Information and Majorization Theory for Fermionic Phase-Space Distributions

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 anticommuting nature of Grassmann variables allows us to provide simple expressions for the Glauber P, 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 then 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. Our 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 and are physically relevant.