Linear and Multilinear Algebra. Linear maps preserving rank of tensor products of matrices

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1 Linear maps preserving rank of tensor products of matrices Journal: Manuscript ID: GLMA-0-0 Manuscript Type: Original Article Date Submitted by the Author: -Aug-0 Complete List of Authors: Zheng, Baodong; Harbin Institute of Technology, Department of Mathematics Xu, JinLi; Harbin Institute of Technology, Department of Mathematics Fošner, Ajda; University of Primorska, Faculty of Management Keywords: Linear preserver, tensor product, rank of a matrix

2 Page of LINEAR MAPS PRESERVING RANK OF TENSOR PRODUCTS OF MATRICES BAODONG ZHENG, JINLI XU, AND AJDA FOŠNER Abstract Let M n be the algebra of all n n complex matrices We characterize linear maps φ : M m m l M m m l satisfying rank φ A A l = rank A A l for all A k M mk, k =,, l 00 Math Subj Class: A0, A, A Key words: Linear preserver, tensor product, rank of a matrix Introduction the Main Theorem Let M n be the set of all n n complex matrices In quantum physics, quantum states of a system with n physical states are represented as density matrices, ie, positive semi-definite matrices with trace one Suppose A M m B M n are the states of two quantum systems Then their tensor Kronecker product A B M m M n will be the state in the bipartite joint system Density matrices in M m M n M mn that can be written as a convex combination of product states are separable states It is easy to see that S M mn is separable if only if it is a convex combination of P P, where P M m P M n are rank one orthogonal projections In [] it was shown that a linear map sending the set of separable states onto itself has a very nice structure Namely, it has the form or m = n A B ϕ A ϕ B A B ϕ B ϕ A, where ϕ j, j =,, has the form X U j XUj or X U jx t Uj for some unitary U M m U M n Here, Y t denotes the transpose of a square matrix Y Y the conjugate transpose of Y Motivated by the above observations, we characterize linear maps on M mn which preserve rank of tensor products of matrices In particular, we show that such a map has the form Corresponding author A B Uτ A τ BV

3 Page of BAODONG ZHENG, JINLI XU, AND AJDA FOŠNER for some invertible U, V M mn, where τ k, k =,, is either the identity map X X or the transposition map X X t More generally, we consider linear maps preserving rank of tensor products of matrices on multipartite systems M m M ml = M m m l, l The research on linear preserver problems has a long history One may see [] its references for results on linear preserver problems Furthermore, for the background on rank preserver problems without the tensor structure we refer the reader to [0,, ] It is well-known that a map φ : M n M n satisfies rank φa = rank A for all A M n if only there exist invertible matrices U, V M n such that either or φa = UAV, A M n, φa = UA t V, A M n One can also see, for example, [, ] for results on rank preservers on tensor products of matrices Moreover, for some recent research on linear preserver problems on tensor spaces arising in quantum information science we refer the reader to [,,,,,, ] Continuing this line of investigations, we observe the structure of rank preservers on multipartite systems Before writing our main theorem, let us introduce some basic definitions fix the notation First of all, throughout the paper, l n, m,, m l are positive integers with r = n m m l 0 For an integer k, I k denotes the k k identity matrix, 0 k the k k zero matrix, E k, i, j k, the k k matrix whose all entries are equal to zero except for the i, j-th entry which is equal to one As usual, we use the notation Diaga,, a k to denote the k k diagonal matrix with diagonal entries a,, a k We say that a linear map φ : M m m l M n preserves rank of tensor products of matrices if φ satisfies rank φ A A l = rank A A l for all A k M mk, k =,, l We call a linear map π on M m m l canonical, if π A A l = τ A τ l A l for all A k M mk, k =,, l, where τ k : M mk M mk, k =,, l, is either the identity map X X or the transposition map X X t In this case we write π = τ τ l Our main result reads as follows

4 Page of PRESERVING RANK OF TENSOR PRODUCTS OF MATRICES Main Theorem A linear map φ : M m m l M n preserves rank of tensor products of matrices if only if there exist invertible matrices U, V M n a canonical map π on M m m l such that φ A A l = U π A A l 0 n m m l V for all A k M mk, k =,, l In particular, if n = m m l, then for all A A l M m m l φ A A l = Uπ A A l V Remark If n < m m l φ : M m m l M n is a linear map, then rank I m m l = m m l rank φ I m m l n < m m l So, φ does not satisfy for all A A l M m m l Remark Let us point out that in our Main Theorem we characterize maps which preserve rank of tensor products of matrices that the resulting map may not preserve rank of all matrices in M m m l More precisely, if l >, then the map φ of the form may not satisfy rank X = rank φx for all X M m m l For example, let n = m m l Denote φ : A A A l A t A A l X = E m E m + E m E m + E m E m + E m E m Y = E m E m + E m E m + E m E m + E m E m Of course, X, Y M m m It is also easy to see that φ X I m m l = Y I m m l rank X I m m l rank φ X I m m l = rank Y I m m l In a special case, when l = m = m =, we have X = Y = φ X =

5 Page of BAODONG ZHENG, JINLI XU, AND AJDA FOŠNER Lemmas Before proving our main result, let us write several lemmas which we will need in the sequel The first one is a well-known theorem from the paper [] Lemma [, Theorem ] Let m, n, s be positive integers with ms n Suppose that X k M n, k =,, m, are complex matrices satisfying m rank X k = s rank X k = ms Then there exist invertible matrices U, V M n such that X k = U E m I s 0 n ms V, k =,, m Lemma Let m, m, n be positive integers with k m m m n Suppose that A M m is an invertible matrix X M n such that rank λe m A 0 n m m + X = m k= for all λ C Then X is of the form 0 k m X 0 X = X X X, 0 X 0 n km where X M m Proof Let X = X X X X X X, X X X with X M k m X M m Then rank X X X X λa + X X = m X X X for all λ C Since A is invertible there exist infinitely many complex numbers λ such that det λa + X 0 here, det λa + X denotes the determinant of a matrix λa + X For those scalars λ, rank of the matrix X X X X λa + X X X X λa + X X 0 X X λa + X X is equal to m, hence, X = X λa + X X Let adj λa + X be the adjoint matrix of λa + X Then det λa + X X = X adj λa + X X

6 Page of PRESERVING RANK OF TENSOR PRODUCTS OF MATRICES Recall that det λa + X is a polynomial of λ with degree m,, thus, each entry of the left matrix above is either a polynomial of λ with degree m or zero On the other h, adj λa + X is a m m matrix formed by taking the transpose of the cofactor matrix of a given matrix λa + X In particular, its entries are polynomials of λ with degree at most m So, each position of the right matrix above is a polynomial of λ with degree at most m This yields that X = 0 Similarly, X = 0, X = 0, X = 0 Lemma Let m, m, n, i, j be positive integers with i j m m m n Suppose that A, A M m are invertible matrices X M n satisfying rank λe m ii A 0 n m m + X = m, rank λe m jj A 0 n m m + X = m for all λ C Then there exist Y, Z M m such that X = E m Y + E m ji Z 0 n m m Proof Applying lemma, the result follows Lemma Let π be a canonical map on M m m l U, V M m m l If Uπ A A l V = 0 for all A k M mk, k =,, l, then U = 0 or V = 0 Proof By the assumption, it is easy to see that UXV = 0 for all X M m m l, thus, either U = 0 or V = 0, as desired Recall that if a matrix U M kh commutes with I k S for all real symmetric S M h, then U has the form W I h with W M k This simple observation will be used in the proof of our last lemma Lemma Let π, π be canonical maps on M m m k U, V M m m l invertible matrices If π A A l U = V π A A l for all A k M mk, k =,, l, then π = π U = V = λi m m l for some nonzero scalar λ C Proof Since π, π are canonical maps on M m m l, we can write π = τ τ l π = η η l, where each of the maps τ k η k, k =,, l, is either the identity map X X or the transposition map X X t Clearly, π i I m m l = I m m l, i =, Hence, U = V To prove that U is a scalar multiple of the identity map we use the induction on l The case l = is obvious So, let l assume that the statement holds true for l Note that for any real symmetric S M ml, we have π i I m m l S =

7 Page of BAODONG ZHENG, JINLI XU, AND AJDA FOŠNER I m m l S, i =, This implies that U commutes with I m m l S for all real symmetric S M ml,, thus, U = W I ml for some matrix W M m m l Since U is invertible, W has to be invertible as well Let us define linear maps π = τ τ l π = η η l It is clear that π π are canonical maps on M m m l Let Y = A A l M m m l be an arbitrary matrix Then π Y W I ml = π Y I ml U = Uπ Y I ml = W π Y I ml, thus, π Y W = W π Y Consequently, by the induction hypothesis, W = λi m m l for some nonzero scalar λ Therefore, U = λi m m l π X = λ π X U = λ Uπ X = π X for all X = A A l M m m l Proof of the Main Theorem Since the sufficiency part of the Main Theorem is clear, we consider only the necessity part So, throughout this section, we assume that φ : M m m l M n is a linear map which satisfies Using the induction on l, we prove that φ is of the form The case l = is just the corollary of Theorem in [0] So, assume that l > that the expected result holds true for l Claim If F,, F m M m rank are rank one matrices with m F k = m, k= then there exist invertible matrices U, V M n such that for all k =,, m, φ F k X = U E m π k X 0 r V for all X = X X l M m m l, where π k are canonical maps on M m m l Proof Since it follows that rank rank F k I m m l = m m l m F k I m m l = m m m l k= rank φ F k I m m l = m m l

8 Page of PRESERVING RANK OF TENSOR PRODUCTS OF MATRICES rank φ F k I m m l = m m m l m k= By Lemma, there exist invertible matrices U, V M n such that φ F k I m m l = U E m I m m l 0 r V, k =,, m After composing φ by the linear transformation X U XV, we may assume that φ F k I m m l = E m I m m l 0 r, k =,, m For k =,, m, let us define maps L k : M m m l M n by L k X = φ F k X It follows from the property of φ that L k are linear maps which preserve rank of tensor products of matrices Thus, applying the induction hypothesis on L k, we conclude that there exist invertible matrices R k, S k M n such that φ F k X = R k E m π k X 0 r S k for all X = X X l M m m l, where π k is a canonical map on M m m l It follows form that E m I m m l 0 r = R k E m I m m l 0 r S k Set R k R k R k S R k = R k R k R k k S k S k, S k = S k R k R k R k S k S k, S k S k S k where R k, Sk M k m m l R k, Sk M m m l Using, we obtain R k R k R k 0 I m m l 0 = R k R k R k S k S k S k 0 I m m l 0 S R k R k R k k S k S k S k S k S k More precisely, R k Sk R k Sk R k Sk 0 I m m l 0 = R k Sk R k Sk R k Sk R k Sk R k Sk R k Sk Since R k Sk = I m m l it follows that R k = Rk = Sk = Sk = 0 Therefore, R k 0 R k S R k = R k R k R k k S k S k, S k = 0 S k R k 0 R k 0 S k S k S k

9 Page of BAODONG ZHENG, JINLI XU, AND AJDA FOŠNER This, together with, yields φ F k X = E m Rπ k k X S k 0 r for all X = X X l M m m l Set T = Diag R,, Rm, I r Then for all k =,, m X = X X l M m m l, φ F k X = T E m π k X 0 r T According to Claim, we may assume that for all k =,, m X = X X l M m m l, φ E m X = E m π k X 0 r, where π k are canonical maps on M m m l Claim We prove that π i = π j = π for all i < j m Moreover, there exist scalars λ 0 such that φ E m X = λ τ E m π X 0 r, for all X = X X l M m m l, where τ is either the identity map Y Y or the transposition map Y Y t Proof For the sake of the simplicity, let us assume that i = j = Denote F = E m + E m, F = E m + E m, F k = E m for k =,, m Then, clearly, matrices F k, k =,, m, satisfy the assumptions in Claim, thus, there exist invertible matrices U, V M n such that for all k =,, m X = X X l M m m l, φ F k X = U E m π k X 0 r V, where π k are canonical maps on M m m l Set U = U U U U U U V = V V V V V V, U U U V V V where U, U, V, V M m m l U, V M n m m l Then, using, we obtain φ E m X = φ F X φ E X = = U π X V π X U π X V U π X V U π X V U π X V U π X V U π X V U π X V U π X V 0 φ E m X = φ F X φ E X =

10 Page of PRESERVING RANK OF TENSOR PRODUCTS OF MATRICES = U π X V U π X V U π X V U π X V U π X V π X U π X V U π X V U π X V U π X V for all X = X X l M m m l Note that for any invertible matrix G k M mk, k =,, l, any scalar ε C, matrices E m + εe m G G l E m + εe m G G l are of rank m m l Therefore, using, we conclude that matrices E m π G G l 0 r + εφ E m G G l, E m π G G l 0 r + εφ E m G G l have rank m m l This yields, by Lemma, that φ E m X = , X M m m l, where sts for some m m l m m l complex matrix It follows from that U π X V = 0 for all X = X X l M m m l, by Lemma, either U = 0 or V = 0 Case I Assume first that U = 0 Using, 0,, we obtain U π X V = π X U π X V = π X for all X = X X l M m m l Writing I m m l instead of X in the above equalities, we get U V = I m m l U V = I m m l Hence, U π X = π X V π X V = U π X Using 0, we conclude that φe m X = E m π X V V 0 r X = φe m E m U U π X 0 r, hence, π X V V = U U π X for all X = X X l M m m l Since rank φe m I m m l = m m l it follows that V V U U are invertible matrices, by Lemma, we have π = π V V = λ I m m l for some λ 0 Thus, φe m X = for all X = X X l M m m l λ E m π X 0 r

11 Page 0 of BAODONG ZHENG, JINLI XU, AND AJDA FOŠNER Case II If V = 0, then, using a similar approach as in Case I, we show that there exists a nonzero scalar λ C such that φe m X = λ E m π X 0 r for all X = X X l M m m l Using the above observations, we complete the proof of our Main Theorem First, by, we may assume that for all i, j m, there exist scalars λ 0 a canonical map π on M m m l such that φ E m X = λ τ E m π X 0 r for all X = X X l M m m l, where τ is either the identity map Y Y or the transposition map Y Y t According to,, rank E m ii + E m + E m ji + E m jj I m m l = m m l, we conclude that rank E m ii + λ τ E m + λ ji τ ji E m ji + E m jj I m m l = m m l This yields that λ λ ji = τ = τ ji for all i < j m Assume for a moment that m that i, j, k m are distinct positive integers Then, according to,, rank E m ii + E m + E m ik I m m l = m m l, rank E m + E m ik + E m jj + E m jk I m m l = m m l, we obtain rank E m ii + λ τ E m rank λ τ E m + λ ik τ ik + λ ik τ ik E m ik E m ik + E m jj + λ jk τ jk = m m l I m m l = m m l E m jk I m m l = Therefore, λ λ jk = λ ik, τ ik = τ, τ = τ jk This yields that τ = τ = τ for all i < j m Now, writing we have T = Diag, λ,, λ m Im m l Ir, φ X X = T τ X π X 0 r T for all X M m X = X X l M m m l The proof of the Main Theorem is completed

12 Page of PRESERVING RANK OF TENSOR PRODUCTS OF MATRICES Remark Let k =,, m m l let us denote R k := {A A l M m m l : rank A A l = k} In our Main Theorem we characterize maps φ on M m m l which leave all the sets R k invariant, ie, φ R k R k, k =,, m m l Here, it is also interesting to study linear maps φ such that holds true just for some fixed k For example, we would like to determine the structure of linear maps φ : M m m l M m m l such that φa A l is a rank one matrix whenever matrices A i M mk, i =,, l, have rank one, ie, φ R R Even for l = this is still an open problem Acknowledgements The authors wish to thank Prof Chi-Kwong Li Nung-Sing Sze for their introducing guiding the topic Jinli Xu was supported by the National Natural Science Foundation Grants of China Grant No the Natural Science Foundation of Heilongjiang Province of China Grant No A00 This research was done when Ajda Fošner Jinli Xu were attending the Summer Research Workshop on Quantum Information Science 0 at Taiyuan University of Technology They gratefully acknowledged the support kind hospitality from the host university References [] D Ž Djokovič, Linear transformations of tensor products preserving a fixed rank, Pacific J Math 0, [] A Fošner, Z Huang, C-K Li, Y- T Poon, N-S Sze, Linear maps preserving the higher numerical ranges of tensor products of matrices, Linear Multilinear Algebra 0, pages DOI:000/00000 [] A Fošner, Z Huang, C-K Li, N-S Sze, Linear maps preserving Ky Fan forms Schatten norms of tensor products of matrices,, SIAM J Matrix Anal Appl 0, [] A Fošner, Z Huang, C-K Li, N-S Sze, Linear maps preserving numerical radius of tensor products of matrices, J Math Anal Appl 0 0, [] A Fošner, Z Huang, C-K Li, N-S Sze, Linear preservers quantum information science, 0, pages DOI: 000/ [] S Friedl, C-K Li, Y-T Poon, N-S Sze, The automorphism group of separable states in quantum information theory, J Math Phys 0, pages DOI: 00/0 [] N Johnston, Characterizing operations preserving separability measures via linear preserver problems, 0, [] C-K Li, S Pierce, Linear preserver problems, Amer Math Monthly 0 00, 0 [] C-K Li, Y-T Poon, N-S Sze, Linear preservers of tensor product of unitary orbits, product numerical range, Linear Algebra Appl 0, -0 [0] C-K Li, L Rodman, P Šemrl, Linear transformations between matrix spaces that map one rank specific set into another, Linear Algebra Appl 00, 0

13 Page of BAODONG ZHENG, JINLI XU, AND AJDA FOŠNER [] M H Lim, Rank Tensor Rank Preservers, Linear Multilinear Algebra, [] R Loewy, Linear mappings which are rank-k nonincreasing, Linear Multilinear Algebra, [] G Matsaglia, G P H Styan, Equalities inequalities for ranks of matrices, Linear Multilinear Algebra, [] L Molnár, Selected preserver problems on algebraic structures of linear operators on function spaces, Lecture Notes in Mathematics, Vol, Berlin 00 Baodong Zheng, Department of Mathematics, Harbin Institute of Technology, Harbin 000, P R China address: zbd@hiteducn Jinli Xu, Department of Mathematics, Harbin Institute of Technology, Harbin 000, P R China address: jclixv@qqcom Ajda Fošner, Faculty of Management, University of Primorska, Cankarjeva, SI-0 Koper, Slovenia address: ajdafosner@fm-kpsi