Presents the basic postulates of quantum mechanics in terms of the density matrix instead of the usual state vector formalism in the case of an isolated system. Extends it to the case of open systems with the help of the reduced density matrix formalism, and to the case of an imperfect state preparation described by a statistical mixture. Introduces the concept of quantum state purity to characterize the degree of mixture of the state, and shows that one can always "purify" a density matrix by going into a Hilbert space of larger dimension.

Appendix G: interaction between a monochromatic field and two-level atom. The problem is treated first in the case of a classical field and a quantum two-level system (semiclassical approach): It is characterised by a rotation of the Bloch vector (Rabi ocillation) and allows us to generate any qubit state by applying a field of well-controlled duration and amplitude. One then includes spontaneous emission to the model, and finally obtains the set of Bloch equations that are used in many different problems of light–matter interaction. One then considers the full quantum case of cavity quantum electrodynamics (CQED), where the field is single mode and fully quantum: this is the Jaynes–Cummings Hamiltonian approach, which is fully solvable when one negelcts spontaneous emission: quantum oscillations and revivals are predicted. Damping is then introduced in the model, and two regimes of strong and weak couplings are predicted in this case.

Experimental chapter that presents experimental devices that allow us to detect individual quantum systems and observe quantum jumps occurring at random times. Described: superconducting single photon detectors, detection of arrays of ions and atoms, the shelving technique that allows us to measure the quantum state of the single atom, state selective field ionization of single Rydberg atoms, detection of single molecules on a surface by confocal microscopy, articial atoms in circuit quantum electrodynamics (cQED)

Appendix B: generalized postulates of quantum mechanics, valid for open systems and general measurements.

Theoretical chapter devoted to the domain of parameter estimation by quantum measurements. It first details the implications of the Heisenberg inequality and gives the expression of the quantum Cramér–Rao bound, a limit that is optimized over all data processing strategies and measurements on a given parameter-dependent quantum state. Measurement optimization over different quantum states and different optical modes in which the quantum state is defined is also discussed in various situations. The chapter then focusses on the measurement-induced perturbation bydiscussing the Heisenberg microscope and the Ozawa inequality, then on different implementations of quantum nondemolition (QND) measurements using the crossed-Kerr effect in quantum optics and opto-mechanics.

Theoretical chapter devoted to the detailed description of continuous variable (CV) systems by consideringthe "phase space," that is spanned by position and momentum for massive particles, quadratures for a quantum electromagnetic field, and phase and charge for electrical circuits. It introduces tools like the Glauber, Husimi, or Dirac phase–space functions, and in more details the Wigner function, that are convenient to describe CV quantum states and their time evolution using the Moyal equation. The chapter gives examples of Wigner functions and their time evolution in the presence of dissipation. It then defines symplectic quantum maps that are simple and important cases of Hamiltonian evolution and are simply related to the covariance matrix containingvariances and correlations. It details the characterization of the quantum processes using the Williamson reduction and Bloch–Messiah decomposition. It discusses Gaussian and non-Gaussian states and the specific measurement procedures for CV states, such as homodyne and double homodyne detection. It introduces the EPR entangled state and, finally, describes how to characterize entanglement and unconditionally teleport Gaussian quantum states.

Experimental chapter describing different experiments allowing us to accurately measure physical parameters, such as very small concentrations of atomic species by intensity monitoring, the observation of gravitational waves using giant interferometers, transverse positioning of light beams, transition frequencies of metrological interest using laser frequency combs, and magnetometry of ultra-small magnetic fields using superconducting quantum interference devices (SQUIDs).

Appendix K: this appendix is an introduction to another very active and promising domain of quantum physics, named circuit quantum electrodynamics (cQED), dealing with the quantum properties of macroscopic objects consisting of superconducting electrical circuits. In an LC circuit the energy is quantized, and charge and flux are two quantum canonical conjugate quantities that do not commute. Their quantum fluctuations are bound by a Heisenberg inequality. A Josephson junction inserted in the circuit introduces strong nonlinearities in the system, which breaks the equidistance between the energy levels and makes the circuit look like a qubit, called a transmon. Cooling at mK temperatures is necessary to have quantum effects dominate over thermal effects. The circuit is embedded in a resonant cavity, and the system bears many analogies with cavity QED and Jaynes–Cummings formalism for coupled photons and atoms. One can perform nondestructive read-out and control of the transmon, as well as phase-sensitive, quantum-limited amplification, with nonlinearities that are much stronger than the ones used in quantum optics.