Tight uniform continuity bounds for quantum entropies: conditional entropy, relative entropy distance and energy constraints

TL;DR

This paper derives tight continuity bounds for quantum entropies, including conditional entropy and relative entropy, under energy constraints, improving understanding of their stability in quantum information theory.

Contribution

It provides the first tight continuity bounds for quantum conditional entropy and relative entropy distance, extending to infinite-dimensional systems with energy constraints.

Findings

Tight bounds for Alicki-Fannes continuity of conditional von Neumann entropy.

New proofs with tighter bounds for relative entropy of entanglement and its regularization.

Continuity bounds for von Neumann entropy under energy constraints in infinite-dimensional systems.

Abstract

We present a bouquet of continuity bounds for quantum entropies, falling broadly into two classes: First, a tight analysis of the Alicki-Fannes continuity bounds for the conditional von Neumann entropy, reaching almost the best possible form that depends only on the system dimension and the trace distance of the states. Almost the same proof can be used to derive similar continuity bounds for the relative entropy distance from a convex set of states or positive operators. As applications we give new proofs, with tighter bounds, of the asymptotic continuity of the relative entropy of entanglement, $E_R$, and its regularization $E_R^\infty$, as well as of the entanglement of formation, $E_F$. Using a novel "quantum coupling" of density operators, which may be of independent interest, we extend the latter to an asymptotic continuity bound for the regularized entanglement of formation, aka entanglement cost, $E_C=E_F^\infty$. Second, analogous continuity bounds for the von Neumann entropy and conditional entropy in infinite dimensional systems under an energy constraint, most importantly systems of multiple quantum harmonic oscillators. While without an energy bound the entropy is discontinuous, it is well-known to be continuous on states of bounded energy. However, a quantitative statement to that effect seems not to have been known. Here, under some regularity assumptions on the Hamiltonian, we find that, quite intuitively, the Gibbs entropy at the given energy roughly takes the role of the Hilbert space dimension in the finite-dimensional Fannes inequality.