In a joint work with Tiju Cherian John we proved the following: Let
$$ \begin{split} \rho = \sum_{i\in \mathcal{I}}r_i |u_i\rangle \langle u_i |, \quad r_i\geq 0,\quad \sum_{i\in \mathcal{I}} r_i = 1,\quad \{u_i\}{i\in \mathcal{I}} \text{ is an o.n. basis of } \mathcal{H}\\ \sigma = \sum{j\in \mathcal{I}}s_j |v_j\rangle \langle v_j|, \quad s_j\geq0, \quad \sum_{j\in \mathcal{I}} s_j = 1, \quad \{v_j\}_{j\in \mathcal{I}}\text{ is an o.n. basis of } \mathcal{H}\end{split} $$
be the spectral decompositions of two density operators $\rho$ and $\sigma$ on some Hilbert space $\mathcal{H}$. The cardinality of $\mathcal{I}$ is equal to the dimension of $\mathcal{H}$. Define the probability measures $P$ and $Q$ on the set $\mathcal{I}\times \mathcal{I}$ by
$$ \begin{split} P(i,j) &= r_i | \langle u_i | v_j \rangle |^2,\\ Q(i,j)& = s_j | \langle u_i | v_j\rangle|^2, \quad \forall (i,j)\in \mathcal{I}\times\mathcal{I}. \end{split} $$
Then, for every convex or concave function $f:(0,\infty) \to \mathbb{R}$ we have that
$$ D_f(\rho|| \sigma)= D_f(P || Q), $$
where $D_f(\rho || \sigma)$ is the quantum $f$-divergence from $\rho$ to $\sigma$ and $D_f (P|| Q)$ is the classical $f$-divergence from $P$ to $Q$.
Recall the definition of the classical $f$-Divergence between two discrete probability distributions $P$ and $Q$ defined on a countable set $\Omega$, where $f:(0, \infty) \to \mathbb{R}$ is a convex or concave function:
$$ D_f(P \| Q) =\sum_{\{\omega : P(\omega)Q(\omega)>0\} } Q(\omega) f\left(\frac{P(\omega)}{Q(\omega)}\right)+f(0) Q(P=0)+f^{\prime}(\infty) P(Q=0), $$
where
$$ f(0) = \lim_{x \to 0} f(x) \quad \text{and} \quad f’(\infty)= \lim_{x \to \infty}\limits \frac{f(x)}{x}. $$
In order to obtain the Kullback-Leibler divergence, we choose $f(x)=x\log x$, in which case $D_f(P ||Q)$ is simply denoted by $D(P || Q).$ Then,
$$ f(0)=0 \quad \text{and} \quad f’(\infty)= \infty. $$
Thus,
$$ D(P||Q) = \left\{ \begin{array}{ll} \sum_{\{\omega \in \Omega: P(\omega)Q(\omega)>0\} }\limits Q(\omega) f\left(\frac{P(\omega)}{Q(\omega)}\right) & \text{if } P\ll Q\\ \infty & \text{otherwise}\end{array} . \right. $$
Thus,
$$ D(P||Q) = \left\{ \begin{array}{ll} \sum_{\{\omega \in \Omega: P(\omega)Q(\omega)>0\} }\limits P(\omega) \log \left(\frac{P(\omega)}{Q(\omega)}\right) & \text{if } P\ll Q\\ \infty & \text{otherwise}\end{array} . \right. $$
This expression can be alternatively written as