The Converse Part of The Theorem for Quantum Hoeffding Bound

TL;DR

This paper proves the converse part of the quantum Hoeffding bound theorem, completing the theoretical foundation for quantum hypothesis testing by establishing the bound's validity.

Contribution

It provides the missing converse proof for the quantum Hoeffding bound, complementing existing work on the direct part, thus fully establishing the theorem.

Findings

The quantum Hoeffding bound is now fully proven, including the converse part.

The proof is based on methods similar to those used in the quantum Chernoff bound.

This work solidifies the theoretical understanding of asymptotic quantum hypothesis testing.

Abstract

We prove the converse part of the theorem for quantum Hoeffding bound on the asymptotics of quantum hypothesis testing, essentially based on an argument developed by Nussbaum and Szkola in proving the converse part of the quantum Chernoff bound. Our result complements Hayashi's proof of the direct (achievability) part of the theorem, so that the quantum Hoeffding bound has now been established.