We present necessary and sufficient conditions for stabilizing a discrete-time LTI plant in the mean squared sense when a sensor transmits the plant state information to a remotely placed controller across a Gaussian product channel. The Gaussian product channel models a continuous-time waveform Gaussian channel, where the encoder transmits information to the receiver across multiple noisy paths. The conditions presented are in terms of the power at the transmitter, noise variances and the unstable eigenvalues of the plant. It is known that linear coding schemes may lead to overly restrictive stabilizability conditions in such scenarios. We present a nonlinear coding scheme and present the resulting sufficient stabilizability conditions. We then prove the necessity of the conditions using information theoretic tools.