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Induced Quantum Divergence: A New Lens on Communication and Source Coding

Gilad Gour

TL;DR

This work introduces the induced divergence, a smoothed, generalized quantum divergence derived from parent relative entropies, designed to replace the hypothesis testing divergence in position-based decoding. It preserves data-processing and monotonicity while smoothly interpolating between parent relative entropy and min-relative entropy, enabling tighter single-shot bounds, especially for the sandwiched Rényi case with $\alpha\in[0,2]$. The induced divergence refines the position-based decoding lemma and extends its applicability to broader state classes, yielding sharper achievability results for classical communication over quantum channels and quantum state redistribution. These results offer a new analytical tool for fundamental single-shot quantum information protocols and highlight potential broader uses in quantum information science and resource theories.

Abstract

This paper introduces the induced divergence, a new quantum divergence measure that replaces the hypothesis testing divergence in position-based decoding, simplifying the analysis of quantum communication and state redistribution while yielding tighter achievability bounds. Derived from a parent quantum relative entropy, it retains key properties such as data processing inequality and Löwner monotonicity. Like the hypothesis testing divergence, it depends on a smoothing parameter and interpolates between the parent relative entropy (as the smoothing parameter approaches one) and the min-relative entropy (as it approaches zero), the latter holding when applied to the sandwiched Rényi relative entropy of order $α\in[0,2]$. This framework refines the position-based decoding lemma, extending its applicability to a broader class of states and improving decoding success probabilities. Two key applications are considered: classical communication over quantum channels, where the induced divergence improves lower bounds on the distillable communication rate, and quantum state redistribution, where it leads to sharper bounds on communication costs. These results provide new insights into fundamental single-shot quantum information protocols and enhance existing analytical techniques.

Induced Quantum Divergence: A New Lens on Communication and Source Coding

TL;DR

This work introduces the induced divergence, a smoothed, generalized quantum divergence derived from parent relative entropies, designed to replace the hypothesis testing divergence in position-based decoding. It preserves data-processing and monotonicity while smoothly interpolating between parent relative entropy and min-relative entropy, enabling tighter single-shot bounds, especially for the sandwiched Rényi case with . The induced divergence refines the position-based decoding lemma and extends its applicability to broader state classes, yielding sharper achievability results for classical communication over quantum channels and quantum state redistribution. These results offer a new analytical tool for fundamental single-shot quantum information protocols and highlight potential broader uses in quantum information science and resource theories.

Abstract

This paper introduces the induced divergence, a new quantum divergence measure that replaces the hypothesis testing divergence in position-based decoding, simplifying the analysis of quantum communication and state redistribution while yielding tighter achievability bounds. Derived from a parent quantum relative entropy, it retains key properties such as data processing inequality and Löwner monotonicity. Like the hypothesis testing divergence, it depends on a smoothing parameter and interpolates between the parent relative entropy (as the smoothing parameter approaches one) and the min-relative entropy (as it approaches zero), the latter holding when applied to the sandwiched Rényi relative entropy of order . This framework refines the position-based decoding lemma, extending its applicability to a broader class of states and improving decoding success probabilities. Two key applications are considered: classical communication over quantum channels, where the induced divergence improves lower bounds on the distillable communication rate, and quantum state redistribution, where it leads to sharper bounds on communication costs. These results provide new insights into fundamental single-shot quantum information protocols and enhance existing analytical techniques.
Paper Structure (23 sections, 15 theorems, 180 equations, 2 figures, 4 tables)

This paper contains 23 sections, 15 theorems, 180 equations, 2 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notations
  4. The Sandwiched Rényi Relative Entropy
  5. The Collision Relative Entropy
  6. The Hypothesis Testing and Information Spectrum Divergence
  7. Rényi Mutual Information
  8. Measured Rényi Relative Entropy
  9. Extensions to the Channel Domain
  10. The Induced Divergence
  11. Relative Entropies with Extended Domain
  12. The Induced Divergence: Definition and Basic Properties
  13. The Induced Rényi Divergence
  14. Enhanced Position-Based Decoding Lemma
  15. Classical Communication over a Quantum Channel
  16. ...and 8 more sections

Key Result

Lemma 1

$\;$

Figures (2)

  • Figure 1: Heuristic description of quantum state redistribution. System $B'$ is a replica of $A'$ on Bob's side.
  • Figure 2: An $(\varepsilon,q)$-eQSR. The LOSE superchannel $\Theta$ make use of the entangled state $\varphi$.

Theorems & Definitions (34)

  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Corollary 1
  • proof
  • Definition 2
  • Lemma 3
  • proof
  • ...and 24 more