  We study the entropic Gromov-Wasserstein and its unbalanced version between
(unbalanced) Gaussian distributions with different dimensions. When the metric
is the inner product, which we refer to as inner product Gromov-Wasserstein
(IGW), we demonstrate that the optimal transportation plans of entropic IGW and
its unbalanced variant are (unbalanced) Gaussian distributions. Via an
application of von Neumann's trace inequality, we obtain closed-form
expressions for the entropic IGW between these Gaussian distributions. Finally,
we consider an entropic inner product Gromov-Wasserstein barycenter of multiple
Gaussian distributions. We prove that the barycenter is a Gaussian distribution
when the entropic regularization parameter is small. We further derive a
closed-form expression for the covariance matrix of the barycenter.
