These lectures will provide the mathematical formalism, concepts and tools required to model quantum open systems (quantum systems coupled to a dissipative environment), and address the main physical questions raised in their description. A particular attention will be paid to the notions of entropy and Gibbs states, of bi-partite systems and (de-)coherence and of Markovian evolution and Lindblad operators.
Chapter 1: Tool box and basics
* Mathematical framework, Functional Calculus for matrices
* Quantum formalism, Density matrices, Pure states, Gibbs states
* Von Neumann entropy, relative entropy and their properties
* Quantum trajectories, two-time measurement protocols
Chapter 2: Bi-partite systems
* By-partite systems, tensor products, partial traces
* Purifications, Schmidt decomposition, Entropy (in-)equalities
* Subadditivity of entropy and Landauer's bound
Chapter 3: Dynamics
* Markovian approximation of Quantum Dynamics
* CPTP maps and Markovian semi-groups
* Lindblad generators and their properties
* Entropy production
Prerequisites:
Quantum Mechanics M1
Statistical physics M1
References:
Exploring the quantum (Haroche & Raimond, Cambridge University Press)
Quantum measurement and control (Wiseman & Milburn, Cambridge University Press)
Quantum computation (Online lectures by John Preskill)