\DD(\FFF)$ is a density operator on the fermion Fock space, and let $p_1,p_2,\ldots$ denote the eigenvalues of its 1-pdm. If $S(\Delta) < \infty$ then $$S( \Delta \| \Gamma_{\Delta} )  \equals - \sum p_j \log p_j - \sum (1-p_j) \log(1-p_j)  - S(\Delta)\ .$$

Nonfreeness as relative entropy mimimum {#Nonfreeness as relative entropy mimimum}
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The nonfreeness $\mathfrak{C}(\Delta)$ of a many-fermion state $\Delta$ is equal to the minimum of its entropy relative to all free reference states. To prove this we will use the inequality $$\label{an entropy inequality}
  S( A ) \ \le\  -\Tr(A \log B)$$ for two density operators on the same Hilbert space. In case $S(A) < \infty$, then $$\label{if entropy is finite}
S( A \| B) = -  \Tr ( A \log B ) - S(A)$$ and (\[an entropy inequality\]) follows immediately from the fact that $S( A \| B) \ge 0$.

\[superproposition\] Suppose $\Delta \in \DD(\FFF)$ and let $\Gamma_{\Delta}$ denote the unique free density operator with the same $1$-pdm as $\Delta$. Then, for all free density operators $\Gamma $, $$\label{superprop-equation}
     S(\Delta\|\Gamma_{\Delta}) \ + \ S(\Gamma_{\Delta}\|\Gamma ) \ = \ S(\Delta\|\Gamma  ),$$ and therefore $$\label{superprop-inequality}
     S(\Delta\|\Gamma_{\Delta}) \ \le \ S(\Delta\|\Gamma  )$$ with equality only if $S(\Delta\|\Gamma_\Delta) = \infty$ or 