Let $X \sim \mathcal{N}( \mu, \Sigma )$ and $Y \sim \mathcal{N}( \nu, \Gamma )$, where both $\Sigma$ and $\Gamma$ are positive-definite. Define two distributions $\mathbb{P}$ and $\mathbb{Q}$, where $\mathbb{P}$ is the law of $XX^T$ and $\mathbb{Q}$ is the law of $YY^T$.
Is there a closed-form formula for the KL-divergence between $\mathbb{P}$ and $\mathbb{Q}$? How about an upper bound that is sharper than $D_{KL}(X, Y)$?