Linear and Multilinear Algebra

Linear preservers and quantum information science

Ajda Fošner, Zejun Huang, Chi-Kwong Li, Nung-Sing Sze

Published Dec 11, 2012• 29 citations

Abstract

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.

DOI: 10.1080/03081087.2012.740029

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Linear and Multilinear Algebra

Linear preservers and quantum information science

Ajda Fošner, Zejun Huang, Chi-Kwong Li, Nung-Sing Sze

Published Dec 11, 2012• 29 citations

Publisher

Abstract

DOI: 10.1080/03081087.2012.740029

Related papers

Linear preservers and quantum information science

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

Let $m,n\ge 2$ be positive integers, $M_m$ the set of $m\times m$ complex matrices and $M_n$ the set of $n\times n$ complex matrices. Regard $M_{mn}$ as the tensor space $M_m\otimes M_n$ . 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 $M_{mn}$ . It is shown that a linear map $ϕ: M_{mn} \rightarrow M_{mn}$ satisfying $|A\otimes B| = |ϕ(A\otimes B)|$ 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 $ϕ$ has the form $A\otimes B \mapsto U(φ_1(A) \otimes φ_2(B))V$ , where $φ_i(X)$ is either the identity map $X \mapsto X$ or the transposition map $X \mapsto X^t$ . The results are extended to tensor space $M_{n_1} \otimes ... \otimes M_{n_m}$ of higher level. The connection of the problem to quantum information science is mentioned.Let $m,n\ge 2$ be positive integers, $M_m$ the set of $m\times m$ complex matrices and $M_n$ the set of $n\times n$ complex matrices. Regard $M_{mn}$ as the tensor space $M_m\otimes M_n$. 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 $M_{mn}$. It is shown that a linear map $ϕ: M_{mn} \rightarrow M_{mn}$ satisfying $$|A\otimes B| = |ϕ(A\otimes B)|$$ 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 $ϕ$ has the form $A\otimes B \mapsto U(φ_1(A) \otimes φ_2(B))V$, where $φ_i(X)$ is either the identity map $X \mapsto X$ or the transposition map $X \mapsto X^t$. The results are extended to tensor space $M_{n_1} \otimes ... \otimes M_{n_m}$ of higher level. The connection of the problem to quantum information science is mentioned.

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

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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.

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In quantum technologies, quantum channels are essential for transmitting quantum states. Quantum channels usually introduce noise, thereby reducing the information in a state. These are mathematically described by completely positive trace-preserving linear maps that represent the generic evolution of quantum systems and are also special cases of Hermitian-preserving trace-preserving (HPTP) linear maps. Concatenating a quantum channel with another quantum channel makes it noisier and degrades its information and resource preservability. In this work, we demonstrate a physically meaningful way to compare a pair of quantum channels using Hermitian-preserving trace-preserving linear maps. Precisely, given a pair of quantum channels $Λ_1, Λ_2$ and an arbitrary unknown input state, we show that the output state of the quantum channel $Λ_2$ from the pair can be uniquely identified from the output statistics of the other channel $Λ_1$ from the pair, using some POVM, iff $Λ_2$ can be obtained from $Λ_1$ by concatenating it with a Hermitian-preserving trace-preserving linear map that is not necessarily positive. In such cases, $Λ_2$ may not always be obtained from $Λ_1$ through post-processing. The relation between these channels is a preorder, and we study its characterization in this work. In other words, we provide an operational interpretation of this preorder induced by HPTP linear maps in a communication scenario. Furthermore, we characterize the difficulty of implementing $Λ_2$ given that $Λ_1$ has already been implemented via a quantifier, namely, physical implementability. We also illustrate the implications of our results for the incompatibility of quantum devices through an example. In short, we provide valuable insights about the relevance of Hermitian-preserving trace-preserving linear maps in physically motivated settings.In quantum technologies, quantum channels are essential for transmitting quantum states. Quantum channels usually introduce noise, thereby reducing the information in a state. These are mathematically described by completely positive trace-preserving linear maps that represent the generic evolution of quantum systems and are also special cases of Hermitian-preserving trace-preserving (HPTP) linear maps. Concatenating a quantum channel with another quantum channel makes it noisier and degrades its information and resource preservability. In this work, we demonstrate a physically meaningful way to compare a pair of quantum channels using Hermitian-preserving trace-preserving linear maps. Precisely, given a pair of quantum channels $Λ_1, Λ_2$ and an arbitrary unknown input state, we show that the output state of the quantum channel $Λ_2$ from the pair can be uniquely identified from the output statistics of the other channel $Λ_1$ from the pair, using some POVM, iff $Λ_2$ can be obtained from $Λ_1$ by concatenating it with a Hermitian-preserving trace-preserving linear map that is not necessarily positive. In such cases, $Λ_2$ may not always be obtained from $Λ_1$ through post-processing. The relation between these channels is a preorder, and we study its characterization in this work. In other words, we provide an operational interpretation of this preorder induced by HPTP linear maps in a communication scenario. Furthermore, we characterize the difficulty of implementing $Λ_2$ given that $Λ_1$ has already been implemented via a quantifier, namely, physical implementability. We also illustrate the implications of our results for the incompatibility of quantum devices through an example. In short, we provide valuable insights about the relevance of Hermitian-preserving trace-preserving linear maps in physically motivated settings.

quant-pharXiv:2512.07822Dec 9, 20250 citations

Let

m,n\ge 2

be positive integers,

M_m

the set of

m\times m

complex matrices and

M_n

the set of

n\times n

complex matrices. Regard

M_{mn}

as the tensor space

M_m\otimes M_n

. 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}

. It is shown that a linear map

ϕ: M_{mn} \rightarrow M_{mn}

satisfying

|A\otimes B| = |ϕ(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

ϕ

has the form

A\otimes B \mapsto U(φ_1(A) \otimes φ_2(B))V

, where

φ_i(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 ... \otimes M_{n_m}

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

For a positive integer

n\geq 2

, let

M_{n}

be the set of

n\times n

complex matrices and

H_{n}

the set of Hermitian matrices in

M_{n}

. We characterize injective linear maps

{\it\phi}:H_{m_{1}\cdots m_{l}}\rightarrow H_{n}

satisfying $$\begin{eqnarray}\text{rank}(A_{1}\otimes \cdots \otimes A_{l})=1\Longrightarrow \text{rank}({\it\phi}(A_{1}\otimes \cdots \otimes A_{l}))=1\end{eqnarray}$$ for all

A_{k}\in H_{m_{k}}

k=1,\dots ,l

, where

l,m_{1},\dots ,m_{l}\geq 2

are positive integers. The necessity of the injectivity assumption is shown. Moreover, 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.

Λ_1, Λ_2

and an arbitrary unknown input state, we show that the output state of the quantum channel

Λ_2

from the pair can be uniquely identified from the output statistics of the other channel

Λ_1

from the pair, using some POVM, iff

Λ_2

can be obtained from

Λ_1

by concatenating it with a Hermitian-preserving trace-preserving linear map that is not necessarily positive. In such cases,

Λ_2

may not always be obtained from

Λ_1

through post-processing. The relation between these channels is a preorder, and we study its characterization in this work. In other words, we provide an operational interpretation of this preorder induced by HPTP linear maps in a communication scenario. Furthermore, we characterize the difficulty of implementing

Λ_2

given that

Λ_1

has already been implemented via a quantifier, namely, physical implementability. We also illustrate the implications of our results for the incompatibility of quantum devices through an example. In short, we provide valuable insights about the relevance of Hermitian-preserving trace-preserving linear maps in physically motivated settings.