A note on the \(p \rightarrow q\) norms of 2-positive maps
type: Item
instance of: scholarly article
title: A note on the \(p \rightarrow q\) norms of 2-positive maps
zbMATH Open document ID: 1163.47015
author: Koenraad M. R. Audenaert
published in: Linear Algebra and its Applications
publication date: 2009-02-10T00:00:00Z
review text: This paper is concerned with a mathematical question raised by \textit{C.\,King} and \textit{M.\,B.\thinspace Ruskai} [Quantum Inf.\ Comput.\ 4, No.\,6--7, 500--512 (2004; Zbl 1162.47307)], namely: Is the \(p \to q\) norm of a completely positive map \(\Phi\) acting between Schatten \(p\) and \(q\) classes of Hermitian operators, \(\|\Phi\|_{p \to q}=\sup_{A=A^*}\frac{\|\Phi (A)\|_q}{\|A\|_p}\), equal to the \(p \to q\) norm of the map when acting between Schatten classes of general, not necessarily Hermitian, operators? A general proof has been given by \textit{J.\,Watrous} [Quantum Inf.\ Comput.\ 5, No.\,1, 57--67 (2005; Zbl 1162.47308)] for any completely positive map and for all values \(p,q\geq 1\). In this paper, the author presents an alternative proof, which shows that the statement holds more generally for all 2-positive maps.
reviewed by: Miyeon Kwon
Mathematics Subject Classification ID: 47B10
Mathematics Subject Classification ID: 47L90
Mathematics Subject Classification ID: 81Q10
Mathematics Subject Classification ID: 81P10
Mathematics Subject Classification ID: 82B10
zbMATH DE Number: 5505640
zbMATH Keywords: 2-positive map
zbMATH Keywords: Schatten norm
zbMATH Keywords: maximal output purity
zbMATH Keywords: quantum channel
MaRDI profile type: Publication
full work available at URL: j.laa.2008.09.040
OpenAlex ID: W2058858559
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cites work: States, effects, and operations. Fundamental notions of quantum theory. Lectures in mathematical physics at the University of Texas at Austin. Ed. by A. Böhm, J. D. Dollard and W. H. Wootters
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DOI: 10.1016/J.LAA.2008.09.040
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description: scientific article; zbMATH DE number 5505640