2012/08/06 by Ajda Fošner, Zejun Huang, Fosner, Ajda +5 · 1 citation
Computer Science · Mathematics · #15A18 #15A69 #15A86 #15B57 #Advanced Topics in Algebra #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Graph theory and applications #Matrix Theory and Algorithms #Quantum Physics (quant-ph)
paper · pdf · doi:10.48550/arxiv.1208.1076
openalex publication_date 2012/08/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let m,n≥ 2 be positive integers, Mm the set of m× m complex matrices and Mn the set of n× n complex matrices. Regard Mmn as the tensor space Mm⊗ Mn. Suppose |⋅| is the Ky Fan k-norm with 1 ≤ k ≤ mn, or the Schatten p-norm with 1 ≤ p ≤ ∞ (p≠ 2) on Mmn. It is shown that a linear map ϕ: Mmn → Mmn satisfying |A⊗ B| = |ϕ(A⊗ B)| for all A ∈ Mm and B ∈ Mn if and only if there are unitary U, V ∈ Mmn such that ϕ has the form A⊗ B ↦ U(φ1(A) ⊗ φ2(B))V, where φi(X) is either the identity map X ↦ X or the transposition map X ↦ Xt. The results are extended to tensor space Mn1 ⊗ ... ⊗ Mnm of higher level. The connection of the problem to quantum information science is mentioned.