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Why do we need quantization to describe vision? What are the quadratureoperators of the electromagnetic field? Is it possible to measure them? Whatare the characteristic functions useful for? In this brief tutorial we providethe theoretical tools needed to describe the generation, manipulation andcharacterization of optical quantum states and of the main passive (beamsplitters) and active (squeezers) devices involved in experiments, such as theHong-Ou-Mandel interferometer and the continuous-variable quantumteleportation. We also introduce the concept of operator ordering and thedescription of a system by means of the $p$-ordered characteristic functions.Then we focus on the quasi-probability distributions and, in particular, on therelation between the marginals of the Wigner function and the outcomes of thequadrature operator measurement. Finally, we introduce the balanced homodynedetection to measure the quadrature operator and the homodyne tomography as atool for characterizing quantum optical states also in the presence of non-unitquantum efficiency.

Introduction to generation, manipulation and characterization of optical quantum states