Note on a Family of Monotone Quantum Relative Entropies

TL;DR

This paper investigates the limits of a family of monotone quantum relative entropies for infinite-dimensional operators, establishing conditions under which the limit matches the expected trace expression.

Contribution

It proves the convergence of the finite-dimensional approximations of quantum relative entropy to a well-defined limit in infinite dimensions, affirming a conjecture about their equivalence under certain conditions.

Findings

Limit of the monotone quantum relative entropy exists and is independent of the projection sequence.

The limit equals the trace of the difference involving the convex function and its derivative.

Equality with the trace expression holds when operators are sufficiently regular and trace-class.

Abstract

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.