[Paper Review] Inequalities for the quantum Renyi divergences with applications to compound coding problems

Milán Mosonyi|arXiv (Cornell University)|Oct 28, 2013

Quantum Information and Cryptography32 references3 citations

TL;DR

This paper establishes two-sided bounds between conventional and recently introduced quantum Rényi divergences, demonstrating their interchangeability near α=1. This enables simpler, more efficient proofs of key quantum information theorems, including quantum Stein’s lemma with composite null hypotheses, universal source compression, and the classical capacity of compound quantum channels, leveraging a newly proven weak quasi-concavity property of the new divergences.

ABSTRACT

We show two-sided bounds between the conventional quantum R\'enyi divergences and the new notion of R\'enyi divergences introduced recently in M\uller-Lennert, Dupuis, Szehr, Fehr and Tomamichel, J. Math. Phys. 54, 122203, (2013), and Wilde, Winter, Yang, arXiv:1306.1586. The bounds imply that the two versions can be used interchangeably near alpha=1, and hence one can benefit from the best properties of both when proving coding theorems in the case of asymptotically vanishing error. We illustrate this by giving short and simple proofs of the quantum Stein's lemma with composite null-hypothesis, universal source compression, and the achievability part of the classical capacity of compound quantum channels. Apart from the above interchangeability, we benefit from a weak quasi-concavity property of the new Renyi divergences that we also establish here.

Motivation & Objective

• To establish quantitative relationships between conventional quantum Rényi divergences and the newer Rényi divergences introduced by Müller-Lennert et al. and Wilde et al.
• To demonstrate that the two divergence definitions can be used interchangeably in the asymptotic regime with vanishing error, particularly near α=1.
• To leverage this interchangeability to provide concise and transparent proofs for fundamental coding theorems in quantum information theory.
• To establish a weak quasi-concavity property of the new Rényi divergences, which supports their analytical use in coding theorems.
• To unify and streamline existing proofs for quantum Stein’s lemma with composite null hypotheses, universal source compression, and classical capacity of compound quantum channels.

Proposed method

• Derives two-sided bounds between the conventional quantum Rényi divergence and the newer version introduced in Müller-Lennert et al. and Wilde et al., focusing on the behavior near α=1.
• Uses the bounds to show that both divergence definitions yield equivalent asymptotic results in the limit of vanishing error, enabling interchangeable use in coding theorems.
• Applies the interchangeability to re-derive known results—quantum Stein’s lemma with composite null hypothesis, universal source compression, and classical capacity of compound channels—using simpler, more direct arguments.
• Establishes a weak quasi-concavity property for the new Rényi divergences, which supports their use in variational formulations and optimization problems common in coding theorems.
• Employs standard tools from quantum information theory, including the quantum relative entropy and the sandwiched Rényi divergence, within a unified framework.

Experimental results

Research questions

• RQ1 How do the conventional and recently introduced quantum Rényi divergences relate to each other in the neighborhood of α=1?
• RQ2 Can the two versions of Rényi divergences be used interchangeably in asymptotic coding theorems with vanishing error?
• RQ3 What analytical advantages does the new Rényi divergence offer in proving quantum coding theorems?
• RQ4 Does the new Rényi divergence satisfy useful convexity or concavity properties that facilitate proof simplification?
• RQ5 Can the interchangeability of divergences streamline proofs of fundamental results like quantum Stein’s lemma with composite hypotheses or the classical capacity of compound channels?

Key findings

• Two-sided bounds are established between conventional and new quantum Rényi divergences, proving their asymptotic equivalence near α=1.
• The two divergence definitions can be used interchangeably in the asymptotic regime with vanishing error, enabling the use of the most analytically convenient version in different parts of a proof.
• The new Rényi divergences satisfy a weak quasi-concavity property, which supports their use in variational characterizations and optimization problems.
• Short and conceptually simple proofs are provided for quantum Stein’s lemma with a composite null hypothesis, universal source compression, and the classical capacity of compound quantum channels.
• The results demonstrate that the new Rényi divergences are not only mathematically well-behaved but also practically advantageous for deriving coding theorems in quantum information theory.