R\'enyi-Holevo inequality from $\alpha$-$z$-R\'enyi relative entropies

TL;DR

This paper introduces a new quantum information inequality based on $oldsymbol{ ext{α-}z}$-Rényi relative entropies, extending classical bounds to quantum communication channels and providing insights into their fundamental limits.

Contribution

It establishes the Holevo-Rényi inequality using $oldsymbol{ ext{α-}z}$-Rényi relative entropies, unifying known quantities and advancing quantum information bounds.

Findings

Derived a quantum bound for the $oldsymbol{ ext{α}}$-mutual information.

Provided new insights into the limits of quantum communication channels.

Unified various quantum relative entropies within a single inequality.

Abstract

We investigate bounds in the transmission of classical information through quantum systems. Our focus lies in the generalized Holevo theorem, which provides a single-letter Holevo-like inequality from arbitrary quantum distance measures. Through the introduction of the $\alpha$-$z$-R\'enyi relative entropies, which comprise known relevant quantities such as the R\'enyi relative entropy and the sandwiched R\'enyi relative entropy, we establish the Holevo-R\'enyi inequality. This result leads to a quantum bound for the $\alpha$-mutual information, suggesting new insights into communication channel performance and the fundamental limits for reliability functions in memoryless multi-letter communication channels.