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.