Linear maps preserving numerical radius of tensor products of matrices

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Title:	Linear maps preserving numerical radius of tensor products of matrices
Authors:	ID Fošner, Ajda (Author) ID Huang, Zejun (Author) ID Li, Chi-Kwong (Author) ID Sze, Nung-Sing (Author)
Files:	http://dx.doi.org/10.1016/j.jmaa.2013.05.030
Language:	English
Work type:	Not categorized
Typology:	1.01 - Original Scientific Article
Organization:	IAM - Andrej Marušič Institute
Abstract:	V članku so karakterizirane linearne preslikave na tenzorskem produktu kompleksnih matrik, ki ohranjajo numerični radij.
Keywords:	matematika, teorija matrik, kompleksne matrike, linearni ohranjevalci, numerični rang, numerični radij, tenzorski produkt
Year of publishing:	2013
Number of pages:	str. 183-189
Numbering:	Vol. 407, iss. 2
PID:	20.500.12556/RUP-133
ISSN:	0022-247X
UDC:	512.643
COBISS.SI-ID:	16648025
Publication date in RUP:	15.10.2013
Views:	5471
Downloads:	103
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Language:	Unknown
Abstract:	Let ▫$m,n ge 2$▫ be positive integers. Denote by ▫$M_m$▫ the set of ▫$m times m$▫ complex matrices and by ▫$w(X)$▫ the numerical radius of a square matrix ▫$X$▫. Motivated by the study of operations on bipartite systems of quantum states, we show that a linear map ▫$phi colon M_{mn} to M_{mn}$▫ satisfies ▫$$w(phi(Aotimes B)) = w(A otimes B)quad text{for all } A in M_m text{ and } B in M_n$$▫ if and only if there is a unitary matrix ▫$U in M_{mn}$▫ and a complex unit ▫$xi$▫ such that ▫$$phi(A otimes B) = xi U(varphi_1(A) otimes varphi_2(B))U^ast quad text{for all } A in M_m text{ and } B in M_n$$▫ where ▫$varphi_k$▫ is the identity map or the transposition map ▫$X mapsto X_t$▫ for ▫$k = 1,2$▫, and the maps ▫$varphi_1$▫ and ▫$varphi_2$▫ will be of the same type if ▫$m,n ge 3$▫. In particular, if ▫$m,n ge 3$▫, the map corresponds to an evolution of a closed quantum system (under a fixed unitary operator), possibly followed by a transposition. The results are extended to multipartite systems.
Keywords:	mathematics, matrix theory, complex matrices, linear preservers, numerical range, numerical radius, tensor product

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