  We derive an upper bound for the efficiency of estimating entries in the
inverse covariance matrix of a high dimensional distribution. We show that in
order to approximate an off-diagonal entry of the density matrix of a
$d$-dimensional Gaussian random vector, one needs at least a number of samples
proportional to $d$. Furthermore, we show that with $n \ll d$ samples, the
hypothesis that two given coordinates are fully correlated, when all other
coordinates are conditioned to be zero, cannot be told apart from the
hypothesis that the two are uncorrelated.
