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Our main goal in this paper is to extend to any system of coupled quadraticHamiltonians some properties known for systems of quantum harmonic oscillatorsrelated with the Brownian Quantum Motion model. In a first part we get a rathergeneral formula for the purity (or the linear entropy) in a short timeapproximation. In a second part we establish a master equation (or aFokker-Planck type equation) for the time evolution of the reduced matrixdensity for bilinearly coupled quadratic Hamiltonians.  The Hamiltonians and the bilinear coupling can be time dependent.  Moreover we give an explicit formula for the solution of this master equationso that the time evolution of the reduced density at time $t$ is connected withthe reduced density at initial time $t_0$ for $t_0 \leq t <t_0 +t_c$ where$t_c\in ]0, \infty]$ is a critical time but reversibility is lost for $t \geqt_0 +t_c$.

Time Evolution of States for Open Quantum Systems. The quadratic case