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An intuitive proof of the data processing inequality

Normand J. Beaudry, Renato Renner

TL;DR

The paper presents an intuitive, self-contained proof of the data processing inequality (DPI) for quantum systems by first proving DPI for the smooth min-entropy in a one-shot setting. It then derives the DPI for the von Neumann entropy by establishing a new, shorter quantum asymptotic equipartition property (QAEP) and applying the i.i.d. limit. This decomposition clarifies the DPI as a fundamental property of smooth entropies plus a QAEP step, with potential benefits for teaching and applying DPI in quantum information tasks. The work also consolidates essential smooth-entropy properties and bounds that support the proofs and their broader use in one-shot information theory.

Abstract

The data processing inequality (DPI) is a fundamental feature of information theory. Informally it states that you cannot increase the information content of a quantum system by acting on it with a local physical operation. When the smooth min-entropy is used as the relevant information measure, then the DPI follows immediately from the definition of the entropy. The DPI for the von Neumann entropy is then obtained by specializing the DPI for the smooth min-entropy by using the quantum asymptotic equipartition property (QAEP). We provide a new, simplified proof of the QAEP and therefore obtain a self-contained proof of the DPI for the von Neumann entropy.

An intuitive proof of the data processing inequality

TL;DR

The paper presents an intuitive, self-contained proof of the data processing inequality (DPI) for quantum systems by first proving DPI for the smooth min-entropy in a one-shot setting. It then derives the DPI for the von Neumann entropy by establishing a new, shorter quantum asymptotic equipartition property (QAEP) and applying the i.i.d. limit. This decomposition clarifies the DPI as a fundamental property of smooth entropies plus a QAEP step, with potential benefits for teaching and applying DPI in quantum information tasks. The work also consolidates essential smooth-entropy properties and bounds that support the proofs and their broader use in one-shot information theory.

Abstract

The data processing inequality (DPI) is a fundamental feature of information theory. Informally it states that you cannot increase the information content of a quantum system by acting on it with a local physical operation. When the smooth min-entropy is used as the relevant information measure, then the DPI follows immediately from the definition of the entropy. The DPI for the von Neumann entropy is then obtained by specializing the DPI for the smooth min-entropy by using the quantum asymptotic equipartition property (QAEP). We provide a new, simplified proof of the QAEP and therefore obtain a self-contained proof of the DPI for the von Neumann entropy.

Paper Structure

This paper contains 7 sections, 12 theorems, 40 equations, 1 table.

Table of Contents

  1. Introduction
  2. Smooth Min-Entropy
  3. Data processing inequality
  4. General Data Processing Inequality
  5. Specialized Data Processing Inequality
  6. Quantum Asymptotic Equipartition Property
  7. General Properties of Smooth Entropies

Key Result

Lemma 4.1

Let $\rho\in S_{=}(\mathcal{H}_{AB})$. Then

Theorems & Definitions (12)

  • Lemma 4.1: Lower bound on the conditional smooth min-entropy
  • Lemma 4.2: Upper bound on the conditional smooth min-entropy
  • Lemma 5.1: Chain rule
  • Lemma 5.2: Lower bound on the smooth min-entropy
  • Lemma 5.3: Upper bound on the $0^{th}$ order Rényi entropy
  • Lemma 5.4: Non-conditional QAEP for smooth min-entropy
  • Lemma 5.5: Non-conditional QAEP for $0^{th}$-order Rényi entropy
  • Lemma 5.6: Relation of conditional von Neumann and conditional smooth min-entropy
  • Lemma A.1: tomamichel09 Purified distance under CP trace non-increasing maps
  • Lemma A.2: berta10 Purified distance relation
  • ...and 2 more