Linear preservers and quantum information science

Abstract

Abstract. In this paper, 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}, ϕj 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 bipartite (quantum) systems and are extended to multipartite systems.2010 Math. Subj. Class.: 15A69, 15A86, 15B57, 15A18.

A note on linear preservers of certain ranks of tensor products of matrices

“…Very recently, Xu et al [7] gave a complete characterization of linear maps between spaces of Hermitian matrices that send tensor product of idempotent matrices to idempotent matrices. Linear maps on tensor products of real vector spaces of Hermitian matrices that preserve the spectral radius, higher numerical range and Ky Fan (Schatten) norms of tensor products of matrices were studied in [8][9][10], respectively.…”

Linear rank preservers of tensor products of rank one matrices

“…Since then, lots of linear preservers have been characterized, see [4,13] and their references. In particular, Marcus and Moyls [19] determined linear maps that send rank one matrices to rank one matrices, which have the form A → M AN or A → M A T N for some nonsingular matrices M and N .…”

Linear maps preserving numerical radius of tensor products of matrices

“…Thus, φ maps Hermitian matrices to Hermitian matrices and preserves numerical radius, which is equivalent to spectral radius for Hermitian matrices. By Theorem 3.3 in [2], we conclude that φ has the asserted form on Hermitian matrices and, hence, on all matrices in M mn . However, if m, n ≥ 3, then, by Example 2.1, neither the map A ⊗ B → A ⊗ B t nor the map A ⊗ B → A t ⊗ B will preserve the numerical range.…”

“…In the following, we first determine the structure of linear preservers of numerical range using the above example and the results in [2].…”

“…Nevertheless, one may be able to extract useful information about the bipartite system by focusing on the set of tensor product matrices. In particular, in the study of linear operators φ : M mn → M mn on bipartite systems, the structure of φ can be determined by studying φ(A ⊗ B) with A ∈ M m , B ∈ M n ; see [1,2,3,7] and their references.…”