A relation between completely bounded norms and conjugate channels

TL;DR

This paper establishes a fundamental relation between a quantum channel and its conjugate, linking their Schatten and completely bounded norms, and uses this to prove norm multiplicativity.

Contribution

It introduces a novel relation between quantum channels and their conjugates, providing an alternative proof for the multiplicativity of specific norms.

Findings

The $p o p$ Schatten norm equals the $1 o p$ completely bounded norm of the conjugate channel.

The relation leads to an alternative proof of norm multiplicativity.

The results deepen understanding of quantum channel norms and their properties.

Abstract

We show a relation between a quantum channel $\Phi$ and its conjugate $\Phi^C$, which implies that the $p\to p$ Schatten norm of the channel is the same as the $1\to p$ completely bounded norm of the conjugate. This relation is used to give an alternative proof of the multiplicativity of both norms.