Please use this identifier to cite or link to this item: http://hdl.handle.net/10397/9584
Title: Linear maps preserving numerical radius of tensor products of matrices
Authors: Fosner, A
Huang, Z
Li, CK
Sze, NS 
Keywords: Complex matrix
Linear preserver
Numerical radius
Numerical range
Tensor product
Issue Date: 2013
Publisher: Academic Press
Journal: Journal of mathematical analysis and applications 
Abstract: Let m, n ≥ 2 be positive integers. Denote by Mm the set of m × 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 φ{symbol} : Mm n → Mm n satisfies w (φ{symbol} (A ⊗ B)) = w (A ⊗ B) for all A ∈ Mm and B ∈ Mn if and only if there is a unitary matrix U ∈ Mm n and a complex unit ξ such that φ{symbol} (A ⊗ B) = ξ U (φ1 (A) ⊗ φ2 (B)) U* for all A ∈ Mm and B ∈ Mn, where φk is the identity map or the transposition map X {mapping} Xt for k = 1, 2, and the maps φ1 and φ2 will be of the same type if m, n ≥ 3. In particular, if m, n ≥ 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.
URI: http://hdl.handle.net/10397/9584
ISSN: 0022-247X
EISSN: 1096-0813
DOI: 10.1016/j.jmaa.2013.05.030
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