LINEAR MAPS PRESERVING NUMERICAL RADIUS OF TENSOR PRODUCTS OF MATRICES

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1 LINEAR MAPS PRESERVING NUMERICAL RADIUS OF TENSOR PRODUCTS OF MATRICES AJDA FOŠNER, ZEJUN HUANG, CHI-KWONG LI, AND NUNG-SING SZE Abstract. Let m, n 2 be positive integers. Denote by M m the set of m m complex matrices and by w(x) the numerical radius of a square matrix X. Motivated by the study of operations on bipartite systems of quantum states, we show that a linear map φ : M mn M mn satisfies w(φ(a B)) = w(a B) for all A M m and B M n if and only if there is a unitary matrix U M mn and a complex unit ξ such that φ(a B) = ξu(ϕ 1(A) ϕ 2(B))U for all A M m and B M n, where ϕ k is the identity map or the transposition map X X t for k = 1, 2, and the maps ϕ 1 and ϕ 2 will be of the same type if m, n 3. In particular, if m, n 3, the map corresponds to an evolution of a closed quantum system (under a fixed unitary operator), possibly followed by a transposition. The results are extended to multipartite systems Math. Subj. Class.: 15A69, 15A86, 15A60, 47A12. Key words: Complex matrix, linear preserver, numerical range, numerical radius, tensor product. 1. Introduction and preliminaries Let M n be the set of n n complex matrices for any positive integer n. For A M n, define (and denote) its numerical range and numerical radius by W (A) = { } u Au : u C n, u u = 1 and w(a) = sup{ µ : µ W (A)}, respectively. The study of numerical range and numerical radius has a long history and is still under active research. Moreover, there are many generalizations motivated by pure and applied topics; see [4, 5, 6]. By the convexity of the numerical range, W (A) = {tr (Auu ) : u C n, u u = 1} = {tr (AX) : X D n }, where D n is the set of density matrices (positive semidefinite matrices with trace one) in M n. In particular, in the study of quantum physics, if A M n is Hermitian corresponding to an observable and if quantum states are represented as density matrices, then W (A) is the set of all possible measurements under the observables and w(a) is a bound for the measurement. If A = A 1 + ia 2, where A 1, A 2 M n are Hermitian, then W (A) is the set of the joint measurement of quantum states under the two observables corresponding to A 1 and A 2. 1

2 2 AJDA FOŠNER, ZEJUN HUANG, CHI-KWONG LI, AND NUNG-SING SZE Suppose m, n 2 are positive integers. Denote by A B the tensor (Kronecker) product of the matrices A M m and B M n. If A and B are observables of two quantum systems, then A B is an observable of the composite bipartite system. Of course, a general observable on the composite system corresponds to C M mn, and observable of the form A B with A M m, B M n 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 matrices. In particular, in the study of linear operators φ : M mn M mn on bipartite systems, the structure of φ can be determined by studying φ(a B) with A M m, B M n ; see [1, 2, 3, 7] and their references. In this paper, we determine the structure of linear maps φ : M mn M mn satisfying w(a B) = w(φ(a B)) for all A M m and B M n. We show that for such a map there is a unitary matrix U M mn and a complex unit ξ such that φ(a B) = ξu(ϕ 1 (A) ϕ 2 (B))U for all A M m and B M n, where ϕ k is the identity map or the transposition map X X t for k = 1, 2, and the maps ϕ 1 and ϕ 2 will be of the same type if m, n 3. In particular, if m, n 3, the map corresponds to an evolution of a closed quantum system (under a fixed unitary operator), possibly followed by a transposition. The study of linear maps on matrices or operators with some special properties are known as preserver problems; for example, see [9] and its references. In connection to preserver problems on bipartite quantum systems, it is quite common that if one considers a linear map φ : M mn M mn and imposes conditions on φ(a B) for A M m, B M n, then the partial transpose maps A B A B t and A B A t B are admissible preservers. It is interesting to note that for numerical radius preservers and numerical range preservers in our study, if m, n 3, then the partial transpose maps are not allowed and that the (linear) numerical radius preserver φ on M mn will be of the standard form X ξv XV or X ξv X t V for some complex unit ξ and unitary V M mn. This is the first example of such results in this line of study. It would be interesting to explore more matrix invariant or quantum properties that the structure of preservers on M mn can be completely determined by the behavior of the map on the small class of matrices of the form A B M mn with A M m, B M n. In our study, we also determine the linear map φ : M mn M mn such that W (A B) = W (φ(a B)) for all (A, B) M m M n. We will denote by X t the transpose of a matrix X M n and X the conjugate transpose of a matrix X M n. The n n identity matrix will be denoted by I n. Let E (n) ij M n be the matrix whose (i, j)-entry is equal to one and all the others are equal to zero. We simply write E ij = E (n) ij if the size of the matrix is clear. We will prove our main result on bipartite systems in Section 2 and extend the results to multipartite systems in Section 3.

3 LINEAR MAPS PRESERVING NUMERICAL RADIUS OF TENSOR PRODUCTS OF MATRICES 3 2. Bipartite systems The following example is useful in our discussion. Example 2.1. Suppose m, n 3. Let A = X O m 3 and B = X O n 3 with X = Then A B is unitarily similar to 0 1 and A B t is unitarily similar to 0 1/ /2 O mn 7, O mn 7. 0 One readily checks (see also [10]) that W (A B) and W (A B t ) = W (A t B) are circular disks centered at the origin with radii w(a B) and w(a B t ), respectively. Moreover, we have w(a B) = λ max (A B + (A B) t )/2 = 4.25 > = λ max (A B t + (A B t ) t )/2 = w(a B t ). In the following, we first determine the structure of linear preservers of numerical range using the above example and the results in [2]. Theorem 2.2. The following are equivalent for a linear map φ : M mn M mn. (a) W (φ(a B)) = W (A B) for any A M m and B M n. (b) There is a unitary matrix U M mn such that φ(a B) = U(ϕ 1 (A) ϕ 2 (B))U for all A M m and B M n, where ϕ k is the identity map or the transposition map X X t for k = 1, 2, and the maps ϕ 1 and ϕ 2 will be of the same type if m, n 3. Proof. Suppose (b) holds. If m, n 3, then the map has the form C UCU or C UC t U. Thus, the condition (a) holds. If m = 2, then A t and A are unitarily similar for every A M 2. So, W (A B) = W (A t B) for any B M n. Hence, the condition (a) holds. Similarly, if n = 2, then (a) holds. Conversely, suppose that W (φ(a B)) = W (A B) for all A M m and B M n. Assume for the moment that A B M mn is a Hermitian matrix. Then W (φ(a B)) = W (A B) R. This yields that φ(a B) is a Hermitian matrix, as well. Thus, φ maps Hermitian matrices to Hermitian matrices and preserves numerical radius, which is equivalent to spectral radius for

4 4 AJDA FOŠNER, ZEJUN HUANG, CHI-KWONG LI, AND NUNG-SING SZE Hermitian matrices. By Theorem 3.3 in [2], we conclude that φ has the asserted form on Hermitian matrices and, hence, on all matrices in M mn. However, if m, n 3, then, by Example 2.1, neither the map A B A B t nor the map A B A t B will preserve the numerical range. So, the last statement about ϕ 1 and ϕ 2 holds. Next, we turn to linear preservers of the numerical radius. We need the following (well-known) lemma to prove our result. We include a short proof for the sake of the completeness. Lemma 2.3. Let A M n with w(a) = x Ax = 1 for some unit x C n. Then for any unitary U M n with x being its first column, there exists some y C n 1 such that (1) U AU = x 1 y Ax. y Proof. Write (x Ax) 1 A = G + ih with G and H Hermitian. Then the largest eigenvalue of G is 1 with x as its corresponding eigenvector and U GU = [1] G 1 for some G 1 M n 1. Moreover, the (1,1)-entry of iu HU is 0 since w(a) = 1. Since U HU is a Hermitian matrix we have 0 y iu HU =. Thus, U y AU has the claimed form. Theorem 2.4. The following are equivalent for a linear map φ : M mn M mn. (a) w(φ(a B)) = w(a B) for any A M m and B M n. (b) There is a unitary matrix U M mn and a complex unit ξ such that φ(a B) = ξu(ϕ 1 (A) ϕ 2 (B))U for all A M m and B M n, where ϕ k is the identity map or the transposition map X X t for k = 1, 2, and the maps ϕ 1 and ϕ 2 will be of the same type if m, n 3. Proof. The implication (b) (a) can be verified readily. Now, suppose (a) holds and let B ij = φ(e ii E jj ) for 1 i m, 1 j n. According to the assumptions, for all 1 i m, 1 j n, there is a unit vector u ij C mn and a complex unit ξ ij such that u ij B iju ij = ξ ij. We will first show that there exists a unitary matrix U M mn and complex units ξ ij such that B ij = ξ ij U(E ii E jj )U, 1 i m, 1 j n. We divide the proof into several claims. Claim 1. Suppose (i, j) (r, s). If u C mn is a unit vector such that u B ij u = 1, then B rs u = 0. Proof. Suppose u is a unit vector such that u B ij u = e iθ. Let ξ = u B rs u. We first consider the case if i = r. For any µ C with µ 1, (2) 1 = w(e ii (E jj + µe ss )) = w(b ij + µb rs ) u (B ij + µb rs )u = e iθ + µξ. Then we must have ξ = 0. Otherwise, e iθ + µξ = e iθ + e iθ ξ > 1 if one chooses µ = e iθ ξ/ ξ. Suppose ξ = 0. Then the inequality in (2) become equality and w(b ij +µb rs ) = u (B ij +µb rs )u =

5 LINEAR MAPS PRESERVING NUMERICAL RADIUS OF TENSOR PRODUCTS OF MATRICES 5 1. By Lemma 2.3, there is y µ C mn 1 such that ( U B ij U + µu B rs U = U (B ij + µb rs )U = e iθ 1 y µ y µ where U is a unitary matrix with u as its first column. Since the above equation holds for any µ C with µ 1, the matrix U B rs U must have the form. So, U 0 B rs U is a matrix with zeros in its first column and row, or equivalently, B rs u = 0, as desired. Similarly, we can prove the case if j = s. Now we consider the case when i r and j s. By the previous argument, we have B is u = B rj u = 0. Then for any µ C with µ 1, 1 = w((e ii + E rr ) (E jj + µe ss )) = w(b ij + B rj + µ(b is + B rs )) u (B ij + B rj + µ(b is + B rs ))u = u B ij u + µu B rs u = e iθ + µξ. It follows that ξ = u B rs u = 0 and hence w(b ij + B rj + µ(b is + B rs )) = u B ij u = 1. By Lemma 2.3, we conclude that (B is + B rs )u = 0 and thus, B rs u = 0. ), Claim 2. Suppose (i, j) (r, s). If u ij, u rs C mn are two unit vectors such that u ij B iju ij = u rsb rs u rs = 1, then u ij u rs = 0. Proof. Suppose u rs = αu ij +βv with α = u ij u rs and β = u rsv, where v is a unit vector orthogonal to u ij. Notice that α 2 + β 2 = 1. By Claim 1, u ij B rs = 0 and B rs u ij = 0 and so 1 = u rsb rs u rs = β 2 v B rs v β 2 w(b rs ) = β 2 1. Thus, β = 1 and hence u ij u rs = α = 0. Claim 3. Let U = [u 11 u 1n u 21 u 2n u m1 u mn ]. Then U U = I mn and U B ij U = ξ ij (E ii E jj ) for all 1 i m and 1 j n. Proof. By Claim 2, {u ij : 1 i m, 1 j n} forms an orthonormal basis. Thus, U U = I mn. Next by Claim 1, u rsb ij u kl = 0 for all (r, s) and (k, l), except the case when (r, s) = (k, l) = (i, j). Therefore, the result follows. According to our assumptions and by Claims 1, 2, 3, we see that up to some unitary similarity φ(e ii E jj ) = ξ ij (E ii E jj ) for 1 i m, 1 j n and some complex units ξ ij. Now, for any unitary X M m, using the same arguments as above, there exists some unitary U X and some complex units µ ij such that φ(xe ii X E jj ) = µ ij U X (E ii E jj )UX

6 6 AJDA FOŠNER, ZEJUN HUANG, CHI-KWONG LI, AND NUNG-SING SZE for all 1 i m and 1 j n. So, φ(xe ii X E jj ) is a unit multiple of rank one Hermitian matrix with numerical radius one. Thus, φ(xe ii X E jj ) = µ ij xx for some unit vector x C mn. Note also that φ(i m E jj ) = D E jj for some diagonal unitary matrix D. If γ > 0, then w(φ((xe ii X + γi m ) E jj )) = 1 + γ. Furthermore, there exists a unit vector u C mn such that u x = 1 and u (D E jj )u = 1. From the second equality, u must have the from u = û e j for some unit vector û C m. Therefore, u x = 1 implies x = ˆx e j for some unit vector ˆx C m. Thus, φ(xe ii X E jj ) has the form R i,x E jj for some R i,x M m. Since this is true for any 1 i m and unitary X M m, we have φ(a E jj ) = ϕ j (A) E jj for all matrices A M m and some linear map ϕ j. Clearly, ϕ j preserves numerical radius and, hence, has the form A ξ j W j AW j or A ξ j W j A t W j for some complex unit ξ j and unitary W j M m. In particular, ϕ j (I m ) = ξ j I m and φ(i mn ) = I m D for some diagonal matrix D M n. Using the same arguments as above, we can show that φ(e ii B) = E ii ϕ i (B) for all matrices B M n and some linear map ϕ i of the form B ξ i Wi B W i or B ξ i Wi B t Wi, where ξ i is a complex unit and W i M n a unitary matrix. Therefore, we have ϕ i (I n ) = ξ i I n and φ(i mn ) = D I n for some diagonal matrix D M m. Since φ(i mn ) = D I n = I m D, we conclude that φ(i mn ) = ξi mn for some complex unit ξ. For the sake of the simplicity, let us assume that φ(i mn ) = I mn. Then φ(e ii E jj ) = E ii E jj for all 1 i m, 1 j n. For any Hermitian matrices A M m and B M n, suppose their spectral decompositions are A = XD 1 X and B = Y D 2 Y. φ(i mn ) = I mn, one sees that there exists a unitary matrix U X,Y Repeating the above argument and using the assumption such that φ(xe ii X Y E jj Y ) = U X,Y (XE ii X Y E jj Y )U X,Y, 1 i m, 1 j n, and, hence, φ(a B) = U X,Y (A B)UX,Y. So, φ maps Hermitian matrices to Hermitian matrices and preserves numerical range on the tensor product of Hermitian matrices. Thus, by the same argument as in the proof of Theorem 2.2, φ has the asserted form on Hermitian matrices and, hence, on all matrices in M mn. If m, n 3, we can use Example 2.1 to conclude that ϕ 1 and ϕ 2 should both be the identity map, or both be the transpose map. The proof is completed. 3. Multipartite systems In this section we extend Theorem 2.2 and Theorem 2.4 to multipartite systems M n1 M nm, m 2.

7 LINEAR MAPS PRESERVING NUMERICAL RADIUS OF TENSOR PRODUCTS OF MATRICES 7 Theorem 3.1. Let n 1,..., n m 2 be positive integers and N = m j=1 n j. The following are equivalent for a linear map φ : M N M N. (a) W (φ(a 1 A m )) = W (A 1 A m ) for any (A 1,..., A m ) M n1 M nm. (b) There is a unitary matrix U M N such that (3) φ(a 1 A m ) = U(ϕ 1 (A 1 ) ϕ m (A m ))U for all (A 1,..., A m ) M n1 M nm, where ϕ k is the identity map or the transposition map X X t for k = 1,..., m, and the maps ϕ j are of the same type for those j s such that n j 3. Proof. The sufficient part is clear. Hermitian matrix A = A 1 A m with A j H nj For the converse, as in the proof of Theorem 2.2, consider for j = 1,..., m. By [2, Theorem 3.4], φ has the asserted form on Hermitian matrices and, hence, on all matrices in M n1 M nm. However, if n i, n j 3 with i < j, let A i = X O ni 3 and A j = X O nj 3, where X is defined as in Example 2.1, and A k = E 11 M nk for k i, j. Then w(a 1 A m ) = 4.25 and w(a 1 A j 1 A t j A j+1 A m ) = Thus, W (A 1 A m ) W (A 1 A j 1 A t j A j+1 A m ), and we see that the last statement about ϕ k holds. Theorem 3.2. Let n 1,..., n m 2 be positive integers and N = m j=1 n j. The following are equivalent for a linear map φ : M N M N. (a) w(φ(a 1 A m )) = w(a 1 A m ) for any (A 1,..., A m ) M n1 M nm. (b) There is a unitary matrix U M N and a complex unit ξ such that (4) φ(a 1 A m ) = ξu(ϕ 1 (A 1 ) ϕ m (A m ))U for all (A 1,..., A m ) M n1 M nm, where ϕ k is the identity map or the transposition map X X t for k = 1,..., m, and the maps ϕ j are of the same type for those j s such that n j 3. Proof. The sufficiency part is clear. We verify the necessity. First of all, one can use similar arguments as in the proof of Theorem 2.4 (see Claim 1, Claim 2, and Claim 3) to show that up to some unitary similarity, φ(e i1 i 1 E i2 i 2 E imi m ) = ξ i1 i 2 i m (E i1 i 1 E i2 i 2 E imi m ) for all 1 i k n k with 1 k m and some complex units ξ i1 i 2 i m. Below we give the details of the proof for the case M n1 M n2 M n3. One readily extends the arguments to the general case. Denote B ijk = φ(e ii E jj E kk ) for 1 i n 1, 1 j n 2, 1 k n 3 and let N = n 1 n 2 n 3. According to the assumptions, for any 1 i n 1, 1 j n 2, 1 k n 3, there is a unit vector u ijk C N and a complex unit ξ ijk such that u ijk B ijku ijk = ξ ijk.

8 8 AJDA FOŠNER, ZEJUN HUANG, CHI-KWONG LI, AND NUNG-SING SZE Claim 4. Suppose (i, j, k) (r, s, t). If u C N is a unit vector such that u B ijk u = 1, then B rst u = 0. Proof. Suppose u is a unit vector such that u B ijk u = e iθ. Let ξ = u B rst u. First, assume that δ ir + δ js + δ kt = 2, where δ ab equals to 1 when a = b and zero otherwise. In other words, we assume that exactly two of the three sets {i, r}, {j, s}, {k, t} are singletons. assume that i = r, j = s, and k t. For any µ C with µ 1, 1 = w(b ijk + µb rst ) u (B ijk + µb rst )u = e iθ + µξ. Without loss of generality, Then we must have ξ = 0. Furthermore, w(b ijk + µb rst ) = u (B ijk + µb rst )u = 1. By Lemma 2.3, one conclude that U B rst U has the form, where U is a unitary matrix with u as its 0 first column, and hence B rst u = 0. Next, suppose δ ir + δ js + δ kt = 1, say i = r. By the previous case, B isk u = B ijt u = 0. Then for any µ C with µ 1, 1 = w(b ijk + B isk + µ(b ijt + B ist )) u (B ijk + B isk + µ(b ijt + B ist ))u = u B ijk u + µu B ist u = e iθ + µξ. It follows that ξ = u B ist u = 0 and hence w(b ijk + B isk + µ(b ijt + B ist )) = u B ijk u = 1. By Lemma 2.3, we conclude that (B ijt + B ist )u = 0 and thus, B ist u = 0. Finally, suppose δ ir + δ js + δ kt = 0. By the previous cases, B isk u = B rjk u = B rsk u = B ijt u = B ist u = B rjt u = 0. Then for any µ C with µ 1, 1 = w((b ii + B rr ) (B jj + B ss ) (B kk + µb tt )) u ((B ii + B rr ) (B jj + B ss ) (B kk + µb tt ))u = u B ijk u + µu B rst u = e iθ + µξ. Then by a similar argument, one conclude that ξ = 0 and B rst u = 0. Claim 5. Suppose (i, j, k) (r, s, t). If u ijk, u rst C mn are two unit vectors such that u ijk B ijku ijk = u rstb rst u rst = 1, then u ijk u rst = 0. Proof. Suppose u rst = αu ijk + βv with α = u ijk u rst and β = u rstv, where v is a unit vector orthogonal to u ijk. Note that α 2 + β 2 = 1. By Claim 1, u ijk B rst = 0 and B rst u ijk = 0 and so 1 = u rstb rst u rst = β 2 v B rst v β 2 w(b rst ) = β 2 1. Thus, β = 1 and hence u ijk u rst = α = 0. Claim 6. Let U = [u 111 u 11n3 u 121 u 12n3 u 1n2 n 3 u 211 u 2n2 n 3 u n1 11 u n1 n 2 n 3 ]. Then U U = I N and U B ijk U = ξ ijk (E ii E jj E kk ) for all 1 i n 1, 1 j n 2, 1 k n 3.

9 LINEAR MAPS PRESERVING NUMERICAL RADIUS OF TENSOR PRODUCTS OF MATRICES 9 Proof. By Claim 2, {u ijk : 1 i n 1, 1 j n 2, 1 k n 3 } forms an orthonormal basis and thus U U = I N. Next by Claim 1, u rstb ijk u lpq = 0 for all (r, s, t) and (l, p, q), except the case when (r, s, t) = (l, p, q) = (i, j, k). Therefore, the result follows. Similarly, for any unitary X M n1, there exists some unitary U X µ i1 i 2 i m such that and some complex units φ(xe i1 i 1 X E i2 i 2 E imim ) = µ i1 i 2 i m U X (XE i1 i 1 X E i2 i 2 E imim )UX for all 1 i k n k with 1 k m. We see that φ(xe i1 i 1 X E i2 i 2 E imim ) is a rank one matrix with numerical radius one. If γ > 0, then w(φ((xe i1 i 1 X + γi n1 ) E i2 i 2 E imim )) = 1 + γ. Thus, φ(xe i1 i 1 X E i2 i 2 E imim ) has the form R E i2 i 2 E imim ) for some R M n1. Since this is true for any unitary X M n1, we have φ(a E i2 i 2 E imi m ) = ϕ i2 i m (A) E i2 i 2 E imi m for all Hermitian matrices A M n1 and some linear map ϕ i2 i m. Clearly, ϕ i2 i m preserves numerical radius and, hence, has the form A ξ i2 i m W i2 i m AW i 2 i m or A ξ i2 i m W i2 i m A t W i 2 i m for some complex unit ξ i2 i m and unitary W i2 i m M n1. In particular, ϕ i2 i m (I n1 ) = ξ i2 i m I n1 and φ(i N ) = I n1 D 1 for some diagonal matrix D 1 M n2 n m. Given 2 k m and using the same arguments as above, one can show that φ(i N ) = D k1 I nk D k2 for some diagonal matrix D k1 M n1 n k 1 and D k2 M nk+1 n m. Since φ(i N ) = I n1 D 1 = D k1 I nk D k2 for k = 2,..., m, we conclude that φ(i N ) = ξi N for some complex unit ξ. For the sake of the simplicity, let us assume that φ(i N ) = I N. Then φ(e i1 i 1 E i2 i 2 E imim ) = E i1 i 1 E i2 i 2 E imim for all 1 i k n k with 1 k m. For any Hermitian matrix A 1 A m M N, suppose its spectral decomposition is A 1 A m = X 1 D 1 X1 X m D m Xm. Repeating the above argument and using the assumption φ(i N ) = I N, we can conclude that there exists a unitary matrix U X1,...,X m such that φ(x 1 E i1 i 1 X1 X m E imim Xm) = U X1,...,X m (X 1 E i1 i 1 X1 X m E imim Xm)U X 1,...,X m for all 1 i k n k with 1 k m. By linearity, we have φ(a 1 A m ) = U X1,...,X m (A 1 A m )UX 1,...,X m. So, φ maps Hermitian matrices to Hermitian matrices and preserves numerical range on the tensor product of Hermitian matrices. By Theorem 3.1, it has the asserted form on Hermitian matrices and, hence, on all matrices in M n1 n m. If n i, n j 3, we observe the matrices A i = X 0 ni 3

10 10 AJDA FOŠNER, ZEJUN HUANG, CHI-KWONG LI, AND NUNG-SING SZE and A j = X 0 nj 3, where X is defined as in Example 2.1, and A k = E 11 M nk for k i, j, to conclude that ϕ i and ϕ j should both be the identity map, or both be the transpose map. The proof is completed. Acknowledgment This research was supported by a Hong Kong GRC grant PolyU with Sze as the PI. The grant also supported the post-doctoral fellowship of Huang and the visit of Fošner to the Hong Kong Polytechnic University in the summer of She gratefully acknowledged the support and kind hospitality from the host university. Fošner was supported by the bilateral research program between Slovenia and US (Grant No. BI-US/ ). Li was supported by a GRC grant and a USA NSF grant; this research was done when he was a visiting professor of the University of Hong Kong in the spring of 2012; furthermore, he is an honorary professor of Taiyuan University of Technology (100 Talent Program scholar), and an honorary professor of the Shanghai University. References [1] S. Friedland, C.K. Li, Y.T. Poon, and N.S. Sze, The Automorphism Group of Separable States in Quantum Information Theory, Journal of Mathematical Physics 52 (2011), [2] A. Fošner, Z. Huang, C.K. Li, and N.S. Sze, Linear preservers and quantum information science, Linear and Multilinear Algebra, to appear. (Available at arxiv: ) [3] A. Fošner, Z. Huang, C.K. Li, and N.S. Sze, Linear maps preserving Ky Fan norms and Schatten norms of tensor product of matrices, preprint. (Available at arxiv: ) [4] K. E. Gustafson and D. K. M. Rao, Numerical range: The Field of values of linear operators and matrices, Universitext, Springer-Verlag, New York, [5] P.R. Halmos, A Hilbert Space Problem Book (Graduate Texts in Mathematics), 2nd edition, Springer, New York, [6] R.A. Horn and C.R. Johnson, Topics in Matrix Analysis, Cambridge University Press, Cambridge, [7] N. Johnston, Characterizing Operations Preserving Separability Measures via Linear Preserver Problems, Linear and Multilinear Algebra 59 (2011), [8] C.K. Li, Linear operators preserving the numerical radius of matrices, Proc. Amer. Math. Soc. 99 (1987), [9] C.K. Li and S. Pierce, Linear preserver problems, Amer. Math. Monthly 108 (2001), [10] C.K. Li and N.K. Tsing, Matrices with circular symmetry on their unitary orbits and C-numerical ranges, Proc. Amer. Math. Soc. 111 (1991), Ajda Fošner, Faculty of Management, University of Primorska, Cankarjeva 5, SI-6104 Koper, Slovenia address: ajda.fosner@fm-kp.si Zejun Huang, Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong address: huangzejun@yahoo.cn Chi-Kwong Li, Department of Mathematics, College of William and Mary, Williamsburg, VA 23187, USA; Department of Mathematics, University of Hong Kong, Pokfulam, Hong Kong address: ckli@math.wm.edu

11 LINEAR MAPS PRESERVING NUMERICAL RADIUS OF TENSOR PRODUCTS OF MATRICES 11 Nung-Sing Sze, Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Hong Kong address: