# Multiplicative Properties of Quantum Channels

## TL;DR

This paper investigates the multiplicative behavior of quantum channels, revealing their spectral and structural properties, and introduces the multiplicative index to measure their stability, with implications for quantum information theory.

## Contribution

It presents a structure theorem for unital quantum channels, analyzes classes of channels via stabilizing subalgebras, and introduces the multiplicative index as a new measure of channel stability.

## Key findings

- Multiplicative domain of unital channels relates to spectral properties.
- Maps with trivial multiplicative domains are dense among unital channels.
- Applications in quantum information theory are discussed.

## Abstract

In this paper, we study the multiplicative behaviour of quantum channels, mathematically described by trace preserving, completely positive maps on matrix algebras. It turns out that the multiplicative domain of a unital quantum channel has a close connection to its spectral properties. A structure theorem, which reveals the automorphic property of an arbitrary unital quantum channel on a subalgebra, is presented. Various classes of quantum channels (irreducible, primitive etc.) are then analysed in terms of this stabilising subalgebra. The notion of the multiplicative index of a unital quantum channel is introduced, which measures the number of times a unital channel needs to be composed with itself for the multiplicative algebra to stabilise. We show that the maps that have trivial multiplicative domains are dense in completely bounded norm topology in the set of all unital completely positive maps. Some applications in quantum information theory are discussed.