# Upper continuity bound on the quantum quasi-relative entropy

## TL;DR

This paper establishes upper bounds on quantum quasi-relative entropy using trace distance, improving existing bounds for certain states and entropy measures in finite-dimensional quantum systems.

## Contribution

It introduces new upper bounds for quasi-relative entropy applicable to various operator monotone functions and state types, enhancing previous bounds for specific entropy measures.

## Key findings

- Upper bounds for quasi-relative entropy in terms of trace distance.
- Improved bounds for Umegaki and Tsallis relative entropies in finite dimensions.
- Enhanced bounds for states in dimensions larger than four.

## Abstract

We provide an upper bound on the quasi-relative entropy in terms of the trace distance. The bound is derived for two cases: 1) any operator monotone decreasing function and full rank mixed qubit or classical states; 2) a large class of operator monotone decreasing function and any mixed qubit or classical states. Moreover, we derive an upper bound for the Umegaki and Tsallis relative entropies in the case of any finite-dimensional states. The bound for the relative entropy improves the known bounds for some states in any dimensions larger than four. The bound for the Tsallis entropy improves the known bounds.