why is many-body entanglement a hard problem?

- Introduction to tensor networks, matrix-product and projected entangled-pair states.

- Tensor network renormalization

- The multi-scale entanglement renormalization ansatz

- Correlation properties in many-body systems (area laws, Lieb-Robinson bounds, etc.)

- Thermodynamic properties of many-body systems and phases

- (Gauge) symmetries in many-body physics

- Quantum correlations in QFT (replica trick, lattice models, vacuum states, entanglement entropies, mutual information, gauge symmetry)

Quantum Mechanics I and II and ideally Advanced Quantum Theory. A further background in Quantum Field Theory and Statistical Physics is helpful.