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Linear maps preserving $(p,k)$-norms of tensor products of matrices

Published online by Cambridge University Press:  15 December 2023

Zejun Huang
Affiliation:
School of Mathematical Sciences, Shenzhen University, Shenzhen 518060, China e-mail: mathzejun@gmail.com
Nung-Sing Sze
Affiliation:
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong e-mail: raymond.sze@polyu.edu.hk
Run Zheng*
Affiliation:
Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong
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Abstract

Let $m,n\ge 2$ be integers. Denote by $M_n$ the set of $n\times n$ complex matrices and $\|\cdot \|_{(p,k)}$ the $(p,k)$ norm on $M_{mn}$ with a positive integer $k\leq mn$ and a real number $p>2$. We show that a linear map $\phi :M_{mn}\rightarrow M_{mn}$ satisfies

$$ \begin{align*}\|\phi(A\otimes B)\|_{(p,k)}=\|A\otimes B\|_{(p,k)} \mathrm{\quad for~ all\quad}A\in M_m\ \mathrm{and}\ B\in M_n\end{align*} $$

if and only if there exist unitary matrices $U,V\in M_{mn}$ such that

$$ \begin{align*}\phi(A\otimes B)=U(\varphi_1(A)\otimes \varphi_2(B))V \mathrm{\quad for~ all\quad}A\in M_m\ \mathrm{and}\ B\in M_n,\end{align*} $$

where $\varphi _s$ is the identity map or the transposition map $X\mapsto X^T$ for $s=1,2$. The result is also extended to multipartite systems.

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Article
Creative Commons
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

1 Introduction

Throughout this paper, we denote by $M_{m,n}$ and $M_n$ the set of $m\times n$ and $n\times n$ complex matrices, respectively. Denote by $H_n$ the set of all $n\times n$ Hermitian matrices. For two matrices $A=(a_{ij})\in M_m$ and $B\in M_n$ , their tensor product is defined to be $A\otimes B=(a_{ij}B)$ , which is an $mn\times mn$ matrix. We denote by

$$ \begin{align*}M_m\otimes M_n=\{A\otimes B: A\in M_m, B\in M_n\}\end{align*} $$

and

$$ \begin{align*}H_m\otimes H_n=\{A\otimes B: A\in H_m, B\in H_n\}.\end{align*} $$

Suppose $A\in M_{m,n}$ . The singular values of A are always denoted in decreasing order by $s_1(A)\geq \cdots \geq s_{\ell }(A)$ , where $\ell =\min \{m,n\}$ . Given a real number $p\geq 1$ and a positive integer $k\leq \min \{m,n\}$ , the $(p,k)$ -norm of A is defined by

$$ \begin{align*}\|A\|_{(p,k)}=\left[\sum\limits_{i=1}^ks_i^p(A)\right]^{ \frac{1}{p}}.\end{align*} $$

The $(p,k)$ -norm, also known as the Ky Fan $(p,k)$ -norm, was first recognized as a special class of unitarily invariant norms in the study of isometries by Grone and Marcus [Reference Grone and Marcus13] in their notable work from the 1970s. The $(p,k)$ -norms encompass many commonly used norms. For instance, the $(1,k)$ -norm reduces to the Ky Fan k-norm, while the $(p,K)$ -norm, with $K = \min \{m,n\}$ , reduces to Schatten p-norm. Moreover, the Ky Fan $1$ -norm, Ky Fan K-norm, and Shatten $2$ -norm are also known as the spectral norm, the trace norm, and the Frobenius norm, respectively. Some earlier works exploring the fundamental properties of the $(p,k)$ -norm can be found in [Reference Horn and Johnson14, Reference Kidman19, Reference Marcus, Kidman and Sandy24].

In addition to being a generalization of many well-known norms, the $(p,k)$ -norm itself has attracted extensive attention from researchers across various fields, particularly in the study of low-rank approximation (e.g., [Reference Doan and Vavasis5, Reference Jiang, Liu, Qi and Dai16, Reference Tanaka and Nakata30]). The application of the $(p,k)$ -norm in quantum information science has also gained recent attention. Researchers in this field have explored the concept of the twisted commutators of two unitaries and focused on determining the minimum norm value of these twisted commutators. The authors in [Reference Chubb and Flammia3] succeeded in obtaining an explicit closed form for the minimum twisted commutation value with respect to the $(p,k)$ -norm. All these show the growing importance and relevance of the $(p,k)$ -norm across various fields of study.

Linear preserver problems concern the study of linear maps on matrices or operators preserving certain special properties. Since Frobenius gave the characterization of linear maps on $M_n$ that preserve the determinant of all matrices in 1897, a lot of linear preserver problems have been investigated (see [Reference Li and Pierce20, Reference Molnár26] and the references therein).

The study of linear preservers on various matrix norms have been extensively explored since Schur [Reference Schur29] characterized linear maps on $M_n$ that preserve the spectral norm. This was followed by a series of subsequent results [Reference Arazy1, Reference Grone12, Reference Grone and Marcus13, Reference Li and Tsing23, Reference Russo27, Reference Russo28]. Notably, Li and Tsing [Reference Li and Tsing23] provided a complete characterization of linear maps that preserve the $(p,k)$ -norms. They showed that linear maps on $M_{m,n}$ that preserve the $(p,k)$ -norms (except for the Frobenius norm) have the form

$$ \begin{align*}A\mapsto UAV \quad\text{or}\quad A\mapsto UA^TV \quad\text{when}\quad m=n\end{align*} $$

for some unitary matrices $U\in M_m$ and $V\in M_n.$

Traditional linear preserver problems deal with linear maps preserving certain properties of every matrix in the whole matrix space $M_n$ or $H_n$ . Recently, linear maps on $M_{mn}$ or $H_{mn}$ only preserving certain properties of matrices in $M_m\otimes M_n$ or $H_m\otimes H_n$ have been investigated. Friedland et al. [Reference Friedland, Li, Poon and Sze11] provided a characterization of linear maps on $H_m\otimes H_n$ that preserve the set of separable states in bipartite systems. The concept of separability is widely recognized as a fundamental and crucial aspect in the field of quantum information science. Johnston in his paper [Reference Johnston17] examined invertible linear maps on $M_m\otimes M_n$ that preserve the set of rank one matrices with bounded Schmidt rank in both row and column spaces. Additionally, the author investigated linear maps on $M_m\otimes M_n$ that preserve the Schmidt k-norm, a norm induced by states with bounded Schmidt rank, which finds extensive application in the field of quantum information. For more details on the Schmidt k-norm, refer to [Reference Johnston and Kribs18].

Note that $M_m\otimes M_n$ and $H_m\otimes H_n$ are small subsets of $M_{mn}$ and $H_{mn}$ . Researchers know much less information on such linear maps. So it is more difficult to characterize such linear maps. Along this line, linear maps on Hermitian matrices preserving the spectral radius were determined in [Reference Fošner, Huang, Li and Sze8]. Linear maps on complex matrices or Hermitian matrices preserving determinant were studied in [Reference Chooi and Kwa2, Reference Ding, Fošner, Xu and Zheng4, Reference Duffner and Cruz6]. Linear maps on complex matrices preserving numerical radius, k-numerical range, product numerical range, and rank-one matrices were characterized in [Reference Fošner, Huang, Li, Poon and Sze7, Reference Fošner, Huang, Li and Sze9, Reference Huang, Shi and Sze15, Reference Li, Poon and Sze21].

In [Reference Fošner, Huang, Li and Sze10], the authors characterized linear maps on $M_{mn}$ preserving the Ky Fan k-norm and the Schatten p-norm of the tensor products $A\otimes B$ for all $A\in M_m$ and ${B\in M_n}$ . Despite the non-obvious connection to the field of quantum information, from a mathematical perspective, it is undeniably intriguing to consider the linear maps that preserve the $(p,k)$ -norm of tensor products of matrices.

Therefore, in this paper, we aim to characterize linear maps $\phi $ on $M_{mn}$ such that for $p>2$ and $1\leq k\leq mn$ ,

(1.1) $$ \begin{align} \|\phi(A\otimes B)\|_{(p,k)}=\|A\otimes B\|_{(p,k)}\quad \text{for all}\quad A\in M_m\text{ and }B\in M_n. \end{align} $$

The comprehensive characterization in the bipartite systems will be presented in Section 2, while in Section 3, we will extend the results to multipartite systems.

2 Bipartite system

The linear maps on $M_{mn}$ satisfying (1.1) are determined by the following theorem.

Theorem 2.1 Let $m,n \geq 2 $ be integers. Given a real number $p>2$ and a positive integer $ k \leq mn$ , a linear map $\phi : M_{mn}\to M_{mn}$ satisfies

(2.1) $$ \begin{align} \|\phi(A\otimes B)\|_{(p,k)}=\|A\otimes B\|_{(p,k)} \quad \text{for all}\quad A\in M_m\ \text{and}\ B\in M_n, \end{align} $$

if and only if there exist unitary matrices $U,V \in M_{mn}$ such that

(2.2) $$ \begin{align} \phi(A\otimes B)=U(\varphi_1(A)\otimes \varphi_2(B))V \quad \text{for all}\quad A\in M_m\ \text{and}\ B\in M_n, \end{align} $$

where $\varphi _s$ is the identity map or the transposition map $X\mapsto X^T$ for $s=1,2$ .

To prove the theorem, we need some notations and preliminary results. Denote by $\|A\|$ and $A^*$ the Frobenius norm and the conjugate transpose of the matrix A, respectively. Two matrices $A,B\in M_n$ are said to be orthogonal, denoted by $A\perp B$ , if $A^*B=AB^*=0$ . Denote by $E_{ij}\in M_{m,n}$ the matrix whose $(i,j)$ th entry is equal to one and all the other entries are equal to zero.

The eigenvalues of an $n\times n$ Hermitian matrix A are always denoted in decreasing order by $\lambda _1(A)\geq \lambda _2(A)\geq \cdots \geq \lambda _n(A)$ . For $A,B\in H_n$ , we use the notation $A\geq B$ or $B\leq A$ to mean that $A-B$ is positive semidefinite. Let $\mathbb {R}$ be the set of all real numbers. Rearrange the components of $x=(x_1,\ldots ,x_n)\in \mathbb {R}^n$ in decreasing order as $x_{[1]}\geq \cdots \geq x_{[n]}$ . For $x=(x_1,\ldots ,x_n),$ $y=(y_1,\ldots ,y_n)\in \mathbb {R}^n$ , if

$$ \begin{align*}\sum_{i=1}^kx_{[i]}\leq \sum_{i=1}^ky_{[i]} \quad \text{for}\quad k=1,\ldots,n,\end{align*} $$

then we say x is weakly majorized by y and denote by $y\succ _w x$ or $x\prec _w y$ .

Notice that $x\mapsto x^{\gamma } (x\geq 0)$ is a convex function for any real number $ \gamma \geq 1$ . One can easily conclude the following lemma.

Lemma 2.2 Let $a,b\in \mathbb {R}$ . If $-a\leq b\leq a$ , then for any real number $\gamma \geq 1$ ,

(2.3) $$ \begin{align} (a+b)^{\gamma}+(a-b)^{\gamma}\geq 2a^{\gamma}. \end{align} $$

The following lemmas are crucial in our proof.

Lemma 2.3 [Reference McCarthy25, Lemma 2.1]

Let $A\in M_n$ be a positive semidefinite matrix. Then

$$ \begin{align*}x^*A^{\gamma}x\geq (x^*Ax)^{\gamma}\|x\|^{2(1-\gamma)}\quad \text{for all}\quad x\in \mathbb{C}^n \text{and}\ \gamma\geq 1.\end{align*} $$

Lemma 2.4 [Reference Zhan31, Lemma 3.7]

Let $A\in M_n$ be a Hermitian matrix, and let $k \leq n$ be a positive integer. Then

$$ \begin{align*} \sum_{i=1}^k\lambda_i(A)=\max\limits_{U^*U=I_k}\mathrm{tr}(U^*AU)\quad \text{and}\quad \sum_{i=1}^k\lambda_{n-i+1}(A)=\min\limits_{U^*U=I_k}\mathrm{tr}(U^*AU), \end{align*} $$

where $I_k$ is the identity matrix of order k and $U\in M_{n,k}$ .

Lemma 2.5 [Reference Li, Šemrl and Sourour22, Lemma 1]

Let $A, B\in M_n$ . Then $A\perp B$ if and only if there exist $\hat {A}\in M_m $ , $\hat {B}\in M_{n-m}$ and unitary matrices $U,V\in M_n$ such that

$$ \begin{align*}UAV=\hat{A}\oplus 0\quad \text{and}\quad UBV=0\oplus \hat{B}.\end{align*} $$

Lemma 2.6 Let $A, B,C\in M_{n}$ . If $(A+B)\perp C$ and $A\perp B$ , then

$$ \begin{align*}A\perp C\quad \text{and} \quad B\perp C.\end{align*} $$

Proof Since $A\perp B$ , we can apply Lemma 2.5 to conclude that there exist $\hat {A}\in M_m $ , $\hat {B}\in M_{n-m}$ and unitary matrices $U,V\in M_n$ such that

$$ \begin{align*}UAV=\hat{A}\oplus 0\quad \text{and}\quad UBV=0\oplus \hat{B}.\end{align*} $$

Let $UCV$ be partitioned as

$$ \begin{align*}UCV=\begin{bmatrix} C_{11}&C_{12}\\ C_{21}&C_{22} \end{bmatrix} \end{align*} $$

with $C_{11}\in M_m$ and $C_{22}\in M_{n-m}.$ It follows from $(A+B)\perp C$ that

$$ \begin{align*} (U(A+B)V)^{*}(UCV)=0 \quad \text{and}\quad (U(A+B)V)(UCV)^*=0, \end{align*} $$

that is,

$$ \begin{align*}\begin{bmatrix} \hat{A}^*C_{11}&\hat{A}^*C_{12}\\ \hat{B}^{*}C_{21}&\hat{B}^{*}C_{22} \end{bmatrix}=0\quad \text{and}\quad \begin{bmatrix} \hat{A}C_{11}^*&\hat{A}C_{21}^*\\ \hat{B}C_{12}^*&\hat{B}C_{22}^* \end{bmatrix}=0. \end{align*} $$

Then we have

$$ \begin{align*}V^*A^*CV=(UAV)^*(UCV)= \begin{bmatrix} \hat{A}^*C_{11}&\hat{A}^*C_{12}\\ 0&0 \end{bmatrix}=0 \end{align*} $$

and

$$ \begin{align*}UAC^*U^*=(UAV)\ (UCV)^*=\begin{bmatrix} \hat{A}C_{11}^*&\hat{A}C_{21}^*\\ 0&0 \end{bmatrix}=0 .\end{align*} $$

Thus, $A^{*}C=0$ and $AC^{*}=0$ , i.e., $A\perp C$ . Similarly, we can also conclude that $B\perp C$ .

Lemma 2.7 Let $C,D\in M_n$ be two Hermitian matrices such that $-C\leq D\leq C$ and $ k\leq n$ be a positive integer. Then, for any real number $ \gamma \geq 1$ ,

(2.4) $$ \begin{align} \sum_{i=1}^{k}\lambda_i^{\gamma}(C+D)+\sum_{i=1}^{k}\lambda_i^{\gamma}(C-D)\geq 2\sum_{i=1}^k\lambda_i^{\gamma}(C). \end{align} $$

Proof Let $U\in M_n$ be a unitary matrix such that

$$ \begin{align*}U^*CU=\mathrm{diag}(\lambda_1(C),\lambda_2(C),\ldots,\lambda_n({C})).\end{align*} $$

Denote by $u_i$ the ith column of U for $i=1,\ldots ,n$ . Let $\hat {U}=[u_1,u_2,\ldots ,u_k].$ Then, applying Lemma 2.4, we have

$$ \begin{align*} \sum_{i=1}^k\lambda_{i}^{\gamma}(C+D)=\sum_{i=1}^k\lambda_{i}\big((C+D)^{\gamma}\big)\geq \mathrm{tr}(\hat{U}^*(C+D)^{\gamma}\hat{U}) \end{align*} $$

and

$$ \begin{align*}\quad \sum_{i=1}^k\lambda_{i}^{\gamma}(C-D)=\sum_{i=1}^k\lambda_{i}\big((C-D)^{\gamma}\big)\geq \mathrm{tr}(\hat{U}^*(C-D)^{\gamma}\hat{U}).\end{align*} $$

Since $-C\leq D\leq C$ , we have

$$ \begin{align*}-x^*Cx\leq x^*Dx\leq x^*Cx \quad\text{ for all}\quad x\in \mathbb{C}^{n}.\end{align*} $$

By Lemma 2.3, we have

$$ \begin{align*} u_i^{*}(C+D)^{\gamma}u_{i}\geq (u_i^*(C+D)u_i)^{\gamma} \quad \text{and} \quad u_i^{*}(C-D)^{\gamma}u_{i}\geq (u_i^*(C-D)u_i)^{\gamma} \end{align*} $$

for $i=1,\ldots ,n$ . Applying Lemma 2.2 with $a=u_i^{*}Cu_i$ and $b=u_i^{*}Du_i$ , we get

$$ \begin{align*} (u_i^{*}(C+D)u_i)^{\gamma}+(u_i^{*}(C-D)u_i)^{\gamma}\geq 2(u_i^{*}Cu_i)^{\gamma}\quad \text{for}\quad i=1,\ldots, n. \end{align*} $$

It follows from the above inequalities that

$$ \begin{align*} \sum_{i=1}^k\lambda_{i}^{\gamma}(C+D)+\sum_{i=1}^k\lambda_{i}^{\gamma}(C-D)\geq& \mathrm{tr}(\hat{U}^*(C+D)^{\gamma}\hat{U})+\mathrm{tr}(\hat{U}^*(C-D)^{\gamma}\hat{U})\\ =&\sum_{i=1}^{k}u_i^{*}(C+D)^{\gamma}u_i+\sum_{i=1}^{k}u_i^{*}(C-D)^{\gamma}u_i \\ \geq&\sum_{i=1}^{k}(u_i^{*}(C+D)u_i)^{\gamma}+\sum_{i=1}^{k}(u_i^{*}(C-D)u_i)^{\gamma} \\ \geq& 2 \sum_{i=1}^{k}(u_i^{*}Cu_i)^{\gamma}=2\sum_{i=1}^k\lambda_i^{\gamma}(C). \\[-42pt] \end{align*} $$

Remark 2.8 The inequality (2.4) can be regarded as a generalization of the inequality (2.3) in Lemma 2.2. It is worth noting that if $-a\leq b\leq a$ , then

$$ \begin{align*}(a+b)^{\gamma}+(a-b)^{\gamma}\leq 2a^{\gamma}\quad \text{for all}\quad 0<\gamma<1.\end{align*} $$

In our attempt to generalize this inequality, we aimed to obtain the following analogous inequality to (2.4):

$$ \begin{align*}\sum_{i=1}^{k}\lambda_i^{\gamma}(C+D)+\sum_{i=1}^{k}\lambda_i^{\gamma}(C-D)\leq 2\sum_{i=1}^k\lambda_i^{\gamma}(C)\quad \text{for all }\quad 0<\gamma<1,\end{align*} $$

where $1\leq k\leq n$ and $C,D\in M_n$ are Hermitian matrices such that $C+D$ and $C-D$ are both positive semidefinite. However, it has been demonstrated that this inequality does not hold in general. A counterexample can be constructed by considering matrices C and D such that $C+D=\mathrm {diag}(1,1,3,3)$ and $C-D=\mathrm {diag}(3,3,1,1)$ . In this case, we observe that $\sum \limits _{i=1}^{2}\lambda _i^{\gamma }(C+D)+\sum \limits _{i=1}^{2}\lambda _i^{\gamma }(C-D)= 4\cdot 3^{\gamma }>2\sum \limits _{i=1}^2\lambda _i^{\gamma }(C)=4\cdot 2^{\gamma }.$

Corollary 2.9 Let $p>2$ be a real number, and let $ k\leq n$ be a positive integer. Then

(2.5) $$ \begin{align} \|A+B\|_{(p,k)}^p+\|A-B\|_{(p,k)}^p\geq 2\sum_{i=1}^k\lambda_i^{\frac{p}{2}}(A^*A+B^*B) \end{align} $$

for all $A,B\in M_n.$

Proof Notice that

$$ \begin{align*}\|A+B\|_{(p,k)}^p=\sum_{i=1}^ks_i^p(A+B)=\sum_{i=1}^k\lambda_i^{\frac{p}{2}}\big((A^*A+B^*B)+(A^*B+B^*A)\big)\end{align*} $$

and

$$ \begin{align*}\|A-B\|_{(p,k)}^p=\sum_{i=1}^ks_i^p(A-B)=\sum_{i=1}^k\lambda_i^{\frac{p}{2}}\big((A^*A+B^*B)-(A^*B+B^*A)\big).\end{align*} $$

Let $C=A^*A+B^*B$ and $D= A^*B+B^*A$ . Then $C+D=(A+B)^*(A+B)$ and $C-D=(A-B)^*(A-B)$ are both positive semidefinite, that is, $-C\leq D\leq C$ . Applying Lemma 2.7, we get (2.5).

Lemma 2.10 Let $A,B\in M_n$ be nonzero matrices, and let $ 2\leq k\leq n$ be an integer. Given a real number $p\geq 1,$ if

$$ \begin{align*}\|A+B\|_{(p,k)}^p=\|A\|_{(p,k)}^p+\|B\|_{(p,k)}^p\quad \text{and}\quad A\perp B,\end{align*} $$

then $\mathrm {rank}(A+B)\leq k.$

Proof With the assumption that $A\perp B$ , we can assume that the largest k singular values of $A+B$ are $s_1(A),\ldots ,s_{\ell }(A),s_1(B),\ldots ,s_{k-\ell }(B)$ for some $0\leq \ell \leq k$ . Then

(2.6) $$ \begin{align} \|A+B\|_{(p,k)}^p=\sum_{i=1}^{\ell}s_i^p(A)+\sum_{i=1}^{k-\ell}s_i^p(B)\leq \sum_{i=1}^{k}s_i^p(A)+\sum_{i=1}^{k}s_i^p(B). \end{align} $$

On the other hand, we have

$$ \begin{align*}\|A+B\|_{(p,k)}^p=\|A\|_{(p,k)}^p+\|B\|_{(p,k)}^p=\displaystyle\sum_{i=1}^{k}s_i^p(A)+\displaystyle\sum_{i=1}^{k}s_i^p(B).\end{align*} $$

Thus, the equality in (2.6) holds, which implies

(2.7) $$ \begin{align} \sum_{i=1}^{\ell}s_i^p(A)=\sum_{i=1}^{k}s_i^p(A)\quad \text{and}\quad \sum_{i=1}^{k-\ell}s_i^p(B)=\sum_{i=1}^{k}s_i^p(B). \end{align} $$

Since A and B are both nonzero, we have

$$ \begin{align*}\displaystyle\sum_{i=1}^{k}s_i^p(A)>0\quad \mathrm{and}\quad \displaystyle\sum_{i=1}^{k}s_i^p(B)>0,\end{align*} $$

which implies $\ell \geq 1$ and $k-\ell \geq 1$ , i.e., $1\leq \ell \leq k-1.$ With (2.7), it follows that

$$ \begin{align*} \sum_{i=\ell+1}^{k}s_i^p(A)=0\quad \text{and} \quad \displaystyle \sum_{i=k-\ell+1}^{k}s_k^p(B)=0, \end{align*} $$

which implies $s_{\ell +1}(A)=0$ and $s_{k-\ell +1}(B)=0.$ Therefore,

$$ \begin{align*}\mathrm{rank}(A)\leq \ell \quad\mathrm{and } \quad\mathrm{rank}(B)\leq k-\ell.\end{align*} $$

Since $A\perp B$ , we have

$$ \begin{align*}\mathrm{rank}(A+B)=\mathrm{rank}(A)+\mathrm{rank}(B)\leq \ell+k-\ell=k.\\[-35pt]\end{align*} $$

Lemma 2.11 Let $A, B\in M_n$ be two positive semidefinite matrices, let $\gamma>1$ be a real number, and let $k\leq n$ be a positive integer. Suppose

(2.8) $$ \begin{align} \sum_{i=1}^{k}\lambda_{i}^{\gamma}(A+\alpha B)\leq \sum_{i=1}^{k}\lambda_i^{\gamma}(A)+\sum_{i=1}^{k}\lambda_i^{\gamma}(\alpha B) \quad \text{for all}\quad 0<\alpha<1 \end{align} $$

and $U^*AU=\mathrm {diag}(\lambda _1(A),\ldots ,\lambda _n(A))$ for some unitary matrix $U\in M_n$ .

  1. (a) If $\lambda _k(A)=0,$ then $A\perp B$ .

  2. (b) If $\lambda _k(A)>0,$ then $U^*BU=0_{k+\ell }\oplus \hat {B}$ with $\hat {B}\in M_{n-k-\ell }$ , where $\ell $ is the largest integer such that $\lambda _{k+\ell }(A)=\lambda _k(A)$ .

Proof Denote the ith diagonal entry of $U^*BU$ by $b_i.$ Then $\lambda _i(A)+\alpha b_{i}$ is the ith diagonal entry of $U^*(A+\alpha B)U$ . It follows that

$$ \begin{align*}\big(\lambda_1(A+\alpha B),\ldots,\lambda_k(A+\alpha B) \big )\succ_w \big (\lambda_{1}(A)+\alpha b_{1},\ldots,\lambda_{k}(A)+\alpha b_{k}\big). \end{align*} $$

Notice that $g(x)=x^{\gamma } (x>0)$ is an increasing convex function when $\gamma>1$ . We can apply Theorem 3.26 in [Reference Zhan31] to obtain

$$ \begin{align*}\big(\lambda_1^{\gamma}(A+\alpha B),\ldots,\lambda_k^{\gamma}(A+\alpha B) \big)\succ_w \big ((\lambda_{1}(A)+\alpha b_{1})^{\gamma},\ldots,(\lambda_{k}(A)+\alpha b_{k})^{\gamma}\big).\end{align*} $$

Thus, $\displaystyle \sum _{i=1}^{k}\lambda _{i}^{\gamma }(A+\alpha B)\geq \displaystyle \sum _{i=1}^{k}(\lambda _i(A)+\alpha b_i)^{\gamma }.$ With the assumption in (2.8), we can conclude that

(2.9) $$ \begin{align} \sum_{i=1}^{k}(\lambda_i(A)+\alpha b_{i})^{\gamma} \leq \sum_{i=1}^{k}\lambda_i^{\gamma}(A)+\sum_{i=1}^{k}\lambda_i^{\gamma}(\alpha B)\quad \text{for all}\quad 0<\alpha<1. \end{align} $$

Let $f(\alpha )=\displaystyle \sum _{i=1}^{k}(\lambda _i(A)+\alpha b_{i})^{\gamma } -\displaystyle \sum _{i=1}^{k}\lambda _i^{\gamma }(A)-\displaystyle \sum _{i=1}^{k}\lambda _i^{\gamma }(\alpha B)$ be a function on $\alpha $ . Then we have

(2.10) $$ \begin{align} f(\alpha)=f(0)+f^{'}(0)\alpha+o(\alpha)= \left[\sum_{i=1}^{k}\lambda_i^{\gamma-1}(A)b_{i}\gamma\right]\alpha+o(\alpha), \end{align} $$

where a function $g(\alpha )=o(\alpha )$ means $ \lim \limits _{\alpha \to 0}\frac {g(\alpha )}{\alpha }=0$ . Since A and B are both positive semidefinite, we have $\lambda _i(A)\geq 0$ and $ b_{i}\geq 0$ for all $i=1,\ldots ,n.$ It follows that $\displaystyle \sum _{i=1}^{k}\lambda _i^{\gamma -1}(A)b_{i}\gamma \geq 0.$ We claim that $\displaystyle \sum _{i=1}^{k}\lambda _i^{\gamma -1}(A)b_{i}\gamma =0$ . Otherwise, $\displaystyle \sum _{i=1}^{k}\lambda _i^{\gamma -1}(A)b_{i}\gamma>0$ leads to $f(\alpha )>0$ when $\alpha>0$ is sufficiently small, which contradicts (2.9). It follows that

$$ \begin{align*}\lambda_i(A)b_{i}=0 \quad \text{for}\quad i=1,\ldots,k.\end{align*} $$

For the case $\lambda _k(A)=0$ , we may assume that $ t$ is the largest integer such that ${\lambda _t(A)>0}$ . Then $U^*AU=\mathrm {diag}(\lambda _1(A),\ldots ,\lambda _t(A))\oplus 0_{n-t}$ and $b_i=0$ for $i=1,\ldots ,t.$ Recall that B is positive semidefinite. Thus, $U^*BU=0_{t}\oplus \hat {B}$ with $\hat {B}\in M_{n-t}.$ It follows that $A\perp B.$

For the case $\lambda _k(A)>0,$ we first have $b_i=0$ for all $i=1,\ldots ,k$ . Since B is positive semidefinite, it follows that $U^*BU=0_k\oplus C$ with $C\in M_{n-k}.$ Recall that $\ell $ is the largest integer such that $\lambda _{k+\ell }(A)=\lambda _k(A)$ . If $\ell =0$ , then the proof is completed. If $\ell>0$ , then for any $i=k+1,\ldots ,k+\ell $ , replacing the role of $\lambda _k(A)+\alpha b_k$ with $\lambda _i(A)+\alpha b_i$ in the above argument, we can conclude $b_i=0$ . Thus, we have $b_i=0$ for $i=1,\ldots ,k+\ell .$ It follows that $U^*BU=0_{k+\ell }\oplus \hat {B}$ with $\hat {B}\in M_{n-k-\ell }$ .

Corollary 2.12 Let $T,S\in M_n$ be two matrices, let $p>2$ be a real number, and let $k\leq n$ be a positive integer. Suppose

$$ \begin{align*}\sum_{i=1}^k\lambda_i^{\frac{p}{2}}(T^*T+x^2S^*S)\leq \sum_{i=1}^k\lambda_i^{\frac{p}{2}}(T^*T)+\sum_{i=1}^k\lambda_i^{\frac{p}{2}}(x^2S^*S)\end{align*} $$

and

$$ \begin{align*}\sum_{i=1}^k\lambda_i^{\frac{p}{2}}(TT^*+x^2SS^*)\leq \sum_{i=1}^k\lambda_i^{\frac{p}{2}}(TT^*)+\sum_{i=1}^k\lambda_i^{\frac{p}{2}}(x^2SS^*)\end{align*} $$

for all $0<x<1,$ and $UTV=\mathrm {diag}(s_1(T),\ldots ,s_n(T))$ for some unitary matrices ${U,V\in M_n}$ .

  1. (1) If $s_k(T)=0,$ then $T\perp S$ .

  2. (2) If $s_k(T)>0,$ then $ USV=0_{k+\ell }\oplus \hat {S}$ with $\hat {S}\in M_{n-k-\ell }$ , where $\ell $ is the largest integer such that $s_{k+\ell }(T)=s_k(T).$

Proof If $s_k(T)=0$ , then $\lambda _k(T^*T)=\lambda _k(TT^*)=0$ . We can use Lemma 2.11 twice to conclude that $T^*T\perp S^*S$ and $TT^*\perp SS^*$ , and hence $T\perp S.$

If $s_k(T)>0$ , then we have

$$ \begin{align*}V^*T^*TV=\mathrm{diag}(s_1^2(T),\ldots,s_{n}^2(T))\quad \text{and}\quad UTT^*U^*=\mathrm{diag}(s_1^2(T),\ldots,s_{n}^2(T)).\end{align*} $$

Notice that $\lambda _{i}(TT^*)=\lambda _{i}(T^*T)=s^2_i(T)$ for $i=1,\ldots ,n$ . Thus,

$$ \begin{align*}\lambda_k(T^*T)=\lambda_k(TT^*)=s_k^2(T)>0\end{align*} $$

and $\ell $ is the largest integer such that

$$ \begin{align*}\lambda_{k+\ell}(TT^*)=\lambda_{k}(TT^*)\quad \text{and} \quad \lambda_{k+\ell}(T^*T)=\lambda_{k}(T^*T).\end{align*} $$

Then we use Lemma 2.11 twice to conclude that

$$ \begin{align*}VS^*SV=0_{k+\ell}\oplus C\quad \text{and}\quad USS^*U^*=0_{k+\ell}\oplus D.\end{align*} $$

It follows that $USV=0_{k+\ell }\oplus \hat {S}.$

The following result originates from the last two paragraphs of the proof of Theorem 2.1 in [Reference Fošner, Huang, Li and Sze10].

Lemma 2.13 Let $\phi :M_{mn}\to M_{mn}$ be a linear map. Suppose for any unitary matrix $X \in M_m$ and integer $1\leq i\leq m$ , there exists a unitary matrix $W_X$ such that

$$ \begin{align*} \phi(XE_{ii}X^*\otimes B)=W_X(E_{ii}\otimes \varphi_{i,X} (B)){W}_X^*\quad \text{for all}\quad B\in M_n, \end{align*} $$

where $\varphi _{i,X}$ is the identity map or the transposition map and $W_I=I_{mn}$ . Then

$$ \begin{align*}\phi(A\otimes B)=\varphi_{1}(A)\otimes \varphi_2(B)\quad \text{for all}\quad A\in M_{m} \text{ and } B\in M_n,\end{align*} $$

where $\varphi _1$ is a linear map and $\varphi _2$ is the identity map or the transposition map.

Proof For any real symmetric matrix $S\in M_n$ and any unitary matrix $X\in M_m$ ,

$$ \begin{align*}{\phi}(I_m\otimes S)=\sum_{i=1}^m{\phi}(XE_{ii}X^*\otimes S)={W}_X(I_m\otimes S){W}_X^*.\end{align*} $$

Since $W_I=I_{mn}$ , it follows that

$$ \begin{align*}{W}_X(I_m\otimes S){W}_X^*=W_I(I_m\otimes S){W}_I^*=I_m\otimes S.\end{align*} $$

Thus, ${W}_X$ commutes with $I_m\otimes S$ for all real symmetric $S\in M_n.$ This yields that ${{W}_X=Z_X\otimes I_n}$ for some unitary matrix $Z_X\in M_m$ , and hence

(2.11) $$ \begin{align} {\phi}(XE_{ii}X^*\otimes B)=(Z_XE_{ii}Z_X^*)\otimes \varphi_{i,X}(B)\quad \text{for all}\quad i=1,\ldots,m\text{ and }B\in M_n. \end{align} $$

Define linear maps $\text {tr}_1 : M_{mn}\to M_{n}$ and $ \text {Tr}_1: M_{mn}\to M_{n}$ as

$$ \begin{align*}\text{tr}_1(A\otimes B)=\text{tr} (A)B\quad \text{and} \quad \text{Tr}_1(A\otimes B)=\text{tr}_1({\phi}(A\otimes B))\end{align*} $$

for all $A\in M_m$ and $B\in M_n.$ The map $\text {tr}_1$ is also called the partial trace function in quantum science. Then

$$ \begin{align*}\text{Tr}_1(XE_{ii}X^*\otimes B)=\varphi_{i,X}(B).\end{align*} $$

Note that $\text {Tr}_1$ is linear and therefore continuous and the set

$$ \begin{align*}\{XE_{ii}X^*\mid 1\leq i\leq m, X\in M_m \text{ is unitary}\}=\{xx^*\in M_m\mid x^*x=1\}\end{align*} $$

is connected. So, all the maps $\varphi _{i, X}$ are the same, and hence we can rewrite (2.11) as

$$ \begin{align*}{\phi}(XE_{ii}X^*\otimes B)=(Z_XE_{ii}Z_X^*)\otimes \varphi_{2}(B)\quad \text{for all}\quad i=1,\ldots,m\text{ and }B\in M_n,\end{align*} $$

where $\varphi _2$ is the identity map or the transposition map. By the linearity of ${\phi }$ , it follows that

$$ \begin{align*}{\phi}(A\otimes B)=\varphi_1(A)\otimes \varphi_{2}(B)\quad \text{for all}\quad A\in M_m\text{ and }B\in M_n\end{align*} $$

for some linear map $\varphi _1.$

Now we are ready to present the proof of Theorem 2.1.

Proof of Theorem 2.1.

Notice that the $(p,k)$ -norm reduces to the spectral norm when $k=1$ . It was shown in [Reference Fošner, Huang, Li and Sze10] that a linear map $\phi $ preserves the spectral norm of tensor products $A\otimes B$ for all $A\in M_m$ and $B\in M_n$ if and only if $\phi $ has form $A\otimes B\mapsto U(\varphi _1(A)\otimes \varphi _2(B))V$ for some unitary matrices $U,V\in M_{mn}$ , where $\varphi _s$ is the identity map or the transposition map for $s=1,2$ . So we need only consider the case when $k\geq 2$ in the following discussion. Since the sufficiency part is clear, we consider only the necessity part.

Suppose a linear map $\phi : M_{mn}\to M_{mn}$ satisfies (2.1) and $k \geq 2$ . We need the following three claims.

Claim 1 For any unitary matrices $X\in M_m\text { and } Y\in M_n$ , we have

$$ \begin{align*}\phi (XE_{ii}X^*\otimes YE_{jj}Y^*)\perp \phi(XE_{ii}X^*\otimes YE_{ss}Y^*)\end{align*} $$

and

$$ \begin{align*}\phi (XE_{jj}X^*\otimes YE_{ii}Y^*)\perp \phi(XE_{ss}X^*\otimes YE_{ii}Y^*)\end{align*} $$

for any possible $i,j,s$ with $j\neq s$ . Moreover,

$$ \begin{align*}\textrm{rank}(\phi(XE_{ii}X^*\otimes Y(E_{jj}+E_{ss})Y^*))\leq k\end{align*} $$

and

$$ \begin{align*}\textrm{rank}(\phi(X(E_{jj}+E_{ss})X^*\otimes YE_{ii}Y^*))\leq k,\end{align*} $$

for any possible $i,j,s$ with $j\neq s $ .

Proof of Claim 1. For simplicity, we denote

$$ \begin{align*}T=\phi (XE_{ii}X^*\otimes YE_{jj}Y^*)\quad \text{and}\quad S=\phi(XE_{ii}X^*\otimes YE_{ss}Y^*).\end{align*} $$

We need to show

$$ \begin{align*}T\perp S \quad\mathrm{and }\quad \mathrm{rank}(T+S)\leq k.\end{align*} $$

With the assumption in (2.1), we have

$$ \begin{align*}\|\phi(XFX^{*}\otimes YGY^{*})\|_{(p,k)}=\|XFX^{*}\otimes YGY^{*}\|_{(p,k)}=\|F\otimes G\|_{(p,k)}\end{align*} $$

for all $F\in M_m$ and $G\in M_n$ . It follows that

$$ \begin{align*}\|T+xS\|_{(p,k)}^p=\|E_{ii}\otimes (E_{jj}+xE_{ss})\|^{p}_{(p,k)}=1+x^p,\end{align*} $$
$$ \begin{align*}\|T-xS\|_{(p,k)}^p=\|E_{ii}\otimes (E_{jj}-xE_{ss})\|^{p}_{(p,k)}=1+x^p,\end{align*} $$
$$ \begin{align*}\|T\|^p_{(p,k)}=1 \quad \text{and}\quad \|xS\|^p_{(p,k)}=x^p\end{align*} $$

for all $0<x<1$ . We can conclude from the above equalities that

(2.12) $$ \begin{align} \|T+xS\|_{(p,k)}^p+\|T-xS\|_{(p,k)}^p=2\|T\|_{(p,k)}^p+2\|xS\|_{(p,k)}^p\quad \text{for all}\quad 0<x<1. \end{align} $$

Applying Corollary 2.9 with $A=T$ and $B=xS$ , we get

(2.13) $$ \begin{align} \|T+xS\|_{(p,k)}^p+\|T-xS\|_{(p,k)}^p\geq 2\sum_{i=1}^k\lambda_i^{\frac{p}{2}}(T^*T+x^2S^*S) \end{align} $$

for all $0<x<1.$ Since $\|T\|_{(p,k)}^p=\displaystyle \sum _{i=1}^k\lambda _i^{\frac {p}{2}}(T^*T)$ and $\|xS\|_{(p,k)}^p=\displaystyle \sum _{i=1}^k\lambda _i^{\frac {p}{2}}(x^2S^*S),$ it follows from (2.12) and (2.13) that

(2.14) $$ \begin{align} \sum_{i=1}^k\lambda_i^{\frac{p}{2}}(T^*T+x^2S^*S)\leq \sum_{i=1}^k\lambda_i^{\frac{p}{2}}(T^*T)+\sum_{i=1}^k\lambda_i^{\frac{p}{2}}(x^2S^*S) \end{align} $$

for all $0<x<1.$

Replacing the role of $(T,S)$ with $(T^*,S^*)$ in the above argument, we have

(2.15) $$ \begin{align} \sum_{i=1}^k\lambda_i^{\frac{p}{2}}(TT^*+x^2SS^*)\leq \sum_{i=1}^k\lambda_i^{\frac{p}{2}}(TT^*)+\sum_{i=1}^k\lambda_i^{\frac{p}{2}}(x^2SS^*) \end{align} $$

for all $0<x<1.$ We claim that $s_k(T)=0$ . Otherwise, suppose $s_k(T)>0$ . Then, by (2.14) and (2.15), we can apply Corollary 2.12 to conclude that there exist unitary matrices $U,V\in M_{n}$ such that

$$ \begin{align*}UTV=\mathrm{diag}(s_1(T),\ldots,s_{mn}(T))\quad \text{and}\quad USV=0_{k+\ell}\oplus \hat{S},\end{align*} $$

where $\ell $ is the largest integer such that $s_{k+\ell }(T)=s_k(T).$ Thus, there exists a sufficiently small number $t>0$ such that the largest k singular values of $T+tS$ are $s_1(T),\ldots ,s_k(T)$ . Since $\|T\|_{(p,k)}^p=\|E_{ii}\otimes E_{jj}\|_{(p,k)}^p=1,$ we have

$$ \begin{align*}\|T+tS\|_{(p,k)}^p=\sum_{i=1}^ks_i^p(T+tS)=\sum_{i=1}^ks_i^p(T)=\|T\|_{(p,k)}^p=1,\end{align*} $$

which contradicts the fact that

$$ \begin{align*}\|T+xS\|_{(p,k)}^p=\|E_{ii}\otimes (E_{jj}+xE_{ss})\|_{(p,k)}^p=1+x^p\quad \text{for all}\quad 0<x<1.\end{align*} $$

So, our claim is correct, that is, $s_k(T)=0.$

Now, applying Corollary 2.12 again, we have $T\perp S.$ Notice that

$$ \begin{align*}\|T+S\|_{(p,k)}^p=\|T\|^p_{(p,k)}+\|S\|_{(p,k)}^p.\end{align*} $$

Applying Lemma 2.10 on S and T, we have

$$ \begin{align*}\mathrm{rank}(T+S)\leq k.\end{align*} $$

Similarly, we can also show that

$$ \begin{align*}\phi (XE_{jj}X^*\otimes YE_{ii}Y^*)\perp \phi(XE_{ss}X^*\otimes YE_{ii}Y^*)\end{align*} $$

and

$$ \begin{align*}\textrm{rank}(\phi(X(E_{jj}+E_{ss})X^*\otimes YE_{ii}Y^*))\leq k\end{align*} $$

for any possible $i,j,s$ with $j\neq s$ .

Claim 2 For any unitary matrices $X\in M_m\text { and } Y\in M_n$ , we have

$$ \begin{align*}\phi (XE_{ii}X^*\otimes Y(E_{jj}+E_{ss})Y^*)\perp \phi(XE_{tt}X^*\otimes Y(E_{jj}+ E_{ss})Y^*)\end{align*} $$

whenever $i\neq t.$

Proof of Claim 2. For simplicity, we denote

$$ \begin{align*}T=\phi (XE_{ii}X^*\otimes Y(E_{jj}+E_{ss})Y^*) \quad \mathrm{and }\quad S=\phi(XE_{tt}X^*\otimes Y(E_{jj} +E_{ss})Y^*).\end{align*} $$

We need to show $S\perp T.$ Applying Corollary 2.9 on T and $xS$ , we get

(2.16) $$ \begin{align} \|T+xS\|_{(p,k)}^p+\|T-xS\|_{(p,k)}^p\geq 2\sum_{i=1}^k\lambda_i^{\frac{p}{2}}(T^*T+x^2S^*S) \end{align} $$

for all $0<x<1.$ With the assumption in (2.1), we have

  1. (i) $\|T+xS\|_{(p,k)}^p+\|T-xS\|_{(p,k)}^p=2\|T\|_{(p,k)}^p+2\|xS\|_{(p,k)}^p$ for the case $k\geq 4$ ;

  2. (ii) $\|T+xS\|_{(p,k)}^p+\|T-xS\|_{(p,k)}^p=2\|T\|_{(p,k)}^p+\|xS\|_{(p,k)}^p$ for the case $k=3$ ;

  3. (iii) $\|T+xS\|_{(p,k)}^p+\|T-xS\|_{(p,k)}^p=2\|T\|_{(p,k)}^p$ for the case $k=2.$

So we can conclude that for any integer $k\geq 2$ ,

(2.17) $$ \begin{align} \nonumber \|T+xS\|_{(p,k)}^p+\|T-xS\|_{(p,k)}^p&\leq 2\|T\|_{(p,k)}^p+2\|xS\|_{(p,k)}^p \\ &=2\sum_{i=1}^k\lambda_i^{\frac{p}{2}}(T^*T)+2\sum_{i=1}^k\lambda_i^{\frac{p}{2}}(x^2S^*S). \end{align} $$

It follows that

(2.18) $$ \begin{align} \sum_{i=1}^k\lambda_i^{\frac{p}{2}}(T^*T+x^2S^*S)\leq \sum_{i=1}^k\lambda_i^{\frac{p}{2}}(T^*T)+\sum_{i=1}^k\lambda_i^{\frac{p}{2}}(x^2S^*S) \end{align} $$

for all $0<x<1.$ The above observations also hold if $(T,S)$ is replaced by $(T^*,S^*)$ , that is,

(2.19) $$ \begin{align} \sum_{i=1}^k\lambda_i^{\frac{p}{2}}(TT^*+x^2SS^*)\leq \sum_{i=1}^k\lambda_i^{\frac{p}{2}}(TT^*)+\sum_{i=1}^k\lambda_i^{\frac{p}{2}}(x^2SS^*) \end{align} $$

for all $0<x<1.$

If $s_k(T)=0$ , then applying Corollary 2.12, we have $T\perp S$ . Otherwise, $s_k(T)>0$ . Notice that Claim 1 implies $\mathrm {rank}(T)\leq k.$ Thus, by (2.18) and (2.19), we can apply Corollary 2.12 to conclude that there exist unitary matrices $U,V\in M_{mn}$ such that

$$ \begin{align*}UTV=\mathrm{diag}(s_1(T),\ldots,s_{k}(T))\oplus 0_{mn-k}\quad \text{and}\quad USV=0_{k}\oplus \hat{S}.\end{align*} $$

It follows that $T\perp S.$ This completes the proof.

Claim 3 For any unitary matrices $X\in M_m$ and $Y\in M_n$ ,

$$ \begin{align*}\phi(XE_{ii}X^*\otimes YE_{jj}Y^*)\perp \phi(XE_{rr}X^*\otimes YE_{ss}Y^*)\quad \text{for any }(i,j)\neq (r,s).\end{align*} $$

Proof of Claim 3. If $i=r$ or $j=s$ , then applying Claim 1 directly, we have

$$ \begin{align*}\phi(XE_{ii}X^*\otimes Y E_{jj}Y^*)\perp \phi (XE_{rr}X^*\otimes YE_{ss}Y^*).\end{align*} $$

Next, we suppose that $i\neq r$ and $j\neq s$ . With Claim 1, we have

(2.20) $$ \begin{align} \phi(XE_{ii}X^*\otimes Y E_{jj}Y^*)\perp \phi (XE_{ii}X^*\otimes YE_{ss}Y^*) \end{align} $$

and

(2.21) $$ \begin{align} \phi(XE_{rr}X^*\otimes Y E_{jj}Y^*)\perp \phi (XE_{rr}X^*\otimes YE_{ss}Y^*). \end{align} $$

With Claim 2, we have

(2.22) $$ \begin{align} \phi(XE_{ii}X^*\otimes Y (E_{jj}+E_{ss})Y^*)\perp \phi (XE_{rr}X^*\otimes Y(E_{jj}+E_{ss})Y^*). \end{align} $$

Applying Lemma 2.6, we conclude from (2.20) and (2.22) that

(2.23) $$ \begin{align} \phi(XE_{ii}X^*\otimes Y E_{jj}Y^*)\perp \phi (XE_{rr}X^*\otimes Y(E_{jj}+E_{ss})Y^*). \end{align} $$

Then, applying Lemma 2.6 again, we can conclude from (2.21) and (2.23) that

$$ \begin{align*}\phi(XE_{ii}X^*\otimes Y E_{jj}Y^*)\perp \phi (XE_{rr}X^*\otimes YE_{ss}Y^*).\end{align*} $$

Now we prove that $\phi $ has the desired form (2.2). For any unitary matrix $Y\in M_n$ , applying Claims 1 and 3, we know

$$ \begin{align*}\mathscr{F}=\{\phi(E_{ii}\otimes YE_{jj}Y^*):i=1,\ldots,m\text{ and }j=1\ldots,n\}\end{align*} $$

is a set of $mn$ orthogonal matrices in $M_{mn}$ . By Claim 1, each matrix in $\mathscr {F}$ has exactly one nonzero singular value, which equals 1. Thus, there exist unitary matrices $U_Y,V_Y\in M_{mn}$ such that

(2.24) $$ \begin{align} \phi(E_{ii}\otimes YE_{jj}Y^*)=U_Y(E_{ii}\otimes E_{jj})V_Y^* \end{align} $$

for all $ i=1,\ldots , m \text { and } j=1,\ldots , n.$ Without loss of generality, we may assume that $U_I=V_I=I_{mn}$ , i.e.,

(2.25) $$ \begin{align} \phi(E_{ii}\otimes E_{jj})=E_{ii}\otimes E_{jj} \end{align} $$

for all $i=1,\ldots , m \text { and } j=1,\ldots , n.$ By (2.24) and (2.25), we have:

  1. (i) $I_{mn}=\phi (I_m\otimes I_n)=U_Y(I_m\otimes I_n)V_Y^*$ ;

  2. (ii) $E_{ii}\otimes I_n=\phi (E_{ii}\otimes I_n)=U_Y(E_{ii}\otimes I_n)V_Y^* \text { for all } i=1,\ldots ,m.$

It follows that $U_Y=V_Y$ and $U_Y$ commutes with $E_{ii}\otimes I_n$ for all $i=1,\ldots , m$ . Therefore, $U_Y$ commutes with $E_{11}\otimes I_n+2E_{22}\otimes I_n+\cdots +mE_{mm}\otimes I_n,$ which implies that $U_Y=\bigoplus \limits _{i=1}^{m}U_{i,Y}$ with unitary matrices $U_{i,Y}\in M_n$ . It follows that

$$ \begin{align*}\phi(E_{ii}\otimes YE_{jj}Y^*)=E_{ii}\otimes U_{i,Y}E_{jj}U_{i,Y}^*.\end{align*} $$

So far, we have showed that for any unitary matrix $Y\in M_n$ , there exists a unitary matrix $U_{i,Y}\in M_n$ depending on i and Y such that

$$ \begin{align*}\phi(E_{ii}\otimes YE_{jj}Y^*)=E_{ii}\otimes U_{i,Y}E_{jj}U_{i,Y}^*\quad \text{for}\quad j=1,\ldots,n.\end{align*} $$

By the linearity of $\phi $ , we conclude from the above equation that for any $i=1,\ldots , m$ , there exists a linear map $\psi _i$ such that

$$ \begin{align*}\phi(E_{ii}\otimes B)=E_{ii}\otimes \psi_{i}( B) \quad \text{for all}\quad B \in M_n.\end{align*} $$

Let $\hat {k}=\min \{k,n\}$ . Then it is easy to check that

$$ \begin{align*}\|\psi_i(B)\|_{(p,\hat{k})}=\|E_{ii}\otimes \psi_i(B)\|_{(p,k)}=\|E_{ii}\otimes B\|_{(p,k)}=\|B\|_{(p,\hat{k})}\end{align*} $$

for all $B\in M_n$ . That is, $\psi _i$ is a linear map on $M_n$ preserving the $(p,\hat {k})$ -norm. Thus, by Theorem 1 in [Reference Li and Tsing23], $\psi _i$ has form $B\mapsto W_i B\widetilde {W}_i$ or $B\mapsto W_i B^T\widetilde {W}_i$ for some unitary matrices $W_i, \widetilde {W}_i\in M_n.$ Let $W=\bigoplus \limits _{i=1}^m W_i$ and $\widetilde {W}=\bigoplus \limits _{i=1}^m\widetilde {W}_i$ . It follows that for any ${i=1,\ldots , m,}$

$$ \begin{align*}\phi(E_{ii}\otimes B)=W(E_{ii}\otimes \varphi_i (B))\widetilde{W}\quad \text{for all}\quad B\in M_n,\end{align*} $$

where $\varphi _i$ is the identity map or the transposition map. Recall that $I_{mn}=\phi (I_m\otimes I_n). $ Thus, we have $\widetilde {W} = W^*$ .

Applying Claim 3 again, we can repeat the same argument above to show that for any unitary matrix $X \in M_m$ and any integer $1\leq i\leq m$ , there exists a unitary matrix $W_X$ such that

$$ \begin{align*} \phi(XE_{ii}X^*\otimes B)=W_X(E_{ii}\otimes \varphi_{i,X} (B)){W}_X^*\quad \text{for all}\quad B\in M_n, \end{align*} $$

where $\varphi _{i,X}$ is the identity map or the transposition map. We may further assume that $W_I=I_{mn}$ . Then, applying Lemma 2.13, we have

$$ \begin{align*}\phi(A\otimes B)=\varphi_1(A)\otimes \varphi_2(B) \quad \text{for all}\quad A\in M_m\text{ and }B\in M_n,\end{align*} $$

where $\varphi _2$ is the identity map or the transposition map and $\varphi _1$ is a linear map on $M_m$ . Let $\widetilde {k}=\min \{k,m\}.$ It is easy to verify that $\varphi _1$ is a linear map on $M_m$ preserving the $(p,\widetilde {k})$ -norm. Hence, $\varphi _1$ also has the form $A\mapsto UAV$ or $A\mapsto UA^TV$ for some unitary matrices $U, V\in M_m$ . This completes the proof.

3 Multipartite system

In this section, we extend Theorem 2.1 to multipartite systems. The proof of the following lemma can be found in the proof of Theorem 3.1 in [Reference Fošner, Huang, Li and Sze10]. For completeness, we present it as follows.

Lemma 3.1 Given an integer $m\geq 2$ , let $n_i \geq 2 $ be integers for $i=1,\ldots ,m$ and ${N=\prod \limits _{i=1}^{m}n_i}$ . Let $\phi :M_N\to M_N$ be a linear map. Suppose for any unitary matrices $X_i\in M_{n_i}$ and any integers $ 1\leq j_i\leq n_i$ with $ 1\leq i\leq m-1$ , there exists a unitary matrix $W_X\in M_N$ depending on $X=(X_1,\ldots ,X_{m-1})$ such that

(3.1) $$ \begin{align} \phi\left (\bigotimes\limits_{i=1}^{m-1}X_iE_{j_ij_i} X_i^*\otimes B\right )= W_{X}\left (\bigotimes\limits_{i=1}^{m-1}E_{j_ij_i}\otimes\varphi_{j_1,\ldots,j_{m-1},X}(B)\right )W_{X}^* \end{align} $$

for all $B\in M_{n_m},$ where $\varphi _{j_1,\ldots ,j_{m-1},X}$ is the identity map or the transposition map and $W_X=I_N$ when $X=(I_{n_1},\ldots ,I_{n_{m-1}})$ . Then

$$ \begin{align*}\phi(A_1\otimes \cdots\otimes A_{m-1}\otimes B)=\varphi_1(A_1\otimes \cdots \otimes A_{m-1})\otimes \varphi_2(B),\end{align*} $$

where $\varphi _1$ is a linear map and $\varphi _2$ is the identity map or the transposition map.

Proof Considering all symmetric real matrices as in the proof of Lemma 2.13, one can conclude that there exists some unitary matrix $Z_{X}$ such that

(3.2) $$ \begin{align} \phi\left (\bigotimes\limits_{i=1}^{m-1}X_iE_{j_ij_i}X_i^*\otimes B\right )= \left(Z_{X}\left (\bigotimes\limits_{i=1}^{m-1}E_{j_ij_i}\right )Z_{X}^*\right) \otimes \varphi_{j_1,\ldots,j_{m-1},X}(B) \end{align} $$

for all $B\in M_{n_m}$ and integers $1\leq j_i\leq n_i $ with $ 1\leq i\leq m-1$ . Define linear maps $\mathrm {tr_1}:M_N\to M_{n_m}$ and $\mathrm {Tr_1}:M_N\to M_{n_m}$ by

$$ \begin{align*}\mathrm{tr_1}(A\otimes B)=\mathrm{tr}(A)B\quad \text{and} \quad \mathrm{Tr_1}(A\otimes B)=\mathrm{tr_1}(\phi(A\otimes B ))\end{align*} $$

for all $A\in M_{n_1\cdots n_{m-1}}$ and $B\in M_{n_m}$ . Then

$$ \begin{align*}\mathrm{Tr_1}\left(\bigotimes\limits_{i=1}^{m-1}X_iE_{j_ij_i}X_i^*\otimes B\right )=\varphi_{j_1,\ldots,j_{m-1},X}(B).\end{align*} $$

Notice that $\text {Tr}_1$ is linear and therefore continuous. Besides, the set

(3.3) $$ \begin{align} \nonumber \left\{\bigotimes\limits_{i=1}^{m-1}X_iE_{j_ij_i}X_i^*\mid 1\leq j_i\leq n_i \text{ and } X_i\in M_{n_i} \text{ is unitary for }i=1,\ldots,m-1\right\} \\ =\left\{\bigotimes\limits_{i=1}^{m-1}x_ix_i^*\mid x_i\in \mathbb{C}^{n_i} \text{ with } x_i^*x_i=1 \text{ for }i=1,\ldots,m-1\right\} \end{align} $$

is connected. So, all the maps $\varphi _{j_1,\ldots ,j_{m-1},X}$ are the same. Then (3.2) can be rewritten as

$$ \begin{align*}\phi\left (\bigotimes\limits_{i=1}^{m-1}X_iE_{j_ij_i}X_i^*\otimes B\right )= \left(Z_{X}\left (\bigotimes\limits_{i=1}^{m-1}E_{j_ij_i}\right )Z_{X}^*\right) \otimes \varphi_{2}(B),\end{align*} $$

where $\varphi _2$ is the identity map or the transposition map. With the linearity of $\phi $ , it follows that

$$ \begin{align*}\phi(A_1\otimes \cdots \otimes A_{m-1}\otimes B)=\varphi_1(A_1\otimes \cdots \otimes A_{m-1})\otimes \varphi_2(B)\end{align*} $$

for some linear map $\varphi _1.$

Theorem 3.2 Given an integer $m\geq 2$ , let $n_i \geq 2 $ be integers for $i=1,\ldots ,m$ and $N=\prod \limits _{i=1}^{m}n_i.$ Then, for any real number $p>2$ and any positive integer $k \leq N,$ a linear map $\phi : M_N \to M_N$ satisfies

(3.4) $$ \begin{align} \|\phi(A_1\otimes\cdots\otimes A_m)\|_{(p,k)}=\|A_1\otimes\cdots\otimes A_m\|_{(p,k)} \end{align} $$

for all $A_i\in M_{n_i}, i=1,\ldots ,m$ , if and only if there exist unitary matrices $U,V \in M_N$ such that

(3.5) $$ \begin{align} \phi(A_1\otimes\cdots\otimes A_m)=U(\varphi_1(A_1)\otimes\cdots\otimes \varphi_m(A_m))V \end{align} $$

for all $A_i\in M_{n_i},i=1,\ldots ,m,$ where $\varphi _i$ is the identity map or the transposition map $A\mapsto A^T $ for $i=1,\ldots ,m$ .

Proof By Theorem 3.2 of [Reference Fošner, Huang, Li and Sze10], we know the result holds for $k=1$ . So we may assume $k\ge 2$ .

We use induction on m. By Theorem 2.1, the result holds for $m=2$ . Now suppose that $m\geq 3$ and the result holds for any $(m-1)$ -partite system. We need to show that the result holds for any m-partite system.

We first show that for any unitary matrices $X_i\in M_{n_i}, i=1,\ldots ,m,$

(3.6) $$ \begin{align} \phi(X_1E_{i_1i_1}X_1^*\otimes\cdots \otimes X_mE_{i_mi_m}X_m^*)\perp\phi(X_1E_{j_1j_1}X_1^*\otimes \cdots\otimes X_mE_{j_mj_m}X_m^*) \end{align} $$

for any $(i_1,\ldots ,i_m)\neq (j_1,\ldots ,j_m).$ Without loss of generality, we need only to prove that (3.6) holds when $X_i=I_{n_i}$ for $i=1,\ldots ,m$ . By Lemma 2.6, it suffices to show that for any integer $1\leq s\leq m,$ we have

(3.7) $$ \begin{align} \nonumber &\phi\left (\bigotimes\limits_{u=1}^{s-1}(E_{i_ui_u}+E_{j_uj_u})\otimes E_{i_si_s}\otimes \bigotimes\limits_{u=s+1}^mE_{i_ui_u}\right ) \\ &\qquad\qquad\qquad\perp \phi\left (\bigotimes\limits_{u=1}^{s-1}(E_{i_ui_u}+E_{j_uj_u})\otimes E_{j_sj_s}\otimes \bigotimes\limits_{u=s+1}^mE_{i_ui_u}\right ) \end{align} $$

for all $\mathrm {i}=(i_1,\ldots ,i_m)$ and $\mathrm {j}=(j_1,\ldots ,j_m)$ with $i_u\neq j_u$ for $ u=1,\ldots , s.$ Denote by $A_s(\mathrm {i,j})$ and $B_s(\mathrm {i,j})$ the two matrices in (3.7) accordingly. It is easy to check that for any integer $1\leq s\leq m$ and real number $0<x<1$ ,

$$ \begin{align*} &\|A_s(\mathrm{i,j})+xB_s(\mathrm{i,j})\|_{(p,k)}^p+\|A_s(\mathrm{i,j}) -xB_s(\mathrm{i,j})\|_{(p,k)}^p\qquad\\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\leq 2\|A_s(\mathrm{i,j})\|^p_{(p,k)}+2\|xB_s(\mathrm{i,j})\|^p_{(p,k)}.\notag \end{align*} $$

Then, applying the same argument as in the proof of Theorem 2.1, we conclude that for any integer $1\leq s \leq m$ and real number $0<x<1$ ,

(3.8) $$ \begin{align} \nonumber &\sum_{i=1}^k\lambda_i^{\frac{p}{2}}\big(A_s^*(\mathrm{i,j})A_s(\mathrm{i,j}\big)+x^2B_s^*(\mathrm{i,j})B_s(\mathrm{i,j})\big) \\ &\qquad\qquad\qquad\qquad\leq \sum_{i=1}^k\lambda_i^{\frac{p}{2}}\big(A^*_s(\mathrm{i,j})A_s(\mathrm{i,j})\big)+\sum_{i=1}^k\lambda_i^{\frac{p}{2}}\big(x^2B_s^*(\mathrm{i,j})B_s(\mathrm{i,j})\big) \end{align} $$

and

(3.9) $$ \begin{align} \nonumber& \sum_{i=1}^k\lambda_i^{\frac{p}{2}}\big(A_s(\mathrm{i,j})A^*_s(\mathrm{i,j})+x^2B_s(\mathrm{i,j})B_s^*(\mathrm{i,j})\big) \\&\qquad\qquad\qquad\qquad\leq \sum_{i=1}^k\lambda_i^{\frac{p}{2}}\big(A_s(\mathrm{i,j})A^*_s(\mathrm{i,j})\big)+\sum_{i=1}^k\lambda_i^{\frac{p}{2}}\big(x^2B_s(\mathrm{i,j})B^*_s(\mathrm{i,j})\big) \end{align} $$

for all $\mathrm {i}=(i_1,\ldots ,i_m)$ and $\mathrm {j}=(j_1,\ldots ,j_m)$ with $i_u\neq j_u$ for $ u=1,\ldots , s.$ Now we distinguish two cases.

Case 1. Suppose $k>2^{m-1}$ . Then

(3.10) $$ \begin{align} \|A_s(\mathrm{i,j})+xB_s(\mathrm{i,j})\|_{(p,k)}^p=2^{s-1}+ a_s x^p\quad \text{ for}\quad 0<x<1, \end{align} $$

where $a_s=2^{s-1}$ for $s=1,\ldots ,m-1$ and $a_m=\min \{k-2^{m-1},2^{m-1}\}$ . We claim that

(3.11) $$ \begin{align} s_k(A_{s}(\mathrm{i,j}))=0 \mathrm{\quad for}\quad s=1,\ldots,m. \end{align} $$

Otherwise, $s_k(A_s(\mathrm {i,j}))>0$ for some $1\leq s\leq m$ . Then, by (3.8) and (3.9), we can use the same argument as in Claim 1 in the proof of Theorem 2.1 to conclude that there exists a sufficiently small $x>0$ such that

$$ \begin{align*}\|A_s(\mathrm{i,j})+xB_s(\mathrm{i,j})\|_{(p,k)}^p=\|A_s(\mathrm{i,j})\|_{(p,k)}^p=2^{s-1},\end{align*} $$

which contradicts (3.10). Thus, (3.11) holds. Applying Corollary 2.12, we have

$$ \begin{align*}A_s(\mathrm{i,j})\perp B_s(\mathrm{i,j}) \mathrm{\quad for}\quad s=1,\ldots,m.\end{align*} $$

Case 2. Suppose $k\leq 2^{m-1}$ . Let $s_0$ be the integer such that $2^{s_0-1}< k\leq 2^{s_0}.$ We can use the same argument as in Case 1 to show that for any integer $1\leq s\leq s_0$ ,

(3.12) $$ \begin{align} A_{s}(\mathrm{i,j})\perp B_s(\mathrm{i,j})\quad\text{ and}\quad s_k(A_s(\mathrm{i,j}))=0 \end{align} $$

for all $\mathrm {i}=(i_1,\ldots ,i_m)$ and $\mathrm {j}=(j_1,\ldots ,j_m)$ with $i_u\neq j_u$ for $u=1,\ldots ,s.$

Next, we use induction on s to prove that for any $s_0+1\leq s \leq m, \mathrm {i}=(i_1,\ldots ,i_m )$ and $\mathrm {j}=(j_1,\ldots ,j_m)$ with $i_u\neq j_u$ for $u=1,\ldots , s,$ there exist unitary matrices $U,V\in M_N$ depending on s and $(\mathrm {i,j})$ such that

(3.13) $$ \begin{align} UA_{s}(\mathrm{i,j})V=I_{2^{s-1}}\oplus 0_{N-2^{s-1}}\quad \text{and}\quad A_{s}(\mathrm{i,j})\perp B_s(\mathrm{i,j}). \end{align} $$

First, by (3.12), we have $A_{s_0}(\mathrm {i,j})\perp B_{s_0}(\mathrm {i,j})$ and there exist unitary matrices $U,V\in M_N$ and an integer $0\leq r< k$ such that

$$ \begin{align*}UA_{s_0}(\mathrm{i,j})V=\mathrm{diag}(a_1,\ldots,a_r)\oplus 0\end{align*} $$

and

$$ \begin{align*}UB_{s_0}(\mathrm{i,j})V=0_r\oplus \mathrm{diag}(b_{r+1},\ldots,b_N),\end{align*} $$

with $a_1\geq \cdots \geq a_r>0$ and $b_{r+1}\geq \cdots \geq b_N\geq 0$ . If $a_1>1$ , then by (3.12), applying Lemma 2.6, we have $s_1\Big (\phi \Big (\bigotimes \limits _{u=1}^mE_{i_ui_u}\Big )\Big )>1$ for some $(i_1,\ldots ,i_m)$ . It follows that $\Big \|\phi \Big (\bigotimes \limits _{u=1}^mE_{i_ui_u}\Big )\Big \|_{(p,k)}>1$ , which contradicts (3.4). Thus, $a_1\leq 1$ , and similarly $b_{r+1}\leq 1.$ Then we have

(3.14) $$ \begin{align} \sum_{j=1}^ra_j^p+\sum_{j=r+1}^kb_j^p\leq k. \end{align} $$

Clearly, $a_1\geq \cdots \geq a_r\geq xb_{r+1}\geq \cdots \geq xb_k$ are the largest k singular values of $A_{s_0}(\mathrm {i,j})+xB_{s_0}(\mathrm {i,j})$ for all $0<x\leq \frac {a_r}{b_{r+1}}$ . Hence,

$$ \begin{align*}\|A_{s_0}(\mathrm{i,j})+xB_{s_0}(\mathrm{i,j})\|^p_{(p,k)}=\sum_{j=1}^ra_j^p+x^p\sum_{j=r+1}^kb_j^p\quad \text{for all}\quad 0< x\leq \frac{a_r}{b_{r+1}}.\end{align*} $$

On the other hand, with (3.4), we have

$$ \begin{align*}\|A_{s_0}(\mathrm{i,j})+xB_{s_0}(\mathrm{i,j})\|^p_{(p,k)}=2^{s_0-1}+x^p(k-2^{s_0-1})\quad \text{for all}\quad 0<x\leq 1.\end{align*} $$

It follows from the above two equations that

$$ \begin{align*}\displaystyle\sum_{j=1}^ra_j^p=2^{s_0-1} \quad\mathrm{and}\quad \displaystyle\sum_{j=r+1}^kb_j^p=k-2^{s_0-1}.\end{align*} $$

Therefore, $ \sum _{j=1}^ra_j^p+ \sum _{j=r+1}^kb_j^p=k,$ i.e., the equality in (3.14) holds, which implies that $a_j=1$ for $j=1,\ldots ,r$ . Notice that $\|A_{s_0}(\mathrm {i,j})\|_{(p,k)}^p=2^{s_0-1}$ . Thus, $r=2^{s_0-1},$ that is, $UA_{s_0}(\mathrm {i,j})V=I_{2^{s_0-1}}\oplus 0_{N-2^{s_0-1}}$ . Hence, (3.13) holds for $s_0$ .

Suppose that (3.13) holds for $s-1$ with $s_0< s\leq m.$ We will show that (3.13) also holds for s. Notice that $A_{s}(\mathrm {i,j})=A_{s-1}(\mathrm {\hat {i},\hat {j}})+B_{s-1}(\mathrm {\hat {i},\hat {j}})$ for some $\mathrm {\hat {i}}=(\hat {i}_1,\ldots ,\hat {i}_m)$ and $\mathrm {\hat {j}}=(\hat {j}_1,\ldots ,\hat {j}_m)$ with $\hat {i}_u\neq \hat {j}_u$ for $u=1,\ldots , s-1.$ By our assumption, we have

$$ \begin{align*}UA_{s-1}(\mathrm{\hat{i},\hat{j}})V=I_{2^{s-2}}\oplus 0_{N-2^{s-2}}\quad\text{and}\quad UB_{s-1}(\mathrm{\hat{i},\hat{j}})V=0_{2^{s-2}}\oplus I_{2^{s-2}}\oplus 0_{N-2^{s-1}}\end{align*} $$

for some unitary matrices $U,V\in M_N$ . It follows that

$$ \begin{align*}UA_{s}(\mathrm{i,j})V=I_{2^{s-1}}\oplus 0_{N-2^{s-1}}.\end{align*} $$

Then, with (3.8) and (3.9), we apply Corollary 2.12 to conclude that

$$ \begin{align*}UB_s(\mathrm{i,j})V=0_{2^{s-1}}\oplus \hat{B}\end{align*} $$

for some $\hat {B}\in M_{N-2^{s-1}}$ . It follows that $A_s(\mathrm {i,j})\perp B_s(\mathrm {i,j}).$ Now we have proved that (3.13) holds for s. Then we can conclude from the above discussion that for any $s=1,\ldots ,m$ ,

$$ \begin{align*}A_s(\mathrm{i,j})\perp B_s(\mathrm{i,j})\end{align*} $$

for all $\mathrm {i}=(i_1,\ldots ,i_m)$ and $\mathrm {j}=(j_1,\ldots ,j_m)$ with $i_u\neq j_u$ for $ u=1,\ldots , s,$ that is, (3.6) holds.

It follows that for any unitary matrix $X_m\in M_{n_m}$ , there exist unitary matrices $U_{X_m}$ and $V_{X_m}$ such that

(3.15) $$ \begin{align} \phi\left(\bigotimes\limits_{i=1}^{m-1} E_{j_ij_i}\otimes X_mE_{j_{m}j_{m}}X_m^*\right)= U_{X_m}(E_{j_1j_1}\otimes \cdots \otimes E_{j_mj_m})V_{X_m}^* \end{align} $$

for all $ j_i=1,\ldots , n_i$ with $1 \leq i\leq m.$ For simplicity, we may assume that $U_{I}=V_{I}=I$ , i.e.,

(3.16) $$ \begin{align} \phi(E_{j_1j_1}\otimes\cdots\otimes E_{j_{m}j_{m}})= E_{j_1j_1}\otimes \cdots \otimes E_{j_mj_m}. \end{align} $$

Then we have $\phi (I_N)=I_N$ . Applying a similar argument as in the last two paragraphs of the proof of Theorem 2.1, one can conclude from (3.15) and (3.16) that there are unitary matrices $W,\widetilde {W}\in M_N$ such that for all $ j_i=1,\ldots ,n_i$ with $1\leq i\leq m-1$ ,

$$ \begin{align*}\phi\left(\bigotimes\limits_{i=1}^{m-1}E_{j_ij_i}\otimes B\right)= W\left(\bigotimes\limits_{i=1}^{m-1}E_{j_ij_i}\otimes \varphi_{j_1,\ldots,j_{m-1}}(B)\right)\widetilde{W},\end{align*} $$

where $\varphi _{j_1,\ldots ,j_{m-1}}$ is the identity map or the transposition map. It follows that ${\phi (I_N)=W\widetilde {W}}$ . Recall that $\phi (I_N)=I_N$ and W and $\widetilde {W}$ are both unitary matrices. We have $\widetilde {W}=W^*.$

Following a similar argument as above, one can show that for any ${X=(X_1,\ldots ,X_{m-1})}$ and any integer $ j_i=1,\ldots , n_i$ with $1\leq i\leq m-1$ , there exists a unitary matrix $W_X\in M_N$ such that

(3.17) $$ \begin{align} \phi\left (\bigotimes\limits_{i=1}^{m-1}X_iE_{j_ij_i}X_i^*\otimes B\right )=W_X\left (\bigotimes\limits_{i=1}^{m-1}E_{j_ij_i}\otimes\varphi_{j_1,\ldots,j_{m-1},X}(B)\right )W_X^* \end{align} $$

for all $B\in M_{n_m},$ where $\varphi _{j_1,\ldots ,j_{m-1},X}$ is the identity map or transposition map. Denote $\mathrm {I}=(I_{n_1},\ldots ,I_{n_{m-1}}).$ For simplicity, we may further assume that $W_{\mathrm {I}}=I_N$ , i.e., for any integer $j_i=1,\ldots , n_i$ with $1\leq i\leq m-1$ ,

$$ \begin{align*} \phi\left (\bigotimes\limits_{i=1}^{m-1}E_{j_ij_i}\otimes B\right )=\bigotimes\limits_{i=1}^{m-1}E_{j_ij_i}\otimes\varphi_{j_1,\ldots,j_{m-1},\mathrm{I}}(B)\quad \text{for all}\quad B\in M_{n_m}. \end{align*} $$

Then we apply Lemma 3.1 to conclude that

$$ \begin{align*}\phi(A_1\otimes \cdots\otimes A_{m-1}\otimes B)=\psi(A_1\otimes \cdots\otimes A_{m-1})\otimes \varphi_m(B)\end{align*} $$

for all $B\in M_{n_m}$ and $A_i\in M_{n_i}$ , $1\leq i\leq m-1$ , where $\varphi _m$ is the identity map or the transposition map and $\psi $ is a linear map. Let $\hat {k}=\min \Big \{k,\prod \limits _{i=1}^{m-1}n_i\Big \}.$ It is easy to check that

$$ \begin{align*}\|\psi(A_1\otimes\cdots \otimes A_{m-1})\|_{(p,\hat{k})}=\|A_1\otimes\cdots \otimes A_{m-1}\|_{(p,\hat{k})}\end{align*} $$

for all $A_i \in M_{n_i}, i=1,\ldots ,m-1.$ Then, by the induction hypothesis, we conclude that there exist unitary matrices $\widetilde {U}$ and $\widetilde {V} $ such that

$$ \begin{align*}\psi(A_1\otimes\cdots \otimes A_{m-1})=\widetilde{U}(\varphi_1(A_1)\otimes \cdots\otimes \varphi_{m-1}(A_{m-1}))\widetilde{V},\end{align*} $$

where $\varphi _i$ is the identity map or the transposition map for $i=1,\ldots ,m-1.$ Therefore, $\phi $ has the desired form and the proof is completed.

4 Conclusion and remarks

In this paper, we determined the structures of linear maps on $M_{mn}$ that preserve the $(p,k)$ -norms of tensor products of matrices for $p>2$ and $1\leq k\leq mn$ . Our study generalized the results in [Reference Fošner, Huang, Li and Sze10] concerning the maps preserving the Ky Fan k-norm and Schatten p-norms. Furthermore, we have also extended the result on bipartite systems to multipartite systems using mathematical induction.

The proofs of the main results in [Reference Fošner, Huang, Li and Sze10] heavily rely on Lemmas 2.2 and 2.7 from the same paper, which specifically address Ky Fan k-norms and Schatten p-norms. However, it is important to note that these two lemmas are not applicable for $(p,k)$ -norms. To overcome this limitation, we derived two inequalities involving the eigenvalues of Hermitian matrices and $(p,k)$ -norms, which are presented in Lemma 2.7 and its corollary. These two inequalities play a crucial role in our proofs.

To characterize linear maps that preserve the $(p, k)$ -norms of tensor products of matrices for $1< p\leq 2$ , we attempted to derive an analogue to the inequality (2.4). However, as demonstrated in Remark 2.8, this analogous inequality does not hold in general. Consequently, the techniques employed in this paper are not able to address the case $1<p\leq 2$ . Therefore, novel approaches need to be introduced to tackle this particular scenario.

Footnotes

The research of Huang was supported by the National Natural Science Foundation of China (Grant No. 12171323), the Science and Technology Foundation of Shenzhen City (Grant No. JCYJ20190808174211224), and the Guangdong Basic and Applied Basic Research Foundation (Grant No. 2022A1515011995). The research of Sze was supported by a Hong Kong RGC grant PolyU 15305719 and a PolyU Internal Research Fund 4-ZZKU.

References

Arazy, J., The isometries of ${C}_p$ . Israel J. Math. 22(1975), 247256.CrossRefGoogle Scholar
Chooi, W. L. and Kwa, K. H., Classical adjoint commuting and determinant preserving linear maps on Kronecker products of Hermitian matrices . Linear Multilinear Algebra 68(2020), 869885.CrossRefGoogle Scholar
Chubb, C. T. and Flammia, S. T., Approximate symmetries of Hamiltonians . J. Math. Phys. 58(2017), 082202.CrossRefGoogle Scholar
Ding, Y., Fošner, A., Xu, J., and Zheng, B., Linear maps preserving determinant of tensor products of Hermitian matrices . J. Math. Anal. Appl. 446(2017), 11391153.CrossRefGoogle Scholar
Doan, X. V. and Vavasis, S., Finding the largest low-rank clusters with Ky Fan $2-k$ -norm and ${\ell}_1$ -norm . SIAM J. Optim. 26(2016), 274312.CrossRefGoogle Scholar
Duffner, M. A. and Cruz, H. F., Rank nonincreasing linear maps preserving the determinant of tensor product of matrices . Linear Algebra Appl. 510(2016), 186191.CrossRefGoogle Scholar
Fošner, A., Huang, Z., Li, C. K., Poon, Y. T., and Sze, N. S., Linear maps preserving the higher numerical range of tensor product of matrices . Linear Multilinear Algebra 62(2014), 776791.CrossRefGoogle Scholar
Fošner, A., Huang, Z., Li, C. K., and Sze, N. S., Linear preservers and quantum information science . Linear Multilinear Algebra 61(2013), 13771390.CrossRefGoogle Scholar
Fošner, A., Huang, Z., Li, C. K., and Sze, N. S., Linear maps preserving numerical radius of tensor product of matrices . J. Math. Anal. Appl. 407(2013), 183189.CrossRefGoogle Scholar
Fošner, A., Huang, Z., Li, C. K., and Sze, N. S., Linear maps preserving Ky Fan norms and Schatten norms of tensor product of matrices . SIAM J. Matrix Anal. Appl. 34(2013), 673685.CrossRefGoogle Scholar
Friedland, S., Li, C. K., Poon, Y. T., and Sze, N. S., The automorphism group of separable states in quantum information theory . J. Math. Phys. 52(2011), 042203.CrossRefGoogle Scholar
Grone, R., Certain isometries of rectangular complex matrices . Linear Algebra Appl. 29(1980), 161171.CrossRefGoogle Scholar
Grone, R. and Marcus, M., Isometries of matrix algebras . J. Algebra 47(1977), 180189.CrossRefGoogle Scholar
Horn, R. A. and Johnson, C. R., Hadamard and conventional submultiplicativity for unitarily invariant norms on matrices . Linear Multilinear Algebra 20(1987), 91106.CrossRefGoogle Scholar
Huang, Z., Shi, S., and Sze, N. S., Linear rank preservers of tensor products of rank one matrices . Linear Algebra Appl. 508(2016), 255271.CrossRefGoogle Scholar
Jiang, W., Liu, J., Qi, H., and Dai, Q., Robust subspace segmentation via nonconvex low rank representation . Inf. Sci. 340(2016), 144158.CrossRefGoogle Scholar
Johnston, N., Characterizing operations preserving separability measures via linear preserver problems . Linear Multilinear Algebra 59(2011), 11711187.CrossRefGoogle Scholar
Johnston, N. and Kribs, D. W., A family of norms with applications in quantum information theory . J. Math Phy. 51(2010), 082202.CrossRefGoogle Scholar
Kidman, K., Stochastic matrices and unitarily invariant norms. Ph.D. Thesis, University of California, Santa Barbara, 1983.Google Scholar
Li, C. K. and Pierce, S., Linear preserver problems . Amer. Math. Mon. 108(2001), 591605.CrossRefGoogle Scholar
Li, C. K., Poon, Y. T., and Sze, N. S., Linear preservers of tensor product of unitary orbits, and product numerical range . Linear Algebra Appl. 438(2013), 37973803.CrossRefGoogle Scholar
Li, C. K., Šemrl, P., and Sourour, A. R., Isometries for Ky Fan norms on block triangular matrix algebras . Arch. Math. 81(2003), 175181.CrossRefGoogle Scholar
Li, C. K. and Tsing, N. K., Some isometries of rectangular complex matrices . Linear Multilinear Algebra 23(1988), 4753.CrossRefGoogle Scholar
Marcus, M., Kidman, K., and Sandy, M., Unitarily invariant generalized matrix norms and Hadamard products . Linear Multilinear Algebra 16(1984), 197213.CrossRefGoogle Scholar
McCarthy, C. A., ${C}_p$ . Israel J. Math. 5(1967), 249271.CrossRefGoogle Scholar
Molnár, L., Selected preserver problems on algebraic structures of linear operators and on function spaces, Lecture Notes in Mathematics, 1895, Springer, Berlin, 2007 Google Scholar
Russo, B., Trace preserving mappings of matrix algebras . Duke Math. J. 36(1969), 297300.CrossRefGoogle Scholar
Russo, B., Isometries of the trace class . Proc. Amer. Math. Soc. 23(1969), 213.Google Scholar
Schur, I., Einige Bemerkungen zur Determinaten Theorie . S.-B. Preuss. Akad. Wiss. Berlin 25(1925), 454463.Google Scholar
Tanaka, M. and Nakata, K., Positive definite matrix approximation with condition number constraint . Optim. Lett. 8(2014), 939947.CrossRefGoogle Scholar
Zhan, X., Matrix theory, Graduate Studies in Mathematics, 147, American Mathematical Society, Providence, RI, 2013.CrossRefGoogle Scholar