existence remark\] Suppose $Q:\HH \longrightarrow \HH$ is a positive-semidefinite contraction operator with finite trace. Then there exists a unique free density operator on $\FFF(\HH)$ with $1$-pdm $Q$.

The von Neumann entropy of a free state is a simple function of its natural occupation numbers $p_i$. The following formula can be established using Proposition \[structural\] in Section \[Free states appendix\].

\[quasifree entropy remark\] If $\Gamma$ is a free state with natural occupation numbers $p_i$, then its von Neumann entropy is $$S(\Gamma) \ = \ -\sum_i p_i \log p_i -\sum_i (1-p_i) \log (1-p_i) \ .$$

Relative entropy correlation functionals {#Relative entropy correlation functionals}
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Recall that $\DD(\FFF)$ denotes the set of density operators on the fermion Fock space $\FFF = \FFF(\HH)$ that represent states of finite average particle number. The $1$-pdm of a density operator $\Delta \in \DD(\FFF)$ is a positive-semidefinite contraction operator on $\HH$ with finite trace. By Remark \[free existence remark\], there exists a unique free density operator with the same $1$-pdm as $\Delta$. We denote this free density operator by $\Gamma_\Delta$. In other words, $\Gamma_\Delta$ denotes the unique free density operator such that $\gamma_{\Gamma_\Delta} = \gamma_\Delta$ (with the notation defined in formula (\[defining property of $1$-pdm\])).

The nonfreeness $\mathfrak{C}(\Delta)$ 