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On Strong Converse Theorems for Quantum Hypothesis Testing and Channel Coding

Hao-Chung Cheng, Li Gao

TL;DR

An alternative one-line proof for this one-shot strong converse bound via the variational expression of measured R\'enyi divergences is demonstrated, showing that the variational expression is a direct consequence of H\"older's inequality.

Abstract

Strong converse theorems refer to the study of impossibility results in information theory. In particular, Mosonyi and Ogawa established a one-shot strong converse bound for quantum hypothesis testing [Comm. Math. Phys, 334(3), 2014], which servers as a primitive tool for establishing a variety of tight strong converse theorems in quantum information theory. In this short note, we demonstrate an alternative one-line proof for this bound via the variational expression of measured Rényi divergences [Lett. Math. Phys, 107(12), 2017]. Then, we show that the variational expression is a direct consequence of Hölder's inequality.

On Strong Converse Theorems for Quantum Hypothesis Testing and Channel Coding

TL;DR

An alternative one-line proof for this one-shot strong converse bound via the variational expression of measured R\'enyi divergences is demonstrated, showing that the variational expression is a direct consequence of H\"older's inequality.

Abstract

Strong converse theorems refer to the study of impossibility results in information theory. In particular, Mosonyi and Ogawa established a one-shot strong converse bound for quantum hypothesis testing [Comm. Math. Phys, 334(3), 2014], which servers as a primitive tool for establishing a variety of tight strong converse theorems in quantum information theory. In this short note, we demonstrate an alternative one-line proof for this bound via the variational expression of measured Rényi divergences [Lett. Math. Phys, 107(12), 2017]. Then, we show that the variational expression is a direct consequence of Hölder's inequality.
Paper Structure (7 sections, 2 theorems, 49 equations)

This paper contains 7 sections, 2 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Rényi Information Measures and Variational Expressions
  4. Strong Converses via Variational Expressions
  5. Randomness-Assisted Classical-Quantum Channel Coding
  6. Historical Remarks and Conclusions
  7. Variational Expressions via Hölder's inequality

Key Result

Lemma 1

Let $\rho$ and $\sigma$ be two density operators satisfying $\mathop{\mathrm{supp}}\nolimits{(\rho)} \subseteq \mathop{\mathrm{supp}}\nolimits{(\sigma)}$. For any order $\alpha\in (0,1)\cup (1,+\infty]$, the following identities hold: For $\alpha>1$, the supremum can be relaxed to $0\leq T\leq \mathds{1}$ such that $T$ is not orthogonal to $\rho$.

Theorems & Definitions (11)

  • Lemma 1: Variational expressions FL13BFT17Hia21Mos23
  • Remark 1
  • Remark 2
  • proof
  • proof : Proof of Proposition \ref{['prop:c-q']}
  • Remark 3
  • proof
  • proof
  • Lemma 2
  • proof
  • ...and 1 more