Purity based continuity bounds for quantum information measures

TL;DR

This paper develops new continuity bounds for quantum information measures based on purity differences, providing dimension-independent bounds and applications to entanglement and quantum Markov chains.

Contribution

It introduces purity-based continuity bounds for various quantum information measures, offering a dimension-free approach and new insights into quantum correlations.

Findings

Continuity bounds for von Neumann entropy depending only on purity and system dimension.

Dimension-free bounds for conditional von Neumann entropy in terms of purity.

Bounds on quantum mutual information, conditional mutual information, and squashed entanglement.

Abstract

In quantum information theory, communication capacities are mostly given in terms of entropic formulas. Continuity of such entropic quantities are significant, as they ensure uniformity of measures against perturbations of quantum states. Traditionally, continuity bounds have been provided in terms of the trace distance, which is a bonafide metric on the set of quantum states. In the present contribution we derive continuity bounds for various information measures based on the difference in purity of the concerned quantum states. In a finite-dimensional system, we establish continuity bounds for von Neumann entropy which depend only on purity distance and dimension of the system. We then obtain uniform continuity bounds for conditional von Neumann entropy in terms of purity distance which is free of the dimension of the conditioning subsystem. Furthermore, we derive the uniform continuity bounds for other entropic quantities like relative entropy distance, quantum mutual information and quantum conditional mutual information. As an application, we investigate the variation in squashed entanglement with respect to purity. We also obtain a bound to the quantum conditional mutual information of a quantum state which is arbitrarily close to a quantum Markov chain.