  We characterize finite-time thermodynamic processes of multidimensional
quadratic overdamped systems. Analytic expressions are provided for heat, work,
and dissipation for any evolution of the system covariance matrix. The
Bures-Wasserstein metric between covariance matrices naturally emerges as the
local quantifier of dissipation. General principles of how to apply these
geometric tools to identify optimal protocols are discussed. Focusing on the
relevant slow-driving limit, we show how these results can be used to analyze
cases in which the experimental control over the system is partial.
