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.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.

Linear and Multilinear AlgebraDec 11, 201229 citations

Maps preserving trace of products and invertible completely positive maps

Huajun Huang, Ming-Cheng Tsai

We show the linearity and injectivity of two maps $\phi_1$ and $\phi_2$ between certain subsets of matrix spaces that satisfy $\operatorname{tr}(\phi_1(A)\phi_2(B))=\operatorname{tr}(AB)$ . We apply it to characterize maps $\phi_i$ ( $i=1, \ldots, m$ ) satisfying $\operatorname{tr} (\phi_1(A_1)\cdots \phi_m(A_m))=\operatorname{tr} (A_1\cdots A_m)$ on maps $\mathcal{S}\to \mathcal{S}$ in which $\mathcal{S}$ is the set of $n$ -by- $n$ general or Hermitian matrices for $m\ge 3$ and positive definite matrices for $m\ge 2$ , and on completely positive maps $M_n\to M_n$ for $m\ge 2$ . Then we give several equivalent characterizations of invertible completely positive maps, that is, $\phi: M_n\to M_k$ and $\psi:M_k\to M_n$ such that $\psi(\phi(A))=A$ for $A\in M_n$ , and provide a comprehensive list of such examples. These maps are connected to the trace-of-product preserving property and are useful in quantum information theory. In addition, we show that a linear map $\phi$ between matrix spaces satisfies $\phi(I)=I$ and $\phi(A^k)=\phi(A)^k$ for a fixed integer $k\ge 2$ if and only if $\phi(A^r)=\phi(A)^r$ for all $r\in {\mathbb Z}_+$ . We characterize maps $\phi, \psi: \mathcal{S} \to \mathcal{S}$ satisfying $\operatorname{tr} (\phi(A)\psi(B))=\operatorname{tr} (AB)$ and $\operatorname{tr} (\phi(A)\psi(B)^k)=\operatorname{tr} (AB^k)$ for a fixed $k\ge 2$ , where $\mathcal{S}$ is the set of general or Hermitian matrices.We show the linearity and injectivity of two maps $\phi_1$ and $\phi_2$ between certain subsets of matrix spaces that satisfy $\operatorname{tr}(\phi_1(A)\phi_2(B))=\operatorname{tr}(AB)$. We apply it to characterize maps $\phi_i$ ($i=1, \ldots, m$) satisfying $$\operatorname{tr} (\phi_1(A_1)\cdots \phi_m(A_m))=\operatorname{tr} (A_1\cdots A_m)$$ on maps $\mathcal{S}\to \mathcal{S}$ in which $\mathcal{S}$ is the set of $n$-by-$n$ general or Hermitian matrices for $m\ge 3$ and positive definite matrices for $m\ge 2$, and on completely positive maps $M_n\to M_n$ for $m\ge 2$. Then we give several equivalent characterizations of invertible completely positive maps, that is, $\phi: M_n\to M_k$ and $\psi:M_k\to M_n$ such that $\psi(\phi(A))=A$ for $A\in M_n$, and provide a comprehensive list of such examples. These maps are connected to the trace-of-product preserving property and are useful in quantum information theory. In addition, we show that a linear map $\phi$ between matrix spaces satisfies $\phi(I)=I$ and $\phi(A^k)=\phi(A)^k$ for a fixed integer $k\ge 2$ if and only if $\phi(A^r)=\phi(A)^r$ for all $r\in {\mathbb Z}_+$. We characterize maps $\phi, \psi: \mathcal{S} \to \mathcal{S}$ satisfying $\operatorname{tr} (\phi(A)\psi(B))=\operatorname{tr} (AB)$ and $\operatorname{tr} (\phi(A)\psi(B)^k)=\operatorname{tr} (AB^k)$ for a fixed $k\ge 2$, where $\mathcal{S}$ is the set of general or Hermitian matrices.

Mar 22, 20210 citations

Tensor Products of Quantum Mappings

Sergey N. Filippov

Journal of Mathematical SciencesNov 20, 20206 citations

We show the linearity and injectivity of two maps

\phi_1

and

\phi_2

between certain subsets of matrix spaces that satisfy

\operatorname{tr}(\phi_1(A)\phi_2(B))=\operatorname{tr}(AB)

. We apply it to characterize maps

\phi_i

(

i=1, \ldots, m

) satisfying

\operatorname{tr} (\phi_1(A_1)\cdots \phi_m(A_m))=\operatorname{tr} (A_1\cdots A_m)

on maps

\mathcal{S}\to \mathcal{S}

in which

\mathcal{S}

is the set of

n

-by-

n

general or Hermitian matrices for

m\ge 3

and positive definite matrices for

m\ge 2

, and on completely positive maps

M_n\to M_n

for

m\ge 2

. Then we give several equivalent characterizations of invertible completely positive maps, that is,

\phi: M_n\to M_k

and

\psi:M_k\to M_n

such that

\psi(\phi(A))=A

for

A\in M_n

, and provide a comprehensive list of such examples. These maps are connected to the trace-of-product preserving property and are useful in quantum information theory. In addition, we show that a linear map

\phi

between matrix spaces satisfies

\phi(I)=I

and

\phi(A^k)=\phi(A)^k

for a fixed integer

k\ge 2

if and only if

\phi(A^r)=\phi(A)^r

for all

r\in {\mathbb Z}_+

. We characterize maps

\phi, \psi: \mathcal{S} \to \mathcal{S}

satisfying

\operatorname{tr} (\phi(A)\psi(B))=\operatorname{tr} (AB)

and

\operatorname{tr} (\phi(A)\psi(B)^k)=\operatorname{tr} (AB^k)

for a fixed

k\ge 2

, where

\mathcal{S}

is the set of general or Hermitian matrices.