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.