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

Ke Li, Quanhua Xu

Abstract

We obtain two new additivity results of quantum channels. The first one is the additivity of the channel Rényi information associated with the sandwiched Rényi divergence of order $α\in[\frac{1}{2},1)$. To prove this, we introduce the completely bounded $1\toα$ quasi-norms for completely positive maps, with $α\in[\frac{1}{2},1)$, 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 $α>1$. The second one is the additivity of the channel dispersion, which is a quantity related to the second-order behavior of quantum information tasks.

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

Abstract

We obtain two new additivity results of quantum channels. The first one is the additivity of the channel Rényi information associated with the sandwiched Rényi divergence of order . To prove this, we introduce the completely bounded quasi-norms for completely positive maps, with , 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 . The second one is the additivity of the channel dispersion, which is a quantity related to the second-order behavior of quantum information tasks.
Paper Structure (6 sections, 10 theorems, 60 equations)

This paper contains 6 sections, 10 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Multiplicativity of Completely Bounded Quasi-norms
  3. Additivity of Quantum Rényi Information
  4. Additivity of the Channel Dispersions
  5. Outlook
  6. Auxiliary Results

Key Result

Theorem 1

Let $\mathcal{M}_1:\mathcal{L}(A_1')\rightarrow\mathcal{L}(B_1)$ and $\mathcal{M}_2:\mathcal{L}(A_2')\rightarrow\mathcal{L}(B_2)$ be completely positive maps. For any $\alpha\in[\frac{1}{2}, 1)$, we have

Theorems & Definitions (20)

  • Theorem 1
  • proof
  • Lemma 2
  • proof
  • Definition 3: MDSFT2013onWWY2014strong
  • Remark 1
  • Theorem 4
  • proof
  • Definition 5
  • Remark 2
  • ...and 10 more