Linear maps preserving r-potents of tensor products of matrices

Linear Algebra and its Applications 520 (2017) 67–76

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Linear Algebra and its Applications www.elsevier.com/locate/laa

Linear maps preserving r-potents of tensor products of matrices Jinli Xu a,∗ , Ajda Fošner b , Baodong Zheng c , Yuting Ding a a

Department of Mathematics, Northeast Forestry University, Harbin 150040, PR China b Faculty of Management, University of Primorska, Cankarjeva 5, SI-6000 Koper, Slovenia c Department of Mathematics, Harbin Institute of Technology, Harbin 150000, PR China

a r t i c l e

i n f o

Article history: Received 26 July 2016 Accepted 14 January 2017 Available online 17 January 2017 Submitted by R. Brualdi

a b s t r a c t Let Mn be the algebra of all n ×n complex matrices and r ≥ 2 a fixed integer. The aim of this paper is to characterize linear maps φ : Mm1 ···ml → Mm1 ···ml such that φ (A1 ⊗ · · · ⊗ Al ) is r-potent whenever A1 ⊗ · · · ⊗ Al is r-potent. © 2017 Elsevier Inc. All rights reserved.

MSC: 15A15 15A69 15A86 15B57 Keywords: Linear preserver Tensor product r-Potent matrix Quantum information science

* Corresponding author. E-mail addresses: [email protected] (J. Xu), [email protected] (A. Fošner), [email protected] (B. Zheng), [email protected] (Y. Ding). http://dx.doi.org/10.1016/j.laa.2017.01.016 0024-3795/© 2017 Elsevier Inc. All rights reserved.

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1. Introduction and the main theorem Let n ≥ 2 be a positive integer and Mn the algebra of all n × n complex matrices. The problem of characterizing linear maps φ on Mn which preserve some subset Π of Mn , i.e., φ (Π) ⊆ Π, has attracted the attention of many mathematicians in the last few decades (see [13,14] and references therein). For example, the case when Π is the set of all singular matrices was considered by Dieudonné [6], the case when Π is a linear group was considered by Dixon [7], and the case when Π is the set of all nilpotent matrices was considered by Botta, Pierce, and Watkins [1]. Motivated by a problem of characterizing local automorphisms and local derivations of some operator algebras, Brešar and Šemrl [3] studied the case when Π is the set of all idempotents in Mn (see also [5]). In [4], the same authors extended this result to linear maps on Mn which preserve r-potent matrices for r ≥ 2. Recall that a matrix A ∈ Mn is r-potent if Ar = A, where r ≥ 2 is an integer. In the recent years a lot of results on r-potent preservers on matrix algebras as well as on operator algebras over infinite dimensional spaces have been obtained (see [18–20] and references therein). It is aim of this paper to continue this work by studying linear maps φ : Mm1 ···ml → Mm1 ···ml which preserve r-potent matrices of tensor products of matrices. Let us point out that our study is mainly theoretical but somehow also related to quantum information science mentioned below. Let m, n ≥ 2 be positive integers. Then for A ∈ Mm and B ∈ Mn , we denote by A ⊗ B ∈ Mm ⊗ Mn their tensor (Kronecker) product. In many pure and applied studies, one considers the tensor product of matrices (see, for example, [2,10,12,16]). Most noticeably, the tensor product is often used in quantum information science [15]. In a quantum system, quantum states are represented as density matrices, i.e., positive semi-definite matrices with trace one. If A ∈ Mm and B ∈ Mn are two quantum states in two quantum systems, then their tensor product A ⊗ B describes the joint state in the bipartite system in which the general states are density matrices in Mm ⊗ Mn ≡ Mmn . A general observable on the bipartite system corresponds to Hermitian matrices C ∈ Mmn and the set of observables of the form A ⊗ B with Hermitian A ∈ Mm , B ∈ Mn is a very small (measure zero) set. Nevertheless, one may be able to extract useful information about the bipartite system by focusing on the set of tensor product of matrices. In particular, in the study of linear operators φ : Mmn → Mmn on bipartite systems, the structure of φ can be determined by studying φ(A ⊗ B) with A ∈ Mm , B ∈ Mn (see [8,9,11] and references therein). More generally, one may also consider tensor states and general states in a multipartite system Mm1 ⊗ · · · ⊗ Mml ≡ Mm1 ···ml , l > 2. Before writing our main theorem, let us introduce basic definitions and fix the notation. First of all, throughout the paper, l and r, n, m1 , . . . , ml ≥ 2 are positive integers with n ≤ m1 · · · ml . For an integer k ≥ 2, Ik (resp., 0k ) denotes the k × k identity matrix (resp., zero matrix) and Pkr denotes the set of all r-potent matrices in Mk , i.e., Pkr = {A ∈ Mk : Ar = A}. In particular, Pk2 = {A ∈ Mk : A2 = A}. We say that a map

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φ : Mm1 ···ml → Mm1 ···ml preserves r-potents of tensor products of matrices if for any A1 ⊗ · · · ⊗ Al ∈ Mm1 ···ml , the image φ (A1 ⊗ · · · ⊗ Al ) is r-potent whenever A1 ⊗ · · · ⊗ Al is r-potent. We call a linear map π on Mm1 ···ml canonical, if π maps A1 ⊗ · · · ⊗ Al to τ1 (A1 ) ⊗ · · · ⊗ τl (Al ), where τk : Mmk → Mmk is either the identity map X → X or the transposition map X → X t , k = 1, . . . , l. Our main result reads as follows. Main Theorem. A linear map φ : Mm1 ···ml → Mn preserves r-potents of tensor products of matrices if and only if either φ = 0 or n = m1 · · · ml and there exist an invertible matrix T ∈ Mn , a scalar λ ∈ C with λr−1 = 1, and a canonical map π on Mm1 ···ml such that φ has the form φ(A1 ⊗ · · · ⊗ Al ) = λT π (A1 ⊗ · · · ⊗ Al ) T −1 , for all Ak ∈ Mmk , k = 1, . . . , l. Taking r = 2, the above theorem generalizes the main theorem in [17]. Corollary 1. A linear map φ : Mm1 ···ml → Mn preserves idempotents of tensor products of matrices if and only if either φ = 0 or n = m1 · · · ml and there exist an invertible matrix T ∈ Mn and a canonical map π on Mm1 ···ml such that φ has the form φ(A1 ⊗ · · · ⊗ Al ) = T π (A1 ⊗ · · · ⊗ Al ) T −1 for all Ak ∈ Mmk , k = 1, . . . , l. 2. Preliminary results In this section we prove several partial results which will be used in the sequel. Lemma 1. Let s and n, u1 , . . . , us ≥ 2 be positive integers with n ≥ u1 + . . . + us and let A = ε1 A1 ⊕ · · · ⊕ εs As ⊕ 0n−(u1 +···+us ) ∈ Mn where Ak ∈ Pu2k and εk ’s are mutually different (r − 1)-roots of unity, k = 1, . . . , s. If a matrix B ∈ Pnr satisfies B 2 = AB = BA, then B = ε1 B1 ⊕ · · · ⊕ εs Bs ⊕ 0n−(u1 +···+us ) for some Bk ∈ Purk , k = 1, . . . , s. Proof. For 1 ≤ k ≤ s, Ak ∈ Pu2k and, thus, there exist an invertible matrix Tk ∈ Muk and a nonnegative scalar vk ≤ uk such that

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Ak = Tk (Ivk ⊕ 0uk −vk ) Tk−1 . Let T = T1 ⊕ · · · ⊕ Ts ⊕ In−(u1 +···+us ) . Then   A = T (ε1 Iv1 ⊕ 0u1 −v1 ) ⊕ · · · ⊕ (εs Ivs ⊕ 0us −vs ) ⊕ 0n−(u1 +···+us ) T −1 . Moreover,   A = T Q ε1 Iv1 ⊕ · · · ⊕ εs Ivs ⊕ 0n−(v1 +···+vs ) (T Q)−1 for some permutation matrix Q ∈ Mn . Let us write ⎡ B11 · · · .. ⎢ .. . ⎢ . B = TQ⎢ ⎣ Bs1 · · · B01 · · ·

B1s .. . Bss B0s

⎤ B10 .. ⎥ . ⎥ −1 ⎥ (T Q) , ⎦ Bs0 B00

where Bkk ∈ Mvk for k = 1, . . . , s and B00 ∈ Mn−(v1 +···+vs ) . Since AB = BA, we have ⎡

ε1 B11 ⎢ .. ⎢ . ⎢ ⎣ εs Bs1 0

··· .. . ··· ···

ε1 B1s .. . εs Bss 0

⎤ ⎡ ε1 B11 ε1 B10 .. ⎥ ⎢ .. . ⎥ ⎢ . ⎥=⎢ εs Bs0 ⎦ ⎣ ε1 Bs1 0 ε1 B01

··· .. . ··· ···

εs B1s .. . εs Bss εs B0s

⎤ 0 .. ⎥ .⎥ ⎥ 0⎦ 0

and, since εk ’s are mutually different scalars, we get −1

B = T Q (B11 ⊕ · · · ⊕ Bst ⊕ B00 ) (T Q)

with Bkk ∈ Pvrk , k = 1, . . . , s. Note that B 2 = AB = BA and, hence,   −1 B 2 = T Q (B11 )2 ⊕ · · · ⊕ (Bss )2 ⊕ (B00 )2 (T Q) ⎡ ⎤ ε1 B11 · · · 0 0 .. ⎢ .. ⎥ . ··· · · ·⎥ ⎢ . −1 = TQ⎢ ⎥ (T Q) . ⎣ 0 · · · εs Bss 0 ⎦ 0 ··· 0 0 This yields that (B00 )2 = 0n−(v1 +···+vs ) and, thus, B00 = (B00 )r = (B00 )2 (B00 )r−2 = 0n−(v1 +···+vs ) . So, we have   −1 B = T Q B11 ⊕ · · · ⊕ Bss ⊕ 0n−(v1 +···+vs ) (T Q)   = T (B11 ⊕ 0u1 −v1 ) ⊕ · · · ⊕ (Bss ⊕ 0us −vs ) ⊕ 0n−(u1 +···+us ) T −1 = T1 (B11 ⊕ 0u1 −v1 ) T1−1 ⊕ · · · ⊕ Ts (Bss ⊕ 0us −vs ) Ts−1 ⊕ 0n−(u1 +···+us ) −1 Writing Bk = ε−1 for k = 1, . . . , s, we get the desired result. 2 k Tk (Bkk ⊕ 0uk −vk ) Tk

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In the following, λ1 , . . . , λr−1 ∈ C are scalars defined by λk = cos

√ 2kπ 2kπ + −1 sin − 1, r−1 r−1

k = 1, . . . , r − 2

and λr−1 = −1. Recall that ⎡

1

λ1

⎢1 ⎢ ⎢ ⎢ .. ⎣.

λ2 .. . 1 λr−1

··· ··· .. . ···

λr−2 1

⎤−1 ⎡

⎥ λr−2 ⎥ 2 ⎥ .. ⎥ . ⎦ r−2 λr−1

⎤ Cr1 ⎢ 1 − λr−1 ⎥ ⎢ C 2 ⎥ ⎢ ⎥ ⎢ r ⎥ 2 ⎢ ⎥=⎢ ⎥ .. ⎢ ⎥ ⎢ .. ⎥ , ⎣ ⎦ ⎣ . ⎦ . r−1 r−1 Cr 1 − λr−1 1 − λr−1 1

⎤

⎡

(1)

 where Cij = ji denotes the binomial coefficient for nonnegative integers i and j. Namely,

r r by the binomial theorem, (1 + λk ) = i=0 Cri λik . On the other hand, for k = 1, . . . , r−1, r we have (1 + λk ) = 1 + λk . This yields that Cr1 λk + Cr2 λ2k + · · · + Crr−1 λr−1 = λk − λrk k and, dividing the equality with λk = 0, we obtain Cr1 + Cr2 λk + · · · Crr−1 λr−2 = 1 − λr−1 k k . Therefore, ⎡

1

⎢1 ⎢ ⎢ ⎢ .. ⎣. 1

λ1

···

λ2 .. .

··· .. .

λr−1

···

⎤ ⎡ ⎤ 1 − λr−1 Cr1 1 ⎥ ⎢ C 2 ⎥ ⎢ 1 − λr−1 ⎥ λr−2 ⎥⎢ r ⎥ ⎢ ⎥ 2 2 ⎥⎢ ⎥ ⎢ ⎥. .. ⎥ ⎢ .. ⎥ = ⎢ .. ⎥ ⎦ . ⎦⎣ . ⎦ ⎣ . r−1 r−1 C λr−2 1 − λ r r−1 r−1 λr−2 1

⎤⎡

Since λk ’s are mutually different, we get (1) as the desired result. Lemma 2. If A, B ∈ Pnr and A + λk B ∈ Pnr for k = 1, . . . , r − 1, then Cr1 B = Ar−1 B + Ar−2 BA + · · · + ABAr−2 + BAr−1 , Cr2 B = Ar−2 B 2 + Ar−3 BAB + · · · + B 2 Ar−2 , .. . Crr−1 B = AB r−1 + BAB r−2 + · · · + B r−2 AB + B r−1 A. Proof. Let

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S1 = Ar−1 B + Ar−2 BA + · · · + ABAr−2 + BAr−1 , S2 = Ar−2 B 2 + Ar−3 BAB + · · · + B 2 Ar−2 , .. . Sr−1 = AB r−1 + BAB r−2 + · · · + B r−2 AB + B r−1 A. Then, for every k = 1, . . . , r − 1, (λk − λrk ) B = λk S1 + λ2k S2 + · · · + λr−1 k Sr−1 , and, since λk = 0, we derive 

 1 − λr−1 B = S1 + λk S2 + · · · + λr−2 k k Sr−1 .

 (k) If we write B = [bij ] and Sk = sij , then the above yields   (1) (2) (r−1) , 1 − λr−1 bij = sij + λk sij + . . . + λr−2 k k sij

1 ≤ i, j ≤ n.

Since this is true for every k = 1, . . . , r − 1, we have ⎡

1 − λr−1 1

⎤

⎡

1

⎢ 1 − λr−1 ⎥ ⎢ 1 ⎢ ⎥ ⎢ 2 ⎥=⎢ bij ⎢ .. ⎢ ⎥ ⎢ .. ⎣ ⎦ ⎣. . 1 − λr−1 r−1

λ1

···

λ2 .. .

··· .. .

1 λr−1

···

⎤ ⎡ (1) ⎤ sij (2) ⎥ r−2 ⎥ ⎢ λ2 ⎥ ⎢ sij ⎥ ⎢ ⎥ ⎥ , .. ⎥ ⎢ .. ⎥ ⎢ . ⎦⎣ . ⎥ ⎦ (r−1) λr−2 sij r−1 λr−2 1

1 ≤ i, j ≤ n,

and, since λk ’s are mutually different, it follows that ⎡

⎤

⎡ 1 λ1 ⎥ ⎢ ⎥ λ2 ⎢1 ⎥ ⎥ = bij ⎢ . .. ⎢. ⎥ ⎣. . ⎦ (r−1) 1 λr−1 sij (1)

sij

⎢ (2) ⎢ sij ⎢ ⎢ . ⎢ .. ⎣

···

λr−2 1

⎤−1 ⎡

··· .. .

⎥ λr−2 ⎥ 2 ⎥ .. ⎥ . ⎦

···

λr−2 r−1

1 − λr−1 1

⎤

⎢ 1 − λr−1 ⎥ ⎢ ⎥ 2 ⎢ ⎥, .. ⎢ ⎥ ⎣ ⎦ . 1 − λr−1 r−1

By (1), we derive ⎡

⎤

⎡ 1 ⎤ Cr bij ⎥ ⎢ 2 ⎥ ⎢ Cr bij ⎥ ⎥ ⎥ ⎢ ⎥, ⎥=⎢ .. ⎥ ⎥ ⎣ ⎦ . ⎦ r−1 (r−1) Cr bij s (1)

sij

⎢ (2) ⎢ sij ⎢ ⎢ . ⎢ .. ⎣ ij

1 ≤ i, j ≤ n,

1 ≤ i, j ≤ n.

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and, hence, for every k = 1, . . . , r − 1, 

(k)



sij



= Crk bij ,

1 ≤ i, j ≤ n.

So, we proved that Sk = Crk B for all k = 1, . . . , r − 1, as desired. 2 Lemma 3. If A ∈ Pn2 , B ∈ Pnr and A + λk B ∈ Pnr for k = 1, . . . , r − 1, then B ∈ Pn2 . Proof. Without loss of generality, we may assume that A  = Iu ⊕ 0n−u for some nonnegB1 B2 , where B1 ∈ Mu and ative integer u ≤ n. According to this, let us write B = B3 B4 B4 ∈ Mn−u . If r = 2, then there is nothing to prove. So, assume that r > 2. By Lemma 2, we have Cr1 B = Ar−1 B + Ar−2 BA + · · · + ABAr−2 + BAr−1    rB1 B2 B1 0 . According to our and, therefore, rB = . This yields that B = 0 0 B3 0 r assumptions, B1 ∈ Pu and, hence, there exist an invertible matrix T ∈ Mu and (r − 1)-roots of unity ε1 , . . . , εs such that 

  B1 = T ε1 Iv1 ⊕ · · · ⊕ εs Ivs ⊕ 0u−(v1 +···+vs ) T −1 , where v1 , . . . , vs are positive integers with v1 + · · · + vs ≤ u. Note that Iu + λk B1 ∈ Pur for all k = 1, . . . , r − 1. So, for every 1 ≤ i ≤ s, r

(1 + λk εi ) = 1 + λk εi ,

k = 1, . . . , r − 1.

Hence, εi = 1 and, consequently, B1 ∈ Pu2 and B ∈ Pn2 .

2

We end this section with the result which can be easily deduced from the proof of the main theorem in [4]. Lemma 4. Let φ : Mm1 ···ml → Mn be a linear map which preserves r-potents of tensor products of matrices. Suppose that projections P = P1 ⊗ · · · ⊗ Pl and Q = Q1 ⊗ · · · ⊗ Ql 2 with Pi , Qi ∈ Pm , i = 1, . . . , l, are orthogonal, (i.e., P Q = QP = 0). Then φ(P )φ(Q) = i φ(Q)φ(P ) = 0. 3. Proof of the main theorem Since the sufficiency part of the theorem is clear, we consider only the necessity part. So, let φ : Mm1 ···ml → Mn be a linear map which preserves r-potents of tensor products of matrices. Then φ (Im1 ···ml ) ∈ Mn is an r-potent matrix and, thus, there exist an

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invertible matrix T ∈ Mn and mutually different (r − 1)-roots of unity ε1 , . . . , εs such that   φ (Im1 ···ml ) = T ε1 Iu1 ⊕ · · · ⊕ εs Ius ⊕ 0n−(u1 +···+us ) T −1 , where u1 , . . . , us are positive integers with u1 + · · · + us ≤ n. Composing φ with the map X → T −1 XT , we may assume that φ (Im1 ···ml ) = ε1 Iu1 ⊕ · · · ⊕ εs Ius ⊕ 0n−(u1 +···+us ) .

(2)

Let us define a chain of sets Γ0 = {Im1 ⊗ · · · ⊗ Iml } ,   2 , Γ1 = P1 ⊗ Im2 ···ml : P1 ∈ Pm 1   2 , i = 1, 2 , Γ2 = P1 ⊗ P2 ⊗ Im3 ···ml : Pi ∈ Pm i .. .

  2 , i = 1, . . . , l . Γl = P1 ⊗ · · · ⊗ Pl : Pi ∈ Pm i In the next step we use the induction on k = 0, 1, . . . , l to prove that φ (X) = ε1 ψ1 (X) ⊕ · · · ⊕ εs ψs (X) ⊕ 0n−(u1 +···+us ) ,

X ∈ Γk ,

(3)

for some ψi (X) ∈ Pu2i , i = 1, . . . , s. Note that the case k = 0 is just (2). So, let k ≥ 1 and assume that our statement holds true for k − 1. For any X = P1 ⊗ · · · ⊗ Pk ⊗ Imk+1 ···ml ∈ Γk , let Y = P1 ⊗ · · · ⊗ Pk−1 ⊗ Imk ⊗ Imk+1 ···ml ∈ Γk−1 . Clearly, X(Y − X) = (Y − X)X = 0 and, by Lemma 4, we derive that φ (X) φ (Y − X) = φ (Y − X) φ (X) = 0. Thus, 2

φ (X) = φ (X) φ (Y ) = φ (Y ) φ (X) . By induction hypothesis, φ(Y ) is of the form φ (Y ) = ε1 ψ1 (Y ) ⊕ · · · ⊕ εs ψs (Y ) ⊕ 0n−(u1 +···+us ) , where ψi (Y ) ∈ Pu2i , i = 1, . . . , s, and, using Lemma 1, we conclude that

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φ (X) = ε1 ψ1 (X) ⊕ · · · ⊕ εs ψs (X) ⊕ 0n−(u1 +···+us ) for some ψi (X) ∈ Puri , i = 1, . . . , s. It remains to prove that ψi (X) ∈ Pu2i , i = 1, . . . , s Let λ1 , . . . , λr−1 ∈ C be scalars defined as in the previous section. Recall that Y + λj X ∈ Γk for every j = 1, . . . , r − 1. This yields that φ (Y ) + λj φ (X) ∈ Pnr ,

j = 1, . . . , r − 1,

and, hence, for every 1 ≤ i ≤ s ψi (Y ) + λj ψi (X) ∈ Puri ,

j = 1, . . . , r − 1.

Using Lemma 3, we see that ψi (X) ∈ Pu2i , i = 1, . . . , s. So, we proved (3). In particular, 2 if Pk ∈ Pm for k = 1, . . . , l, and X = P1 + · · · + Pl , then k φ (X) = ε1 ψ1 (X) ⊕ · · · ⊕ εs ψs (X) ⊕ 0n−(u1 +···+us ) for some ψi (X) ∈ Pu2i , i = 1, . . . , s. Let 1 ≤ i ≤ s and let ψi : Mm1 ···ml → Mui be a linear map such that (3) holds true. Obviously, ψi preserves idempotents of tensor products of matrices and, according to [17, Main theorem], we have two options. If ui < m1 · · · mn , then ψi = 0. On the other hand, if ui = m1 · · · mn , then ui = n since n ≤ m1 · · · mn . In this case φ = εi ψi and there exists a canonical map π on Mm1 ···ml such that ψi (A1 ⊗ · · · ⊗ Al ) = π (A1 ⊗ · · · ⊗ Al ) ,

Ak ∈ Mmk , k = 1, . . . , l

for all Ak ∈ Mmk , k = 1, . . . , l. This completes the proof. Remark 1. Let us point out that we deal just with the case n ≤ m1 · · · ml . It would also be interesting to study linear maps φ : Mm1 ···ml → Mn with n > m1 · · · ml which preserve r-potents of tensor products of matrices. The structure of these kind of maps can be more complex and even for l = 2 or r = 2 this is still an open problem. For example, a possible form of φ is  s   X → T λk πk (X) ⊕ 0n−sm1 ···ml T −1 (4) k=1

where T ∈ Mn is an invertible matrix, for k = 1, · · · , s, πk is canonical map on Mm1 ···ml , λk ∈ C with λr−1 = 1. But, we don’t know whether (4) is the only form of φ. k Remark 2. In our main theorem, r ≥ 2 is a fixed integer. Another open problem is: a characterization of linear maps on bipartite or multipartite systems which preserve potents of tensor products of matrices (recall that a matrix A is said to be potent if A = Ar for some integer r ≥ 2). Again, it does not seem easy to apply our proofs to solve this problem.

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Acknowledgements The authors show great thanks to the referee for his/her careful reading of the paper and valuable comments which greatly improved the readability of the paper. Jinli Xu was supported by the Fundamental Research Funds for the Central Universities (No. 2572015BB19) and the Foundation of Talent Introduction and Double FirstRate for the Northeast Forestry University (No. 1020160016), Ajda Fošner was partially supported by the bilateral research programme between Slovenia and the USA (Grant No. BI-US/14-15-012), Yuting Ding was supported by the Heilongjiang Provincial Natural Science Foundation (No. A201401), the National Nature Science Foundation of China (No. 11501091), and the Postdoctoral Science Foundation of China (No. 2015M571382, No. 2016T90266). References [1] P. Botta, S. Pierce, W. Watkins, Linear transformations that preserve the nilpotent matrices, Pacific J. Math. 104 (1983) 39–46. [2] N. Bourbaki, Elements of mathematics, in: Algebra I, Springer-Verlag, New York, 1989. [3] M. Brešar, P. Šemrl, Mappings which preserve idempotents, local automorphisms, and local derivations, Canad. J. Math. 45 (1993) 483–496. [4] M. Brešar, P. Šemrl, Linear transformations preserving potent matrices, Proc. Amer. Math. Soc. 119 (1993) 81–86. [5] G.-H. Chan, M.-H. Lim, Linear preservers on powers of matrices, Linear Algebra Appl. 162–164 (1992) 615–626. [6] J. Dieudonné, Sur une gén eralisation du groupe orthogonal á quatre variables, Arch. Math. 1 (1949) 282–287. [7] J. Dixon, Rigid embeddings of simple groups in the general linear group, Canad. J. Math. 29 (1977) 384–391. [8] A. Fošner, Z. Huang, C.-K. Li, N.-S. Sze, Linear preservers and quantum information science, Linear Multilinear Algebra 61 (2013) 1377–1390. [9] S. Friedland, C.-K. Li, Y.-T. Poon, N.-S. Sze, The automorphism group of separable states in quantum information theory, J. Math. Phys. 52 (042203) (2011) 1–8. [10] A.K. Jain, Fundamentals of Digital Image Processing, Prentice Hall, New Jersey, 1989. [11] N. Johnston, Characterizing operations preserving separability measures via linear preserver problems, Linear Multilinear Algebra 59 (2011) 1171–1187. [12] S.M. Lane, B. Birkhoff, Algebra, AMS Chelsea, Providence, 1999. [13] C.-K. Li, S. Pierce, Linear preserver problems, Amer. Math. Monthly 108 (2001) 591–605. [14] L. Molnár, Selected Preserver Problems on Algebraic Structures of Linear Operators and on Function Spaces, Lecture Notes in Mathematics, vol. 1895, Springer, Berlin, 2007. [15] M.A. Nielsen, I.L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, Cambridge, 2000. [16] W.-H. Steeb, Y. Hardy, Matrix Calculus and Kronecker Product: A Practical Approach to Linear and Multilinear Algebra, 2nd edition, World Scientific, Singapore, 2011. [17] B. Zheng, J. Xu, A. Fošner, Linear maps preserving idempotents of tensor products of matrices, Linear Algebra Appl. 470 (2015) 25–39. [18] S. Hou, J. Hou, K-potent preserving linear maps, Acta Math. Sci. 22 (2002) 517–525. [19] X. Zhang, C. Cao, Linear k-power/k-potent preservers between matrix spaces, Linear Algebra Appl. 412 (2005) 373–379. [20] H. You, Z. Wang, k-Potence preserving maps without the linearity and surjectivity assumptions, Linear Algebra Appl. 426 (2007) 238–254.