Note on a Family of Monotone Quantum Relative Entropies

Given a convex function $\phi$ 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. $\mathbf{\text{104}}$ (2014), 691-705.] the relative entropy defined by $$\mathcal{H}$(A,B)=$\text{Tr}$ [ \varphi(A) - \varphi(B) - \varphi’(B)(A-B) ]$. Amongst other things, they prove that the so-defined quantity is monotone if and only if ${\phi }^{\mathrm{\prime }}$ is operator monotone. The monotonicity is then used to properly define $\mathcal{H}(A,B)$ for self-adjoint bounded operators acting on an infinite-dimensional Hilbert space by a limiting procedure. More precisely, for an increasing sequence of finite-dimensional projections $\{P_n{\}}_{n=1}^{\mathrm{\infty }}$ with $P_n \to 1$strongly,thelimit$\lim_{n \to \infty} $\mathcal{H}$(P_n A P_n, P_n B P_n)$isshowntoexistandtobeindependentofthesequenceofprojections$\lbrace P_n \rbrace_{n=1}^{\infty}$. The question whether this sequence converges to its “obvious” limit, namely $$\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} ]$\mathrm{.}Iftheoperators$A$and$B$areregularenough,thatis$(A-B)$,$\varphi(A)-\varphi(B)$and$\varphi’(B)(A-B)$aretrace-class,theidentity$$\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.