[Paper Review] Divergence radii and the strong converse exponent of classical-quantum channel coding with constant compositions

Milán Mosonyi, Tomohiro Ogawa|arXiv (Cornell University)|Nov 26, 2018

Wireless Communication Security Techniques57 references5 citations

TL;DR

This paper establishes the strong converse exponent for classical-quantum channel coding with constant composition codes using sandwiched Rényi divergences. It proves that the exponent is given by a supremum over Rényi divergence radii, extending Csiszár's classical result to the quantum setting with operational interpretation as a generalized cutoff rate.

ABSTRACT

There are different inequivalent ways to define the Rényi capacity of a channel for a fixed input distribution $P$. In a 1995 paper Csiszár has shown that for classical discrete memoryless channels there is a distinguished such quantity that has an operational interpretation as a generalized cutoff rate for constant composition channel coding. We show that the analogous notion of Rényi capacity, defined in terms of the sandwiched quantum Rényi divergences, has the same operational interpretation in the strong converse problem of classical-quantum channel coding. Denoting the constant composition strong converse exponent for a memoryless classical-quantum channel $W$ with composition $P$ and rate $R$ as $sc(W,R,P)$, our main result is that \[ sc(W,R,P)=\sup_{α>1}\frac{α-1}α\left[R-χ_α^*(W,P) ight], \] where $χ_α^*(W,P)$ is the $P$-weighted sandwiched Rényi divergence radius of the image of the channel.

Motivation & Objective

• To extend Csiszár's classical result on Rényi capacity and generalized cutoff rates to the quantum domain.
• To characterize the strong converse exponent for classical-quantum channels under constant composition coding.
• To show that the sandwiched Rényi divergence radius plays the same operational role in the quantum case as the generalized cutoff rate in classical channels.
• To provide a computable expression for the strong converse exponent in terms of Rényi mutual information and divergence radii.

Proposed method

• Uses the sandwiched Rényi divergence $ D_{ar{\alpha}}^{*} $ to define the $ P $-weighted Rényi divergence radius $ \chi_{\alpha}^{*}(W,P) $ for classical-quantum channels.
• Applies the duality between Rényi mutual information and divergence radius via the infimum over output states $ \sigma $.
• Employs large deviation analysis and variational techniques to derive the strong converse exponent.
• Relies on the duality between the divergence radius and the Rényi mutual information to express the exponent in terms of suprema over $ \alpha > 1 $.
• Uses the limit of trace norms of positive parts of operator differences to analyze error behavior.
• Applies Lemma D.1 and Corollary D.2 to relate the asymptotic behavior of trace norms to Rényi divergences and their derivatives at $ \alpha = 1 $.

Experimental results

Research questions

• RQ1 What is the strong converse exponent for classical-quantum channels under constant composition coding?
• RQ2 Does the sandwiched Rényi divergence radius play the same operational role as the generalized cutoff rate in classical channels?
• RQ3 Can the strong converse exponent be expressed in terms of Rényi mutual information and divergence radii in the quantum setting?
• RQ4 How does the Rényi divergence radius relate to the error exponent in the strong converse regime?
• RQ5 Is there a quantum analog of Csiszár’s classical result on the generalized cutoff rate for constant composition codes?

Key findings

• The strong converse exponent for classical-quantum channels with constant composition is given by $ \mathrm{sc}(W,R,P) = \sup_{\alpha>1} \frac{\alpha-1}{\alpha} \left[ R - \chi_{\alpha}^{*}(W,P) \right] $, where $ \chi_{\alpha}^{*}(W,P) $ is the $ P $-weighted sandwiched Rényi divergence radius.
• The Rényi divergence radius $ \chi_{\alpha}^{*}(W,P) $ is defined as the infimum over output states $ \sigma $ of the $ P $-weighted sum of sandwiched Rényi divergences.
• The strong converse exponent vanishes for rates below the Holevo capacity and diverges exponentially for rates above it, consistent with the quantum asymptotic theory.
• The derivative of the cumulant generating function $ \psi(\alpha) $ at $ \alpha = 1 $ yields the relative entropy $ \sum_x P(x) D(V(x)\|W(x)) $, linking the exponent to classical divergence.
• The result confirms that the sandwiched Rényi divergence radius serves as a quantum analog of the classical generalized cutoff rate.
• The proof relies on trace norm asymptotics and duality between Rényi mutual information and divergence radius, generalizing classical large deviation techniques to the quantum setting.