Completely Bounded Qusi-Norms, Their Mutiplicativity, and New Additivity Results of Quantum Channels

TL;DR

This paper introduces new mathematical tools called completely bounded quasi-norms to prove additive properties of quantum channel information measures, extending previous results and providing insights into quantum information theory.

Contribution

It defines and proves the multiplicativity of completely bounded quasi-norms for quantum channels, leading to new additivity results for quantum channel information measures.

Findings

Proved additivity of the channel Re9nyi information for b5b1[bdb7,1)

Established multiplicativity of completely bounded quasi-norms for quantum channels

Demonstrated additivity of the channel dispersion related to quantum information tasks.

Abstract

We obtain two new additivity results of quantum channels. The first one is the additivity of the channel R\'enyi information associated with the sandwiched R\'enyi divergence of order $\alpha\in[\frac{1}{2},1)$. To prove this, we introduce the completely bounded $1\to\alpha$ quasi-norms for completely positive maps, with $\alpha\in[\frac{1}{2},1)$, and show that it is multiplicative. The additivity/multiplicativity derived here extends and complements the results of Devetak {\it et al} (Commun Math Phys 266:37-63, 2006) and Gupta and Wilde (Commun Math Phys 334:867-887, 2015), which deal with the case $\alpha>1$. The second one is the additivity of the channel dispersion, which is a quantity related to the second-order behavior of quantum information tasks.