  We put forward several information-theoretic measures for analyzing the
uncertainty of fermionic phase-space distributions using the theory of
supernumbers. In contrast to the bosonic case, the anti-commuting nature of
Grassmann variables allows us to provide simple expressions for the Wigner $W$-
and Husimi $Q$-distributions of the arbitrary state of a single fermionic mode.
It appears that all physical states are Gaussian and thus can be described by
positive or negative thermal distributions (over Grassmann variables). We are
then able to prove several fermionic uncertainty relations, including notably
the fermionic analogs of the (yet unproven) phase-space majorization and Wigner
entropy conjectures for a bosonic mode, as well as the Lieb-Solovej theorem and
the Wehrl-Lieb inequality. The central point is that, although fermionic
phase-space distributions are Grassmann-valued and do not have a
straightforward interpretation, the corresponding uncertainty measures are
expressed as Berezin integrals which take on real values, hence are physically
relevant.
