  We prove that the log-determinant of the covariance matrix obeys the strong
subadditivity inequality for arbitrary tripartite states of multimode
continuous variable quantum systems. This establishes general limitations on
the distribution of information encoded in the second moments of canonically
conjugate operators. The inequality is shown to be stronger than the
conventional strong subadditivity inequality for von Neumann entropy in a class
of pure tripartite Gaussian states. We finally show that such an inequality
implies a strict monogamy-type constraint for joint Einstein-Podolsky-Rosen
steerability of single modes by Gaussian measurements performed on multiple
groups of modes.
