A New Proof of the Direct Part of Stein's Lemma in Quantum Hypothesis Testing

TL;DR

This paper presents a simplified proof of the direct part of Stein's lemma in quantum hypothesis testing using a key operator inequality, avoiding complex prior theorems and providing new insights into quantum statistical methods.

Contribution

It introduces a novel proof technique for the direct part of Stein's lemma in quantum hypothesis testing based on a key operator inequality, bypassing previous complex theorems.

Findings

Simplified proof of the direct part of Stein's lemma

New proof of Hiai-Petz's theorem

Operator inequality as a fundamental tool

Abstract

The direct part of Stein's lemma in quantum hypothesis testing is revisited based on a key operator inequality between a density operator and its pinching. The operator inequality is used to show a simple proof of the direct part of Stein's lemma without using Hiai-Petz's theorem, along with an operator monotone function, and in addition it is also used to show a new proof of Hiai-Petz's theorem.