Synthesis and reconstruction of quantum states

The central question addressed during this lecture was the optimal estimation of the state of a quantum system from measurements (necessarily limited in number and fraught with error) on that system. The fundamentally probabilistic nature of quantum theory led us to recall and use elements of classical probability theory and estimation in this lecture. The problem of estimation is inseparable from that of copying or cloning states in quantum physics. We have therefore also tackled this problem and studied the theoretical limits to cloning fidelity for two-level systems (qubits). The study of cloning naturally led us to address the problem of key exchange security in quantum cryptography. All espionage strategies are based on the interception and copying of qubits exchanged between partners seeking to communicate secretly. Finally, we tackled the problem of state estimation and reconstruction for a harmonic oscillator describing a mode of the electromagnetic field. This is a system with an infinite number of levels, richer than a qubit, which can conveniently be described by its Wigner function. This provides a graphic representation of the system's non-classical properties. The reconstruction of Wigner functions for different states of the field (Fock states with a well-defined number of photons or Schrödinger cat states) has been described, and has given us the opportunity to review a number of recent cavity quantum electrodynamics experiments.