**Gianfranco Cariolaro**, Professor Emeritus of Quantum Communications at the Department of Information Engineering of the University of Padova, will be visiting AQM on *Tue March 7, 2017* and will give a seminar in *Aula Polvani* at *14:30*, entitled

### Unified Theory of Gaussian unitaries and Gaussian states

Abstract

The class of Gaussian unitaries includes three important subclasses: 1) displacement, 2) rotation, and 3) squeezing. These unitaries can be defined in the *n* mode by a simple extension of the definition of the single mode. A fundamental theorem by Ma and Rhodes (1990), vitually ignored in the literature, states that the cascade combination of 1), 2), and 3) allows one to generate the whole class of Gaussian unitaries (and of Gaussian states). This is the key to developing a Unified Theory of Gaussian unitaries and Gaussian states, which allows one to obtain the main tools, as Bogoliubov and symplectic transformations, Block–Messiah reduction, synthesis of a general Gaussian unitary with elementary components, statistics of photon numbers, normally ordered form, and so on. For each of these topics **all results are ****explicitly** obtained in terms of the parameters of 1), 2), and 3), that is, an *n* complex vector, an *n* x *n* rotation matrix (Hermitian), and *n* x *n* squeeze matrix (symmetric).