of $\Delta$ is defined to be the entropy of $\Delta$ relative to $\Gamma_\Delta$, that is, $$\label{nonfreeness intro def} 
\mathfrak{C}(\Delta) \equals  S(\Delta \|\Gamma_\Delta) \ .$$ This equals $\Tr(\Delta \log\Delta) - \Tr(\Delta \log\Gamma_{\Delta})$ provided that $\Tr(\Delta \log\Delta) > -\infty$.

Correlation functionals closely related to nonfreeness are obtained by using other “divergences" instead of the relative entropy to compare the states $\Delta$ and $\Gamma_\Delta$. Using a divergence that enjoys the properties of additivity and monotonicity will yield a correlation functional with those properties. We have in mind the Rényi divergences $$D_\alpha(\rho \| \sigma) \equals \frac{1}{\alpha - 1} \log \Tr (\rho^\alpha \sigma^{1-\alpha})$$ for $0 < \alpha \le 2$ and the “sandwiched" relative Rényi entropies [@Mueller-Lennert...; @WildeWinterYang; @FrankLieb] $$\widetilde{D}_\alpha(\rho \| \sigma)  \equals \frac{1}{\alpha - 1} \log \Tr \Big(\big(\sigma^{\frac{1-\alpha}{2\alpha}} \rho \sigma^{\frac{1-\alpha}{2\alpha}}\big)^\alpha\Big)$$ for $\alpha \ge \tfrac12$. The divergences $D_1$ and $\widetilde{D}_1$ are defined by taking limits $\alpha \longrightarrow 1$ and both equal the relative entropy $S(\rho \| \sigma)$.

For values of $\alpha$ in the appropriate ranges, the correlation functionals $$\begin{aligned}
\mathfrak{C}_\alpha( \Delta )  &=&  D_\alpha(\Delta \|\Gamma_{\Delta} ) \\
\widetilde{\mathfrak{C}}_\alpha( \Delta ) &=&   \widetilde{D}_\alpha(\Delta \|\Gamma_{\Delta} )  \end{aligned}$$ all share the following properties with the nonfreeness functional $\mathfrak{C} = \mathfrak{C}_1 = \widetilde{\mathfrak{C}}_1$ :

[(i)]{} they take only non-negative values, 