Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Some properties and applications of the new quantum $f$-divergences

Salman Beigi, Christoph Hirche, Marco Tomamichel

TL;DR

The paper studies the new quantum f-divergences defined via an integral representation, focusing on deriving alternative trace-like expressions, monotonicity, and convexity properties to unlock their full potential. It develops trace representations for H_α and D_α, proves inequalities comparing the new Renyi divergences to the Petz and sandwiched variants, and uses these tools to provide an alternative achievability proof for the quantum Chernoff bound. Additional integral representations yield a suite of inequalities, including reverse Pinsker-type bounds and χ^2-based controls, and a Taylor expansion framework clarifies higher-order corrections. Overall, the work broadens the analytical toolkit for quantum divergences, enabling tighter bounds in hypothesis testing, state discrimination, and information-processing tasks while clarifying the relationships among various quantum divergence families.

Abstract

Recently, a new definition for quantum $f$-divergences was introduced based on an integral representation. These divergences have shown remarkable properties, for example when investigating contraction coefficients under noisy channels. At the same time, many properties well known for other definitions have remained elusive for the new quantum $f$-divergence because of its unusual representation. In this work, we investigate alternative ways of expressing these quantum $f$-divergences. We leverage these expressions to prove new properties of these $f$-divergences and demonstrate some applications. In particular, we give a new proof of the achievability of the quantum Chernoff bound by establishing a strengthening of an inequality by Audenaert et al. We also establish inequalities between some previously known Renyi divergences and the new Renyi divergence. We further investigate some monotonicity and convexity properties of the new $f$-divergences, and prove inequalities between these divergences for various functions.

Some properties and applications of the new quantum $f$-divergences

TL;DR

The paper studies the new quantum f-divergences defined via an integral representation, focusing on deriving alternative trace-like expressions, monotonicity, and convexity properties to unlock their full potential. It develops trace representations for H_α and D_α, proves inequalities comparing the new Renyi divergences to the Petz and sandwiched variants, and uses these tools to provide an alternative achievability proof for the quantum Chernoff bound. Additional integral representations yield a suite of inequalities, including reverse Pinsker-type bounds and χ^2-based controls, and a Taylor expansion framework clarifies higher-order corrections. Overall, the work broadens the analytical toolkit for quantum divergences, enabling tighter bounds in hypothesis testing, state discrimination, and information-processing tasks while clarifying the relationships among various quantum divergence families.

Abstract

Recently, a new definition for quantum -divergences was introduced based on an integral representation. These divergences have shown remarkable properties, for example when investigating contraction coefficients under noisy channels. At the same time, many properties well known for other definitions have remained elusive for the new quantum -divergence because of its unusual representation. In this work, we investigate alternative ways of expressing these quantum -divergences. We leverage these expressions to prove new properties of these -divergences and demonstrate some applications. In particular, we give a new proof of the achievability of the quantum Chernoff bound by establishing a strengthening of an inequality by Audenaert et al. We also establish inequalities between some previously known Renyi divergences and the new Renyi divergence. We further investigate some monotonicity and convexity properties of the new -divergences, and prove inequalities between these divergences for various functions.
Paper Structure (20 sections, 31 theorems, 190 equations, 1 figure)

This paper contains 20 sections, 31 theorems, 190 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Definitions and preliminary results
  3. Notation
  4. $f$-divergences
  5. Divergences and their derivatives
  6. Inequalities relating the new and old Rényi divergences and their applications
  7. Towards trace representations for the new $f$-divergences
  8. Hellinger divergence for integer $\alpha\geq 2$
  9. Hellinger divergence for $0<\alpha<1$
  10. Other divergences
  11. Inequalities between Rényi divergences
  12. Quantum Chernoff bound
  13. Monotonicity and convexity properties
  14. Ordering of $f$-divergences
  15. Convexity and monotonicity of the Hellinger divergences
  16. ...and 5 more sections

Key Result

Theorem 2.1

Let $f \in {\mathcal{F}}$ and $\rho \ll\gg \sigma$. We have where $\chi^2(\rho\|\sigma) = H_2(\rho\|\sigma)$ is the Hellinger divergence of order $2$.

Figures (1)

  • Figure 1: Plot of the different Rényi divergences (left) and Hellinger divergences (right) defined in the main text over $\alpha$ for the states specified above. For this and all other examples we tried, Conjecture \ref{['Conj:Ha-2int']} appears to hold.

Theorems & Definitions (61)

  • Theorem 2.1: hirche2023quantum
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • ...and 51 more