Quantum Information & Dynamics - WMM9AM88

Objectif(s)

The goal of these lectures is to provide a mathematical description of the quantum formalism in finite dimension and to introduce the mathematical concepts and tools required for the analysis of such quantum systems and their dynamics. On the one hand, we will study the key aspects of quantum information theory. On the other hand, we will describe certain properties of quantum dynamics that need to be taken into account in the implementation of quantum algorithms and that will be applied to emblematic systems. The interaction with an external classical electromagnetic field will also be considered both from a theoretical and a numerical point of view.

Contenu(s)

Part I (9 hours) : Quantum formalism and Functional analysis in finite dimension

• Quantum states, observables, quantum measurement process.
• Spectral theorem, functional calculus, Klein’s inequality
• Entropies, Gibbs state, variational principle.
  Part II (9 hours) : Quantum Information
• Multipartite quantum systems: tensor product spaces, partial trace
• Entanglement characterization and detection: Schmidt decomposition, entanglement witnesses, entanglement criteria
• Quantum channels: representations, output entropies
• Quantum information: no-cloning theorem, additivity problems, quantum algorithms
  Part III (9 hours) : Quantum Dynamics of Open Systems
• Quantum trajectory and associated Markov process
• Discrete and continuous in time Quantum dynamics, decoherence
• Quantum master equation, Markovian approximation, Lindblad dynamics
  Part IV (9 hours including 3 hours of practical sessions in Python) : ODE-PDE modeling and numerical analysis in quantum optics
• Numerical resolution of the Lindblad equation (conservation of the physical properties, splitting methods).
• Maxwell-Bloch model (Maxwell equations, coupling with Bloch equations, Cauchy problem).
• Numerical resolution of Maxwell-Bloch equations (finite difference scheme on staggered grids, stability analysis).

Linear Algebra, Analysis, ODE theory (master 1 level)