Transformations, generation and measurement of photonic states that possess a Gaussian density matrix

In this paper, the authors consider issues of importance in dealing with 1-mode and 2-mode photonic states that possess a Gaussian density matrix. By a Gaussian density matrix or a Gaussian rho, the authors mean an exponential of a quadratic polynomial of the creation and annihilation operators of the modes. The authors discuss the set of all possible linear transformations of N modes into an equal number of modes. The authors propose a device, which is called a covmuter, that can perform any linear transformation in this group. Given a fixed covariance matrix V0 of a Gaussian-rho state, the authors find all covariance matrices V for which there exists a multi-mode linear transformation that takes V0 to V. The authors study the conservation laws obeyed by such transformations. Finally, the authors show that a covmuter can be used both to generate and to measure 1-mode and 2-mode Gaussian-rho states