  We prove a generalization of the strong subadditivity of the von Neumann
entropy for bosonic quantum Gaussian systems. Such generalization determines
the minimum values of linear combinations of the entropies of subsystems
associated to arbitrary linear functions of the quadratures, and holds for
arbitrary quantum states including the scenario where the entropies are
conditioned on a memory quantum system. We apply our result to prove new
entropic uncertainty relations with quantum memory, a generalization of the
quantum Entropy Power Inequality, and the linear time scaling of the
entanglement entropy produced by quadratic Hamiltonians.
