@kodlu: Thanks for your comment, but I'm not sure if you're right. Differential entropy is well-defined and discretization not necessary. If I sample from a multivariate Gaussian and use a non-parametric entropy estimator, the estimate converges to the expected theoretical value as the samples go to infinity. It's only the special case $H(x|x)$ that seems to be tricky for some reason.

Note that this is equivalent to $L(x) = p(x)\int_y \frac{p(y\mid x)}{p(y)} dy$, where all distributions are marginals or conditionals of a joint Gaussian distribution $p(x,y)$. This might be easier to tackle.