The first lecture introduced the lecture by recalling a number of situations in nuclear magnetic resonance and quantum optics, where the coupling of a quantum system with its environment plays an essential role. We emphasized the difference between the study of a set of systems, for which a statistical approach is sufficient, and that of a single quantum system evolving along a stochastic trajectory. The need for a specific formalism to describe the latter situation was underlined. The problem of decoherence, already tackled in previous years, was recalled, through a brief presentation of Schrödinger's cat problem and an analysis of the fundamental problem of measurement in quantum physics. We introduced the formalism of quantum transformations, which allows us to describe the evolution of open systems in a very general way, without having to specify the state of the environment. We emphasized the advantages, from the point of view of quantum transformations, of the density operator, which makes it possible (among other things) :
- introduce the concept of generalized measurement as the effect on an open system of a standard measurement performed in a state space extended to a larger environment ;
- describe the irreversible evolution of the density operator of an open system as the result of a process of generalized measurement corresponding to a leakage of unread information on the system into a wider environment;
- understand an experiment on a single realization of an open system as a continuous generalized measurement of that system;
- describe decoherence-stable states ("pointer states") as the eigenstates of (generally non-Hermitian) operators associated with the generalized measures responsible for the system's evolution;
- to familiarize ourselves with a useful formalism for studying the manipulation and control of decoherence (introduction to the 2004-2005 lecture).
The first lesson continued with a review of the notion of density operator and the Von Neumann measure in quantum physics. In particular, the form ambiguity of the density operator was highlighted, using the example of a two-level system (qubit) represented by a vector in the Bloch sphere. This vector can be decomposed in an infinite number of ways as a sum of vectors whose extremities are on the sphere and which represent pure cases. The lesson ended with a reminder of the notion of entanglement, illustrated by the study of an entangled system of two qubits. Bell states and EPR quantum correlations, Schmidt decomposition and entanglement entropy were recalled. Finally, we generalized Bell states and EPR to systems of any finite dimension and concluded the lesson by discussing the impossibility of any superluminal communication using "instantaneous" quantum correlations.
