Linear maps preserving numerical radius of tensor products of matrices

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Language:	English
Year of publishing:	2013
Typology:	1.01 - Original Scientific Article
Organization:	UP - University of Primorska
UDC:	512.643
COBISS:	16648025
ISSN:	0022-247X
Views:	4393
Downloads:	102
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Secondary language:	Unknown
Secondary title:	Linearni ohranjevalci numeričnega radija na tenzorskem produktu matrik
Secondary 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 ▫$$w(\phi(A\otimes B)) = w(A \otimes B)\quad \text{for all } A \in M_m \text{ and } B \in M_n$$▫ 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.
Secondary keywords:	mathematics;matrix theory;complex matrices;linear preservers;numerical range;numerical radius;tensor product;
Type (COBISS):	Not categorized
Pages:	str. 183-189
Volume:	ǂVol. ǂ407
Issue:	ǂiss. ǂ2
Chronology:	2013
ID:	1476880