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.

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We have included the factor 2πn2 in the integration measure to avoid having it in the probability densities of Gaussian random variables.

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