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Note on a Family of Monotone Quantum Relative Entropies

Given a convex function $φ$ 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 $\mathcal{H}(A,B)=\text{Tr} [ φ(A) - φ(B) - φ'(B)(A-B) ]$. Amongst other things, they prove that the so-defined quantity is monotone if and only if $φ'$ 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 $\text{Tr} [ φ(A)- φ(B) - φ'(B)(A-B) ]$, has been left open. We answer this question in principle affirmatively and show that $\lim_{n \to \infty} \mathcal{H}(P_n A P_n, P_n B P_n) = \text{Tr}[ φ(A) - φ(B) - \frac{\text{d}}{\text{d} α} φ( αA + (1-α)B )|_{α= 0} ]$. If the operators $A$ and $B$ are regular enough, that is $(A-B)$, $φ(A)-φ(B)$ and $φ'(B)(A-B)$ are trace-class, the identity $\text{Tr}[ φ(A) - φ(B) - \frac{\text{d}}{\text{d} α} φ( αA + (1-α)B )|_{α= 0} ] = \text{Tr} [ φ(A)- φ(B) - φ'(B)(A-B) ]$ holds.