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

Let m, n ≥ 2 be integers. Denote by M n the set of n × n complex matrices. Let k · k ( p,k ) be the ( p, k ) norm on M mn with 1 ≤ k ≤ mn and 2 < p < ∞ . We show that a linear map φ : M mn → M mn satisﬁes k φ ( A ⊗ B ) k ( p,k ) = k A ⊗ B k ( p,k ) for all A ∈ M m and B ∈ M n if and only if there exist unitary matrices U, V ∈ M mn such that φ ( A ⊗ B ) = U ( ϕ 1 ( A ) ⊗ ϕ 2 ( B )) V for all A ∈ M m and B ∈ M n , where ϕ s is the identity map or the transposition map X → X T for s = 1 , 2. The result is also extended to multipartite systems.


Introduction
Throughout this paper, we denote by M m,n and M n the set of m × n and n × n complex matrices, respectively.Denote by H n the set of all n × n Hermitian matrices.For two matrices A = (a ij ) ∈ M m and B ∈ M n , their tensor product is defined to be A ⊗ B = (a ij B), which is an mn × mn matrix.We denote by Suppose A ∈ M m,n .The singular values of A are always denoted in decreasing order by s 1 (A) ≥ • • • ≥ s ℓ (A), where ℓ = min{m, n}.Given a real number p ≥ 1 and a positive integer k ≤ min{m, n}, the (p, k)-norm of A is defined by .
The (p, k)-norm, also known as the Ky Fan (p,k)-norm, was first recongnized as a special class of unitarily invariant norms in the study of isometries by Grone and Marcus [13] 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 Knrom, 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 references [14,19,24].
In addition to being a generalation 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., [4,16,30].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 [2] 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 [20,26] and their references.
The study of linear preservers on various matrix norms have been extensively explored since Schur [29] characterized linear maps on M n that preserve the spectral norm.This was followed by a series of subsequent results [1,12,13,23,27,28].Notably, Li and Tsing [23] 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 A → UAV or A → UA T V when m = n for some unitary matrices U ∈ M m and V ∈ 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 ⊗M n or H m ⊗H n have been investigated.
Friedland et al. [11] provided a characterization of linear maps on H m ⊗ 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 [17] examined invertible linear maps on M m ⊗ 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 ⊗ 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 the Schmidt k-norm, refer to [18].
Note that M m ⊗ M n and H m ⊗ 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 [8].Linear maps on complex matrices or Hermitian matrices preserving determinant were studied in [3,5,6].Linear maps on complex matrices preserving numerical radius, k-numerical range, product numerical range and rank-one matrices were characterized in [7,9,15,22].
In [10], the authors characterized linear maps on M mn preserving the Ky Fan k-norm and the Schatten p-norm of the tensor products A ⊗ B for all A ∈ M m and B ∈ 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 φ on M mn such that for p > 2 and 1 ≤ k ≤ mn, (1.1) 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.

Bipartite system
The linear maps on M mn satisfying (1.1) are determined by the following theorem.
Theorem 2.1.Let m, n ≥ 2 be integers.Given a real number p > 2 and a positive integer k ≤ mn, a linear map φ : if and only if there exist unitary matrices U, V ∈ M mn such that where ϕ s is the identity map or the transposition map X → 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 ∈ M n are said to be orthogonal, denoted by A ⊥ B, if A * B = AB * = 0. Denote by E ij ∈ 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 × n Hermitian matrix A are always denoted in decreasing order by λ 1 (A) ≥ λ 2 (A) ≥ • • • ≥ λ n (A).For A, B ∈ H n , we use the notation A ≥ B or B ≤ A to mean that A − B is positive semidefinite.Let R be the set of all real numbers.Rearrange the components of x = (x 1 , . . ., x n ) ∈ R n in decreasing order as then we say x is weakly majorized by y and denote by y ≻ w x or x ≺ w y.
Notice that x → x γ (x ≥ 0) is a convex function for any real number γ ≥ 1.One can easily conclude the following lemma. ( The following lemmas are crucial in our proof. (2.4) Proof.Let U ∈ M n be a unitary matrix such that Denote by u i the i-th column of U for i = 1, . . ., n.Let Û = [u 1 , u 2 . . ., u k ].Then applying Lemma 2.3, we have By Lemma 2.2, we have Remark 2.1.The inequality (2.4) can be regarded as a generalization of the inequality In our attempt to generalize this inequality, we aimed to obtain the following analogous inequality to (2.4): .In this case, we observe that Corollary 2.1.Let p > 2 be a real number and k ≤ n be a positive integer.Then Proof.Notice that 6, we get (2.5).
Lemma 2.7.Let A, B ∈ M n be nonzero matrices and 2 ≤ k ≤ n be an integer.Given a real number p ≥ 1, if and A ⊥ B, Proof.With the assumption that A ⊥ B, we can assume that the largest k singular values of On the other hand, we have Thus, the equality in (2.6) holds, which implies

.7)
Since A and B are both nonzero, we have Since A ⊥ B, we have Lemma 2.8.Let A, B ∈ M n be two positive semidefinite matrices, γ > 1 be a real number and k ≤ n be a positive integer.Suppose and U * AU = diag(λ 1 (A), . . ., λ n (A)) for some unitary matrix U ∈ M n .
Proof.Denote the i-th diagonal entry of U * BU by b i .Then λ i (A) + αb i is the i-th diagonal entry of U * (A + αB)U.It follows that Notice that g(x) = x γ (x > 0) is an increasing convex function when γ > 1.We can apply the Theorem 3.26 in [31] to obtain Thus, λ γ i (αB) be a function on α.Then we have α > 0 is sufficiently small, which contradicts (2.9).It follows that For the case λ k (A) = 0, we may assume that t is the largest integer such that λ t (A) > 0. Then U * AU = diag(λ 1 (A), . . ., λ t (A)) ⊕ 0 n−t and b i = 0 for i = 1, . . ., t. Recall that B is positive semidefinite.Thus, For the case λ k (A) > 0, we first have b i = 0 for all i = 1, . . ., k.Since B is positive semidefinite, it follows that U * BU = 0 k ⊕ C with C ∈ M n−k .Recall that ℓ is the largest integer such that λ k+ℓ (A) = λ k (A).If ℓ = 0, then the proof is completed.If ℓ > 0, then for any i = k + 1, . . ., k + ℓ, replacing the role of λ k (A) + αb k with λ i (A) + αb i in the above argument, we can conclude b i = 0. Thus, we have b i = 0 for i = 1, . . ., k + ℓ.It follows that Corollary 2.2.Let T, S ∈ M n be two matrices, p > 2 be a real number and k ≤ n be a positive integer.Suppose for all 0 < x < 1, and UT V = diag(s 1 (T ), . . ., s n (T )) for some unitary matrices U, V ∈ M n .
(1) If s k (T ) = 0, then T ⊥ S. If s k (T ) > 0, then we have and ℓ is the largest integer such that Then we use Lemma 2.8 twice to conclude that It follows that USV = 0 k+ℓ ⊕ Ŝ.
The following result originates from the last two paragraphs of the proof of Theorem 2.1 in [10].Lemma 2.9.Let φ : M mn → M mn be a linear map.Suppose for any unitary matrix X ∈ M m and integer 1 ≤ i ≤ m, there exists a unitary matrix W X such that where ϕ i,X is the identity map or the transposition map and W I = I mn .Then where ϕ 1 is a linear map and ϕ 2 is the identity map or the transposition map.
Proof.For any real symmetric matrix S ∈ M n and any unitary matrix Since W I = I mn , it follows that Thus, W X commutes with I m ⊗ S for all real symmetric S ∈ M n .This yields that W X = Z X ⊗ I n for some unitary matrix Z X ∈ M m , and hence Note that Tr 1 is linear and therefore continuous and the set is connected.So, all the maps ϕ i,X are the same and hence we can rewrite (2.11) as where ϕ 2 is the identity map or the transposition map.By the linearity of φ, it follows that for some linear map ϕ 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 [10] that a linear map φ preserves the spectral norm of tensor products A ⊗ B for all A ∈ M m and B ∈ M n if and only if φ has form A ⊗ B → U(ϕ 1 (A) ⊗ ϕ 2 (B))V for some unitary matrices U, V ∈ M mn , where ϕ s is the identity map or the transposition map for s = 1, 2. So we need only consider the case when k ≥ 2 in the following discussion.
Since the sufficiency part is clear, we consider only the necessity part.
Suppose a linear map φ : M mn → M mn satisfies (2.1) and k ≥ 2. We need the following three claims.
Claim 1.For any unitary matrices X ∈ M m and Y ∈ M n , we have for any possible i, j, s with j = s.Moreover, for any possible i, j, s with j = s.
Proof of Claim 1.For simplicity, we denote We need to show T ⊥ S and rank(T + S) ≤ k.
With the assumption in (2.1), we have for all F ∈ M m and G ∈ M n .It follows that (p,k) = 1 and xS p (p,k) = x p for all 0 < x < 1.We can conclude from the above equalities that for all 0 < x < 1. (2.12) Applying Corollary 2.1 with A = T and B = xS, we get i (x 2 S * S), it follows from (2.12) and (2.13) that for all 0 < x < 1.
Replacing the role of (T, S) with (T * , S * ) in the above argument, we have 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 2.2 to conclude that there exist unitary matrices U, V ∈ M n such that UT V = diag(s 1 (T ), . . ., s mn (T )) and USV = 0 k+ℓ ⊕ Ŝ, where ℓ is the largest integer such that s k+ℓ (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 ), . . ., s k (T ). Since which contradicts the fact that Similarly, we can also show that for any possible i, j, s with j = s.
Claim 2. For any unitary matrices X ∈ M m and Y ∈ M n , we have Proof of Claim 2. For simplicity, we denote We need to show S ⊥ T. Applying Corollary 2.1 on T and xS, we get for all 0 < x < 1.With the assumption in (2.1), we have So we can conclude that for any integer k ≥ 2, i (x 2 S * S). (2.17) for all 0 < x < 1.The above observations also hold if (T, S) is replaced by (T * , S * ), that is, for all 0 < x < 1.
If s k (T ) = 0, then applying Corollary 2.2 we have T ⊥ S. Otherwise, s k (T ) > 0. Notice that Claim 1 implies rank(T ) ≤ k.Thus, by (2.18) and (2.19), we can apply Corollary 2.2 to conclude that there exist unitary matrices U, V ∈ M mn such that It follows that T ⊥ S.This completes the proof.

Claim 3. For any unitary matrices
Proof of Claim 3. If i = r or j = s, then applying Claim 1 directly we have Next, we suppose that i = r and j = s.With Claim 1, we have With Claim 2, we have Then applying Lemma 2.5 again, we can conclude from (2.21) and (2.23) that Now we prove that φ has the desired form (2.2).For any unitary matrix Y ∈ M n , applying Claim 1 and Claim 3 we know . ., m and j = 1 . . ., n} is a set of mn orthogonal matrices in M mn .By Claim 1, each matrix in F has exactly one nonzero singular value, which equals 1.Thus, there exist unitary matrices for all i = 1, . . ., m and j = 1, . . ., n.Without loss of generality, we may assume that for all i = 1, . . ., m and j = 1, . . ., n.By (2.24) and (2.25), we have So far, we have showed that for any unitary matrix Y ∈ M n , there exists a unitary matrix U i,Y ∈ M n depending on i and Y such that By the linearity of φ, we conclude from the above equation that for any i = 1, . . ., m, there exists a linear map ψ i such that Let k = min{k, n}.Then it is easy to check that for all B ∈ M n .That is, ψ i is a linear map on M n preserving the (p, k) norm.Thus, by Theorem 1 in [23], where ϕ i is the identity map or the transposition map.Recall that I mn = φ(I m ⊗ I n ).Thus, we have W = W * .
Applying Claim 3 again, we can repeat the same argument above to show that for any unitary matrix X ∈ M m and any integer 1 ≤ i ≤ m, there exists a unitary matrix where ϕ i,X is the identity map or the transposition map.We may further assume that W I = I mn .Then applying Lemma 2.9, we have where ϕ 2 is the identity map or the transposition map and ϕ 1 is a linear map on M m .Let k = min{k, m}.It is easy to verify that ϕ 1 is a linear map on M m preserving the (p, k) norm.Hence, ϕ 1 also has the form A → UAV or A → UA T V for some unitary matrices U, V ∈ M m .This completes the proof.

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 [10].For completeness, we present it as follows.
Let φ : M N → M N be a linear map.Suppose for any unitary matrices X i ∈ M n i and any integers 1 for all B ∈ M nm , where ϕ j 1 ,...,j m−1 ,X is the identity map or the transposition map and W X = I N when X = (I n 1 , . . ., I n m−1 ).Then where ϕ 1 is a linear map and ϕ 2 is the identity map or the transposition map.
Proof.Considering all symmetric real matrices as in the proof of Lemma 2.9, one can conclude that there exists some unitary matrix Z X such that Notice that Tr 1 is linear and therefore continuous.Besides, the set is connected.So, all the maps ϕ j 1 ,...,j m−1 ,X are the same.Then (3.2) can be rewritten as where ϕ 2 is the identity map or the transposition map.With the linearity of φ, it follows that for some linear map ϕ 1 .
for all A i ∈ M n i , i = 1, . . ., m, if and only if there exist unitary matrices for all A i ∈ M n i , i = 1, . . ., m, where ϕ i is the identity map or the transposition map A → A T for i = 1, . . ., m.
Proof.By Theorem 3.2 of [10], we know the result holds for k = 1.So we may assume k ≥ 2.
We use induction on m.By Theorem 2.1, the result holds for m = 2. Now suppose that m ≥ 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 for any (i 1 , . . ., i m ) = (j 1 , . . ., j m ).Without loss of generality, we need only to prove that (3.6) holds when X i = I n i for i = 1 . . ., m.By Lemma 2.5, it suffices to show that for any integer 1 ≤ s ≤ m, we have for all i = (i 1 , . . ., i m ) and j = (j 1 , . . ., j m ) with i u = j u for u = 1, . . ., s. Denote by A s (i, j) and B s (i, j) the two matrices in (3.7) accordingly.It is easy to check that for any integer 1 ≤ s ≤ m and real number 0 < x < 1, Then applying the same argument as in the proof of Theorem 2.1, we conclude that for any integer 1 ≤ s ≤ m and real number 0 < x < 1, for all i = (i 1 , . . ., i m ) and j = (j 1 , . . ., j m ) with i u = j u for u = 1, . . ., s.Now we distinguish two cases.
Otherwise, s k (A s (i, j)) > 0 for some 1 ≤ s ≤ 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 A s (i, j) + xB s (i, j) p (p,k) = A s (i, j) p (p,k) = 2 s−1 , which contradicts (3.10).Thus, (3.11) holds.Applying Corollary 2.2, we have A s (i, j) ⊥ B s (i, j) for s = 1, . . ., m.
Next, we use induction on s to prove that for any s 0 + 1 ≤ s ≤ m, i = (i 1 , . . ., i m ) and j = (j 1 , . . ., j m ) with i u = j u for u = 1, . . ., s, there exist unitary matrices U, V ∈ M N depending on s and (i, j) such that UA s (i, j)V = I 2 s−1 ⊕ 0 N −2 s−1 and A s (i, j) ⊥ B s (i, j).(3.13) for some B ∈ M N −2 s−1 .It follows that A s (i, j) ⊥ B s (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, . . ., m, A s (i, j) ⊥ B s (i, j) for all i = (i 1 , . . ., i m ) and j = (j 1 , . . ., j m ) with i u = j u for u = 1, . . ., s, that is, (3.6) holds.
It follows that for any unitary matrix X m ∈ M nm , there exist unitary matrices U Xm and V Xm such that for all j i = 1, . . ., n i with 1 ≤ i ≤ m.For simplicity, we may assume that U I = V I = I, i.e., Following a similar argument as above, one can show that for any X = (X 1 , . . ., X m−1 ) and any integer j i = 1 . . ., n i with 1 ≤ i ≤ m − 1, there exists a unitary matrix W X ∈ M N such that E j i j i ⊗ ϕ j 1 ,...,j m−1 ,X (B) W * X (3.17) for all B ∈ M nm , where ϕ j 1 ,...,j m−1 ,X is the identity map or transposition map.Denote I = (I n 1 , . . ., I n m−1 ).For simplicity, we may further assume that W I = I N , i.e., for any integer j i = 1, . . ., n i with 1 E j i j i ⊗ ϕ j 1 ,...,j m−1 ,I (B) for all B ∈ M nm .
Then we apply Lemma 3.1 to conclude that for all B ∈ M nm and A i ∈ M n i , 1 ≤ i ≤ m − 1, where ϕ m is the identity map or the transposition map and ψ is a linear map.Let k = min k, m−1 i=1 n i .It is easy to check that for all A i ∈ M n i , i = 1, . . ., m − 1.Then by the induction hypothesis, we conclude that there exist unitary matrices U and V such that where ϕ i is the identity map or the transposition map for i = 1, . . ., m − 1. Therefore φ has the desired form and the proof is completed.

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 ≤ k ≤ mn.Our study generalized the results in [10] concerning the maps preserving the Ky Fan k-norm and 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 [10] 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 2.6 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 ≤ 2, we attempted to derive an analogue to the inequality (2.4).However, as demonstrated in Remark 2.1, this analogous inequality does not hold in general.Consequently the techniques employed in this paper are not able to address the case 1 < p ≤ 2. Therefore, novel approaches need to be introduced to tackle this particular scenario.

Lemma 2 . 2 . 1 .Lemma 2 . 5 .Lemma 2 . 6 .
[25, Lemma 2.1] Let A ∈ M n be a positive semidefinite matrix.Then x * A γ x ≥ (x * Ax) γ x 2(1−γ) for all x ∈ C n and γ ≥ Lemma 2.3.[31, Lemma 3.7] Let A ∈ M n be a Hermitian matrix and k ≤ n be a positive integer.Then k i=1 λ i (A) = max U * U =I k tr(U * AU) and k i=1 λ n−i+1 (A) = min U * U =I k tr(U * AU), where I k is the identity matrix of order k and U ∈ M n,k .Lemma 2.4.[21, Lemma 1] Let A, B ∈ M n .Then A ⊥ B if and only if there exist Â ∈ M m , B ∈ M n−m and unitary matrices U, V ∈ M n such that UAV = Â ⊕ 0 and UBV = 0 ⊕ B. Let A, B, C ∈ M n .If (A + B) ⊥ C and A ⊥ B, then A ⊥ C and B ⊥ C. Proof.Since A ⊥ B, we can apply Lemma 2.4 to conclude that there exist Â ∈ M m , B ∈ M n−m and unitary matrices U, V ∈ M n such that UAV = Â ⊕ 0 and UBV = 0 ⊕ B. Let UCV be partitioned as UCV = C 11 C 12 C 21 C 22 with C 11 ∈ M m and C 22 ∈ M n−m .It follows from (A + B) ⊥ C that (U(A + B)V ) * (UCV ) = 0 and (U(A + B)V )(UCV ) * = 0, * C = 0 and AC * = 0, i.e., A ⊥ C. Similarly, we can also conclude that B ⊥ C. Let C, D ∈ M n be two Hermitian matrices such that −C ≤ D ≤ C and k ≤ n be a positive integer.Then for any real number γ ≥ 1, k i=1

where 1 ≤
k ≤ n and C, D ∈ 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 = diag(1, 1, 3, 3) and C − D = diag(3, 3, 1, 1) .10) where a function g(α) = o(α) means lim α→0 g(α) α = 0. Since A and B are both positive semidefinite, we have λ i (A) ≥ 0 and b i ≥ 0 for all i = 1, . . ., n.It follows that k i=1
B) for all i = 1, . . ., m and B ∈ M n .(2.11) Define linear maps tr 1 : M mn → M n and Tr 1 : M mn → M n as tr 1 (A ⊗ B) = tr(A)B and Tr 1 (A ⊗ B) = tr 1 (φ(A ⊗ B)) for all A ∈ M m and B ∈ M n .The map tr 1 is also called the partial trace function in quantum science.Then Tr Now applying Corollary 2.2 again, we have T ⊥ S. Notice thatT + S p (p,k) = T p (p,k) + S p (p,k).Applying Lemma 2.7 on S and T we have rank(T + S) ≤ k.
all B ∈ M nm and integers 1 ≤ j i ≤ n i with 1 ≤ i ≤ m − 1. Define linear maps tr 1 : M N → M nm and Tr 1 : M N → M nm by tr 1 (A ⊗ B) = tr(A)B and Tr 1 (A ⊗ B) = tr 1 (φ(A ⊗ B))

Theorem 3 . 1 .
Given an integer m ≥ 2, let n i ≥ 2 be integers for i = 1, . . ., m and N = m i=1 n i .Then for any real number p > 2 and any positive integer k ≤ N, a linear map φ :

. 16 )− 1 i=1E j i j i ⊗ B = W m− 1 i=1E
Then we have φ(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, W ∈ M N such that for all j i = 1, . . ., n i with 1≤ i ≤ m − 1, φ mj i j i ⊗ ϕ j 1 ,...,j m−1 (B) W ,where ϕ j 1 ,...,j m−1 is the identity map or the transposition map.It follows that φ(I N ) = W W . Recall that φ(I N ) = I N and W and W are both unitary matrices.We have W = W * .