Linear maps preserving numerical radius of tensor products of matrices

Ajda Fošner, Zejun Huang, Chi Kwong Li, Nung Sing Sze

Research output: Journal article publicationJournal articleAcademic researchpeer-review

5 Citations (Scopus)

Abstract

Let m, n≥2 be positive integers. Denote by Mmthe 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:Mmn→Mmnsatisfies w(φ(A⊗B))=w(A⊗B)forall A∈Mm and B∈Mn if and only if there is a unitary matrix U∈Mmnand a complex unit ξ such that φ(A⊗B)=ξU(φ1(A)⊗φ2(B))U*forall A∈Mm and B∈Mn, where φkis the identity map or the transposition map X{mapping}Xtfor k=1, 2, and the maps φ1and φ2will 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.
Original languageEnglish
Pages (from-to)183-189
Number of pages7
JournalJournal of Mathematical Analysis and Applications
Volume407
Issue number2
DOIs
Publication statusPublished - 15 Nov 2013

Keywords

  • Complex matrix
  • Linear preserver
  • Numerical radius
  • Numerical range
  • Tensor product

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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