Joint numerical ranges, quantum maps, and joint numerical shadows
type: Item
instance of: scholarly article
title: Joint numerical ranges, quantum maps, and joint numerical shadows
zbMATH Open document ID: 1266.15034
author: Eugene Gutkin
author: Karol Życzkowski
published in: Linear Algebra and its Applications
publication date: 2013-02-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1207.1227
review text: The present work is concerned with the joint numerical range (JNR), numerical range, quantum maps and their applications in quantum mechanics, especially in the theory of quantum information. Theorem~1 shows that the JNR of an \(m\)-tuple of Hermitian operators is a linear projection of the set of pure quantum states to \(\mathbb R^m\). With any \(m\)-tuple of Hermitian operators, the authors associate a probability measure on \(\mathbb R^m\). This is the joint numerical shadow of the \(m\)-tuple of Hermitian operators; it extends the concept of numerical shadow of a complex operator. Several properties of joint numerical shadows are put forward. The effects of the quantum map \(\Phi\) on \(m\)-tuples of Hermitian operators are examined. Applications of the authors' material in the theory of quantum information are illustrated by examples.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 47A12
Mathematics Subject Classification ID: 81P16
Mathematics Subject Classification ID: 51M15
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 81P45
zbMATH DE Number: 6138423
zbMATH Keywords: joint numerical range
zbMATH Keywords: joint numerical shadow
zbMATH Keywords: affine equivalence
zbMATH Keywords: linear projection
zbMATH Keywords: quantum state
zbMATH Keywords: quantum map
zbMATH Keywords: qubit
zbMATH Keywords: decaying channel
zbMATH Keywords: qutrit
zbMATH Keywords: double flip channel
zbMATH Keywords: quantum information
zbMATH Keywords: Hermitian operators
reviewed by: A. R. Alimov
MaRDI profile type: Publication
Wikidata QID: Q112266920
OpenAlex ID: W2963421010
arXiv ID: 1207.1227
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DOI: 10.1016/J.LAA.2012.10.043
description: scientific article