  I study heat and norm transport in a one-dimensional lattice of linear
Schr\"odinger oscillators with conservative stochastic perturbations. Its
equilibrium properties are the same of the Discrete Nonlinear Schr\"odinger
equation in the limit of vanishing nonlinearity. When attached to external
classical reservoirs that impose nonequilibrium conditions, the chain displays
diffusive transport, with finite Onsager coefficients in the thermodynamic
limit and a finite Seebeck coefficient.
