Continuity bounds for quantum entropies arising from a fundamental entropic inequality

TL;DR

This paper derives a new tight upper bound for the difference in von Neumann entropies of quantum states, leading to refined continuity bounds for quantum entropies and extending to infinite-dimensional systems.

Contribution

It introduces a novel entropic inequality based on the Jordan-Hahn decomposition, refining and implying the Audenaert-Fannes inequality, with applications to quantum conditional and relative entropy bounds.

Findings

Established a tight upper bound for von Neumann entropy differences.

Derived a refined entropic inequality that implies the AF inequality.

Provided continuity bounds for quantum conditional and relative entropies.

Abstract

We establish a tight upper bound for the difference in von Neumann entropies between two quantum states, $\rho_1$ and $\rho_2$. This bound is expressed in terms of the von Neumann entropies of the mutually orthogonal states derived from the Jordan-Hahn decomposition of the difference operator $(\rho_1 - \rho_2)$. This yields a novel entropic inequality that implies the well-known Audenaert-Fannes (AF) inequality. In fact, it also leads to a refinement of the AF inequality. We employ this inequality to obtain a uniform continuity bound for the quantum conditional entropy of two states whose marginals on the conditioning system coincide. We additionally use it to derive a continuity bound for the quantum relative entropy in both variables. Interestingly, the fundamental entropic inequality is also valid in infinite dimensions.