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arXiv:quant-ph/0601071

A relation between completely bounded norms and conjugate channels

Anna Jenčová

Published Jan 11, 2006• 3 citations
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Abstract

We show a relation between a quantum channel $Φ$ and its conjugate $Φ^C$, 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.

DOI
10.48550/arxiv.quant-ph/0601071

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We show a relation between a quantum channel ΦΦΦ and its conjugate ΦCΦ^CΦC, which implies that the p→pp\to pp→p Schatten norm of the channel is the same as the 1→p1\to p1→p completely bounded norm of the conjugate. This relation is used to give an alternative proof of the multiplicativity of both norms.
King and Ruskai asked whether the p→qp\to qp→q norm of a completely positive map ΦΦΦ, acting between Schatten ppp and qqq classes of self-adjoint operators, ∣∣Φ∣∣p→q=max⁡A=A∗∣∣Φ(A)∣∣q∣∣A∣∣p||Φ||_{p\to q} = \max_{A=A^*} \frac{||Φ(A)||_q}{||A||_p}∣∣Φ∣∣p→q​=maxA=A∗​∣∣A∣∣p​∣∣Φ(A)∣∣q​​, is equal to the p→qp\to qp→q norm of that map when acting between Schatten classes of general, not necessarily self-adjoint, operators. The first proof has been given by Watrous. We give an alternative proof of this statement.
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∈[12,1)α\in[\frac{1}{2},1)α∈[21​,1). To prove this, we introduce the completely bounded 1→α1\toα1→α quasi-norms for completely positive maps, with α∈[12,1)α\in[\frac{1}{2},1)α∈[21​,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>1α>1α>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.
Let Φ\PhiΦ be a super-operator, i.e., a linear mapping of the form Φ:L(F)→L(G)\Phi:\mathrm{L}(\mathcal{F})\rightarrow\mathrm{L}(\mathcal{G})Φ:L(F)→L(G) for finite dimensional Hilbert spaces F\mathcal{F}F and G\mathcal{G}G. This paper considers basic properties of the super-operator norms defined by ∥Φ∥q→p=sup⁡{∥Φ(X)∥p/∥X∥q : X≠0}\|\Phi\|_{q\rightarrow p}= \sup\{\|\Phi(X)\|_p/\|X\|_q\,:\,X\not=0\}∥Φ∥q→p​=sup{∥Φ(X)∥p​/∥X∥q​:X=0}, induced by Schatten norms for 1≤p,q≤∞1\leq p,q\leq\infty1≤p,q≤∞. These super-operator norms arise in various contexts in the study of quantum information. In this paper it is proved that if Φ\PhiΦ is completely positive, the value of the supremum in the definition of ∥Φ∥q→p\|\Phi\|_{q\rightarrow p}∥Φ∥q→p​ is achieved by a positive semidefinite operator XXX, answering a question recently posed by King and Ruskai~\cite{KingR04}. However, for any choice of p∈[1,∞]p\in [1,\infty]p∈[1,∞], there exists a super-operator Φ\PhiΦ that is the {\em difference} of two completely positive, trace-preserving super-operators such that all Hermitian XXX fail to achieve the supremum in the definition of ∥Φ∥1→p\|\Phi\|_{1\rightarrow p}∥Φ∥1→p​. Also considered are the properties of the above norms for super-operators tensored with the identity super-operator. In particular, it is proved that for all p≥2p\geq 2p≥2, q≤2q\leq 2q≤2, and arbitrary Φ\PhiΦ, the norm ∥Φ∥q→p\|\Phi \|_{q\rightarrow p}∥Φ∥q→p​ is stable under tensoring Φ\PhiΦ with the identity super-operator, meaning that ∥Φ∥q→p=∥Φ⊗I∥q→p\|\Phi \|_{q\rightarrow p} = \|\Phi \otimes I\|_{q\rightarrow p}∥Φ∥q→p​=∥Φ⊗I∥q→p​. For 1≤p<21\leq p < 21≤p<2, the norm ∥Φ∥1→p\|\Phi\|_{1\rightarrow p}∥Φ∥1→p​ may fail to be stable with respect to tensoring Φ\PhiΦ with the identity super-operator as just described, but ∥Φ⊗I∥1→p\|\Phi\otimes I\|_{1\rightarrow p}∥Φ⊗I∥1→p​ is stable in this sense for III the identity super-operator on L(H)\mathrm{L}(\mathcal{H})L(H) for $\op{dim}(\mathcal{H}) = \op{dim}(\mathcal{F})$. This generalizes and simplifies a proof due to Kitaev \cite{Kitaev97} that established this fact for the case p=1p=1p=1.