On connection between two characteristics of dependence of Gaussian random vectors

Let $\xi$ and $\eta$ be Gaussian random vectors with distributions $\mathscr P_1(dx)$ and $\mathscr P_2(dy)$ respectvely, and with joint distribution $\mathscr P_{12}(dxdy)$. Let $I$ be the information about $\eta$, containing in $\xi$. Set
$$ V=\operatorname{Var}(\mathscr P_{12}-\mathscr P_1\times\mathscr P_2). $$
In the paper connection between $I$ and $V$ is under investigation.