# Simplifying additivity problems using direct sum constructions

## TL;DR

This paper introduces a method using direct sum constructions to simplify and generalize additivity problems in quantum information theory, showing equivalences and implications for various capacities and entanglement measures.

## Contribution

It demonstrates that additivity for arbitrary quantum channels reduces to unital channels and introduces a general tool for deriving additivity results via direct sums.

## Key findings

- Additivity for arbitrary channels is equivalent to unital channels.
- Weak additivity implies strong additivity for convex entanglement monotones.
- Direct sum constructions enable deriving information quantities from summands.

## Abstract

We study the additivity problems for the classical capacity of quantum channels, the minimal output entropy and its convex closure. We show for each of them that additivity for arbitrary pairs of channels holds iff it holds for arbitrary equal pairs, which in turn can be taken to be unital. In a similar sense, weak additivity is shown to imply strong additivity for any convex entanglement monotone. The implications are obtained by considering direct sums of channels (or states) for which we show how to obtain several information theoretic quantities from their values on the summands. This provides a simple and general tool for lifting additivity results.