\sum_i \big( \log(p_i) \langle h_i | \gamma_\Delta h_i \rangle -  \log(1-p_i) (1 -  \langle h_i | \gamma_\Delta f_i \rangle ) \big).\end{aligned}$$ Since $\gamma_\Delta$ is also the $1$-pdm of $\Gamma_\Delta$, the conclusion (\[conclusion\]) follows.

When the von Neumann entropy $S(\Delta) = -\Tr (\Delta \log\Delta)$ is finite we may use the formula $$\label{we may use}
S(\Delta \| \Gamma_{\Delta}) \equals - \Tr(\Delta \log\Gamma_{\Delta}) - S(\Delta)$$ for the relative entropy. This leads to simple formulas for nonfreeness.

\[max ent prop\] Suppose $\Delta \in \DD(\FFF)$ satisfies $S(\Delta) < \infty$. Let $\Gamma_{\Delta}$ denote the unique free density operator with the same $1$-pdm as $\Delta$. Then, $$\label{basic formula}
     S(\Delta\|\Gamma_{\Delta}) \equals S(\Gamma_\Delta)-S(\Delta) \ .$$

By Lemma \[good lemma\], $$-\Tr( \Delta \log \Gamma_{\Delta} )  \equals  -\Tr( \Gamma_{\Delta} \log \Gamma_{\Delta} ) \equals 
    S(\Gamma_\Delta ).$$ Substituting $S(\Gamma_\Delta )$ for $-\Tr( \Delta \log \Gamma_{\Delta} )$ in equation (\[we may use\]) yields (\[basic formula\]).

By Remark \[quasifree entropy remark\], the von Neumann entropy of the free state $\Gamma_\Delta$ is a function of its natural occupation numbers. The natural occupation numbers of $\Gamma_{\Delta}$ are the same as those of $\Delta$, since they have the same $1$-pdm; therefore, using (\[basic formula\]) we obtain the following simple formula for nonfreeness:

\[explicit nonfreeness formula\] Suppose $\Delta \in 