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

Tensor Products of Quantum Mappings

Sergey N. Filippov

Journal of Mathematical SciencesNov 20, 20206 citations

Linear Maps Preserving Ky Fan Norms and Schatten Norms of Tensor Products of Matrices

Ajda Fošner, Zejun Huang, Chi-Kwong Li +1 more

For a positive integer $n$ , let $M_n$ be the set of $n\times n$ complex matrices. Suppose $\|\cdot\|$ is the Ky Fan $k$ -norm with $1 \le k \le mn$ or the Schatten $p$ -norm with $1 \le p \le \infty$ ( $p\ne 2$ ) on $M_{mn}$ , where $m,n\ge 2$ are positive integers. It is shown that a linear map $\phi: M_{mn} \rightarrow M_{mn}$ satisfying $\|A\otimes B\| = \|\phi(A\otimes B)\| for all A \in M_m and B \in M_n$ if and only if there are unitary $U, V \in M_{mn}$ such that $\phi$ has the form $A\otimes B \mapsto U(\varphi_1(A) \otimes \varphi_2(B))V$ , where $\varphi_s(X)$ is either the identity map $X \mapsto X$ or the transposition map $X \mapsto X^t$ . The results are extended to tensor space $M_{n_1} \otimes \cdots \otimes M_{n_m}$ of higher level. The connection of the problem to quantum information science is mentioned.For a positive integer $n$, let $M_n$ be the set of $n\times n$ complex matrices. Suppose $\|\cdot\|$ is the Ky Fan $k$-norm with $1 \le k \le mn$ or the Schatten $p$-norm with $1 \le p \le \infty$ ($p\ne 2$) on $M_{mn}$, where $m,n\ge 2$ are positive integers. It is shown that a linear map $\phi: M_{mn} \rightarrow M_{mn}$ satisfying $\|A\otimes B\| = \|\phi(A\otimes B)\| for all A \in M_m and B \in M_n$ if and only if there are unitary $U, V \in M_{mn}$ such that $\phi$ has the form $A\otimes B \mapsto U(\varphi_1(A) \otimes \varphi_2(B))V$, where $\varphi_s(X)$ is either the identity map $X \mapsto X$ or the transposition map $X \mapsto X^t$. The results are extended to tensor space $M_{n_1} \otimes \cdots \otimes M_{n_m}$ of higher level. The connection of the problem to quantum information science is mentioned.

arXiv:1211.0396SIAM Journal on Matrix Analysis and ApplicationsJan 1, 201315 citations

Completely Positive Matrix Products

Eric Evert, Michael T. Jury, Scott McCullough

Building on recent works that investigate positivity preserving matrix products, we {examine} the class of \JCP (\jcp) matrix products. A bilinear map on the Cartesian product of the space of n by n matrices with itself into m by m matrices is a \jcp matrix product if the natural linear map it induces on the tensor product of the space of n by n matrices with itself into m by m matrices is completely positive. In particular, a matrix product is \jcp if and only if its naturally associated Choi matrix is positive semidefinite. Similarly, a matrix product is \jcp if and only if it admits a Choi-Kraus representation. We use the Choi-Kraus representation of \jcp matrix products to study various basic properties, including positivity lower bounds, commutativity, units, causality, and separability. As examples, we apply our results to the Schur (Hadamard) product and the convolution product.Building on recent works that investigate positivity preserving matrix products, we {examine} the class of \JCP (\jcp) matrix products. A bilinear map on the Cartesian product of the space of n by n matrices with itself into m by m matrices is a \jcp matrix product if the natural linear map it induces on the tensor product of the space of n by n matrices with itself into m by m matrices is completely positive. In particular, a matrix product is \jcp if and only if its naturally associated Choi matrix is positive semidefinite. Similarly, a matrix product is \jcp if and only if it admits a Choi-Kraus representation. We use the Choi-Kraus representation of \jcp matrix products to study various basic properties, including positivity lower bounds, commutativity, units, causality, and separability. As examples, we apply our results to the Schur (Hadamard) product and the convolution product.

arXiv:2607.13251Jul 14, 20260 citations

For a positive integer

n

, let

M_n

be the set of

n\times n

complex matrices. Suppose

\|\cdot\|

is the Ky Fan

k

-norm with

1 \le k \le mn

or the Schatten

p

-norm with

1 \le p \le \infty

(

p\ne 2

) on

M_{mn}

, where

m,n\ge 2

are positive integers. It is shown that a linear map

\phi: M_{mn} \rightarrow M_{mn}

satisfying

\|A\otimes B\| = \|\phi(A\otimes B)\| for all A \in M_m and B \in M_n

if and only if there are unitary

U, V \in M_{mn}

such that

\phi

has the form

A\otimes B \mapsto U(\varphi_1(A) \otimes \varphi_2(B))V

, where

\varphi_s(X)

is either the identity map

X \mapsto X

or the transposition map

X \mapsto X^t

. The results are extended to tensor space

M_{n_1} \otimes \cdots \otimes M_{n_m}

of higher level. The connection of the problem to quantum information science is mentioned.