Note on a Family of Monotone Quantum Relative Entropies

Abstract

Given a convex function ${\varphi}$ and two hermitian matrices A and B, Lewin and Sabin study in (Lett Math Phys 104:691–705, 2014) the relative entropy defined by ${\mathcal{H}(A,B)={\rm Tr} \left[ \varphi(A) - \varphi(B) - \varphi'(B)(A-B) \right]}$. Among 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 bounded self-adjoint operators acting on an infinite-dimensional Hilbert space by a limiting procedure. More precisely, for an increasing sequence of finite-dimensional projections ${\left\lbrace P_n \right\rbrace_{n=1}^{\infty}}$ 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 ${\left\lbrace P_n \right\rbrace_{n=1}^{\infty}}$. The question whether this sequence converges to its "obvious" limit, namely ${{\rm Tr} \left[ \varphi(A)- \varphi(B) - \varphi'(B)(A-B) \right]}$, has been left open. We answer this question in principle affirmatively and show that ${\lim_{n \to \infty} \mathcal{H}(P_n A P_n, P_n B P_n) = {\rm Tr} \left[ \varphi(A) - \varphi(B) - \frac{{\rm d}}{{\rm d} \alpha} \varphi\left( \alpha A + (1-\alpha)B \right)\vert_{\alpha = 0} \right]}$. 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 ${{\rm Tr}\Big[ \varphi(A) - \varphi(B) - \frac{{\rm d}}{{\rm d} \alpha} \varphi\left( \alpha A + (1-\alpha)B \right)\vert_{\alpha = 0} \Big] = {\rm Tr} \Big[ \varphi(A)- \varphi(B) - \varphi'(B)(A-B) \Big]}$ holds.