Zejun Huang

Linear preservers and quantum information science

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 on mn × mn Hermitian matrices such that φ(A ⊗ B) and A ⊗ B have the same spectrum for any m × m Hermitian A and n × n Hermitian B. Such a map has the form A ⊗ B ↦ U(ϕ1(A) ⊗ ϕ2(B))U* for mn × mn Hermitian matrices in tensor form A ⊗ B, where U is a unitary matrix, and for j ∈ {1, 2}, ϕ j is the identity map X ↦ X or the transposition map X ↦ X t . The structure of linear maps leaving invariant the spectral radius of matrices in tensor form A ⊗ B is also obtained. The results are connected to bipartite (quantum) systems and are extended to multipartite systems.

Physical transformations between quantum states

Given two sets of quantum states {A1, …, Ak} and {B1, …, Bk}, represented as sets as density matrices, necessary and sufficient conditions are obtained for the existence of a physical transformation T, represented as a trace-preserving completely positive map, such that T(Ai) = Bi for i = 1, …, k. General completely positive maps without the trace-preserving requirement, and unital completely positive maps transforming the states are also considered.

Linear maps preserving Ky Fan norms and Schatten norms of tensor products of matrices

For a positive integer

Linear maps preserving the higher numerical ranges of tensor products of matrices

n

Digraphs that have at most one walk of a given length with the same endpoints

, let

Extremal digraphs whose walks with the same initial and terminal vertices have distinct lengths

M_n

Linear rank preservers of tensor products of rank one matrices

be the set of

Sizes and transmissions of digraphs with a given clique number

n\times n

Weakening the conditions in some classical theorems on linear preserver problems

complex matrices. Suppose

Factorization of Permutations

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