  The antisymmetry of a fermionic quantum state has a marked effect on its
entanglement properties. Recently, Carlen, Lieb and Reuvers (CLR) studied this
effect, in particular concerning the entropy of the two-body reduced density
matrix of a fermionic state. They conjecture that this entropy is minimized by
Slater determinants. We use tools from quantum information theory to make
progress on their conjecture, proving it when the dimension of the underlying
Hilbert space is not too large. We also derive general properties of the
entropy of the $k$-body reduced density matrix as a function of $k$: It is
concave for all $k$ and non-decreasing for all $k\leq N/2$, where $N$ is the
number of fermionic particles. We can apply these general facts to improve the
bound of CLR in all dimensions.
