Observables, Density Matrix, Reduced Density Matrix, Entanglement Entropy 

Quantum Condensed Matter Physics: Lecture 6
Theoretical physicist Dr Andrew Mitchell presents an advanced undergraduate / introductory Master's level lecture course on Quantum Condensed Matter Physics at University College Dublin. This is a complete and self-contained set of lectures, in which the theory is built up from scratch, and requires only a knowledge of basic quantum mechanics.
In this lecture, I discuss how to calculate physical observables from the solution of a given Hamiltonian. With the eigenstates and energies, we use statistical mechanics to find the thermal expectation value of a quantum mechanical operator. This problem is then formulated in terms of the density matrix. Pure and mixed states are discussed. Finally, we consider measurements on a part of a quantum system, and introduce the concept of the reduced density matrix. The entanglement entropy is calculated and analyzed.
Navigate through the lectures of this course in order using the playlist:
• Quantum condensed matt...
Recommended course textbook: "Many-body quantum theory in condensed matter physics" by Bruss and Flensberg