# Additivity violation of the regularized Minimum Output Entropy

## TL;DR

This paper demonstrates a violation of additivity for the regularized Minimum Output Entropy in quantum channels, using a non-random, infinite-dimensional construction within the commuting-operator framework, advancing understanding in quantum information theory.

## Contribution

It provides the first known example of additivity violation for the regularized minimum output entropy in the commuting-operator setting, extending beyond the one-shot case.

## Key findings

- Established additivity violation in the regularized case
- Constructed a non-random, infinite-dimensional example
- Developed a variant of the Haagerup inequality for free groups

## Abstract

The problem of additivity of the Minimum Output Entropy is of fundamental importance in Quantum Information Theory (QIT). It was solved by Hastings in the one-shot case, by exhibiting a pair of random quantum channels. However, the initial motivation was arguably to understand regularized quantities and there was so far no way to solve additivity questions in the regularized case. The purpose of this paper is to give a solution to this problem. Specifically, we exhibit a pair of quantum channels which unearths additivity violation of the regularized minimum output entropy. Unlike previously known results in the one-shot case, our construction is non-random, infinite dimensional and in the commuting-operator setup. The commuting-operator setup is equivalent to the tensor-product setup in the finite dimensional case for this problem, but their difference in infinite dimensional setting has attracted substantial attention and legitimacy recently in QIT with the celebrated resolutions of Tsirelson's and Connes embedding problem. Likewise, it is not clear that our approach works in the finite dimensional setup. Our strategy of proof relies on developing a variant of the Haagerup inequality optimized for a product of free groups.