# On the quantum Renyi relative entropies and related capacity formulas

## TL;DR

This paper provides an operational interpretation of quantum lpha-relative entropies as generalized cutoff rates and explores their implications for quantum channel capacities, unifying various capacity measures for lpha within a specific parameter range.

## Contribution

It introduces a new operational interpretation of quantum lpha-relative entropies and demonstrates the equivalence of various generalized Holevo capacities for lpha, along with bounds on one-shot capacities.

## Key findings

- Quantum lpha-relative entropies are linked to generalized cutoff rates.
- Various lpha-relative entropy-based capacities coincide for lpha.
- An upper bound on one-shot epsilon-capacity is established.

## Abstract

We show that the quantum $\alpha$-relative entropies with parameter $\alpha\in (0,1)$ can be represented as generalized cutoff rates in the sense of [I. Csiszar, IEEE Trans. Inf. Theory 41, 26-34, (1995)], which provides a direct operational interpretation to the quantum $\alpha$-relative entropies. We also show that various generalizations of the Holevo capacity, defined in terms of the $\alpha$-relative entropies, coincide for the parameter range $\alpha\in (0,2]$, and show an upper bound on the one-shot epsilon-capacity of a classical-quantum channel in terms of these capacities.