Search

Let Ψ : ⨂ i = 1 d H n i → ⨂ i = 1 d H n i be a linear map on the Kronecker product of spaces of Hermitian matrices H n i of size n i ≥ 3. (If d=1, we identify ⨂ i = 1 d H n i with H n 1 .) We establish a condition under which Ψ adj ⨂ i = 1 d A i = adj Ψ ⨂ i = 1 d A i if and only if det Ψ ⨂ i = 1 d A i = det ⨂ i = 1 d A i for all ⨂ i = 1 d A i ∈ ⨂ i = 1 d H n i . Then for d ∈ { 1 , 2 } , we apply this fact to characterize maps Ψ : ⨂ i = 1 d H n i → ⨂ i = 1 d H n i such that Ψ adj ⨂ i = 1 d A i = adj Ψ ⨂ i = 1 d A i ⨂ i = 1 d with some mild conditions. [ABSTRACT FROM AUTHOR]

In this note, we characterize linear maps between spaces of Hermitian matrices over a field with involution which preserve the rank of the identity matrix and the rank of tensor products of rank one Hermitian matrices. It is assumed that the fixed field with respect to the field with involution has at least four elements. We also present some applications of our main result to other preserver problems. [ABSTRACT FROM AUTHOR]

For a positive integer, letbe the set ofcomplex matrices. Supposeare positive integers and. Denote bythe-numerical range of a matrix. It is shown that a linear mapsatisfiesfor allandif and only if there is a unitarysuch that one of the following holds.   For all , ,     and for all , ,where (1)is the identity mapor the transposition map, or (2)andhas the formor. [ABSTRACT FROM PUBLISHER]

In this article, a brief survey of recent results on linear preserver problems and quantum information science is given. In addition, characterization is obtained for linear operators φ onmn × mnHermitian matrices such that φ(A ⊗ B) andA ⊗ Bhave the same spectrum for anym × mHermitianAandn × nHermitianB. Such a map has the formA ⊗ B ↦ U(ϕ1(A) ⊗ ϕ2(B))U* formn × mnHermitian matrices in tensor formA ⊗ B, whereUis a unitary matrix, and forj ∈ {1, 2}, ϕjis the identity map X ↦ Xor the transposition map X ↦ Xt. The structure of linear maps leaving invariant the spectral radius of matrices in tensor formA ⊗ Bis also obtained. The results are connected to bipartite (quantum) systems and are extended to multipartite systems. [ABSTRACT FROM AUTHOR]