  Given a convex function $\varphi$ and two hermitian matrices $A$ and $B$,
Lewin and Sabin study in [M. Lewin, J. Sabin, {\it A Family of Monotone Quantum
Relative Entropies}, Lett. Math. Phys. \textbf{104} (2014), 691-705.] the
relative entropy defined by $\mathcal{H}(A,B)=\text{Tr} [ \varphi(A) -
\varphi(B) - \varphi'(B)(A-B) ]$. Amongst other things, they prove that the
so-defined quantity is monotone if and only if $\varphi'$ is operator monotone.
The monotonicity is then used to properly define $\mathcal{H}(A,B)$ for
self-adjoint bounded operators acting on an infinite-dimensional Hilbert space
by a limiting procedure. More precisely, for an increasing sequence of
finite-dimensional projections $\lbrace P_n \rbrace_{n=1}^{\infty}$ with $P_n
\to 1$ strongly, the limit $\lim_{n \to \infty} \mathcal{H}(P_n A P_n, P_n B
P_n)$ is shown to exist and to be independent of the sequence of projections
$\lbrace P_n \rbrace_{n=1}^{\infty}$. The question whether this sequence
converges to its "obvious" limit, namely $\text{Tr} [ \varphi(A)- \varphi(B) -
\varphi'(B)(A-B) ]$, has been left open. We answer this question in principle
affirmatively and show that $\lim_{n \to \infty} \mathcal{H}(P_n A P_n, P_n B
P_n) = \text{Tr}[ \varphi(A) - \varphi(B) - \frac{\text{d}}{\text{d} \alpha}
\varphi( \alpha A + (1-\alpha)B )|_{\alpha = 0} ]$. If the operators $A$ and
$B$ are regular enough, that is $(A-B)$, $\varphi(A)-\varphi(B)$ and
$\varphi'(B)(A-B)$ are trace-class, the identity $\text{Tr}[ \varphi(A) -
\varphi(B) - \frac{\text{d}}{\text{d} \alpha} \varphi( \alpha A + (1-\alpha)B
)|_{\alpha = 0} ] = \text{Tr} [ \varphi(A)- \varphi(B) - \varphi'(B)(A-B) ]$
holds.
