Fisher information matrix of Husimi distribution

We calculate the Fischer information matrix of Husimi distribution in the Fock Bargmann representation. It turns out that the Fisher information of the position and that of the momentum move in opposite directions, and that a weighted trace of the Fisher information matrix is a constant independent of the Husimi distribution. This may be interpreted as a kind of uncertainty relation (in the spirit of Heisenberg uncertainty principle) from the statistical inference point of view.