A new criterion and a special class of k-positive maps

A new criterion and a special class of k-positive maps Dedicated to Professor Leiba Rodman. Jinchuan Hou Department of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, China. jinchuanhou@aliyun.com Chi-Kwong Li Department of Mathematics, College of William and Mary, Williamsburg, VA 23187, USA; 100 Talent Scholar, Faculty of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, China. ckli@math.wm.edu Yiu-Tung Poon Department of Mathematics, Iowa State University, Ames, IA 50011, USA. ytpoon@iastate.edu Xiaofei Qi Department of Mathematics, Shanxi University, Taiyuan, 030006, China. qixf1980@126.com Nung-Sing Sze Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Hong Kong. raymond.sze@polyu.edu.hk Abstract We study k-positive maps on operators. We obtain new criteria on k-positivity in terms of the k-numerical range, and use it to improve and refine some earlier results on k-positive maps related to the study of quantum information science. We also consider a special class of positive maps extending the construction of Choi on positive maps that are not completely positive. Some open questions on the decomposability of such maps are answered. 2000 Mathematics Subject Classification. 15A69, 15A60, 15B57, 46N50. Keywords. k-positive maps, k-numerical range. 1

1 Introduction Denote by BH, K the set of bounded linear operators from the Hilbert space H to the Hilbert space K, and write BH, K = BH if H = K. Let BH + be the set of positive semidefinite operators in BH. If H and K have dimensions n and m respectively, we identify BH, K with the set M m,n of m n matrices, and write M n,n = M n, and BH + = M + n. A linear map L : BH BK is positive if LBH + BK +. For a positive integer k, the map L is k-positive if the map I k L : M k BH M k BK is positive, where I k LA = LA ij for A = A ij 1 i,j k with A ij BH. A map is completely positive if it is k-positive for every positive integer k. The study of positive maps has been the central theme for many pure and applied topics; for example, see [5, 17, 20, 23, 24, 31]. In particular, the study has attracted a lot of attention of physicists working in quantum information science in recent decades, because positive linear maps can be used to distinguish entanglement of quantum states[14]. There is considerable interest in finding positive maps that are not completely positive, which can be applied to detect entangled states for example, see [1, 3, 4, 7, 8, 9, 10, 12, 16, 18, 19, 21, 26, 27, 28] and the references therein. Completely positive linear maps have been studied extensively by researchers. However, the structure of positive linear maps is still unclear even for the finite dimensional case [6, 11, 17, 25]. In this paper, we obtain a new criterion of k-positivity in terms of the k-numerical range. The result is used to refine a result of Chruściński and Kossakowski[7]. We also consider a new class of positive maps extending the construction of Choi on positive maps that not completely positive. An open question on the decomposability of such maps is answered. The paper is organized as follows. Section 2 summarizes some basic known criteria for k- positive maps. In Section 3, we obtain a new criterion for k-positivity for elementary operators in terms of the k-numerical range. The new criterion is used to refine the results of Chruściński and Kossakowski[7]. In Section 3, we discuss a family of positive maps, called D-type positive maps. They can be viewed as an extension of the construction of Choi on positive maps that are not completely positive. We give a necessary and sufficient condition for such maps to be k-positive. We also consider the decomposability of such maps and answer an open problem of the first and fourth authors. In our discussion, we will use the physicists notation. The adjoint of an operator A is denoted by A. A vector in H will be denoted by x and x is the dual vector of the vector x in the dual space of H. The inner product of two vectors x and y in H will be denoted by y x. If x H and y K, then the rank one operator x y : K H will send a vector z K to x y z = y z x. 2 Preliminaries In this section, we review several equivalent conditions of k-positivity that will be used in the subsequent discussion. Some of these conditions are known; for example, see [5, 12, 21, 30, 31]. Proposition 2.1 Suppose L : BH BK is a linear map continuous under strong operator topology. The following are equivalent. a L is k-positive, i.e., I k L is positive. b I k LP is positive semi-definite for all rank one orthogonal projection P M k BH. 2

c for all orthonormal subset X = { x 1,..., x k } H, the operator matrix defined by L X = L x i x j 1 i,j k is positive semi-definite. Let {E 11, E 12,..., E nn } be the standard basis for M n. Suppose L : M n BK is a linear map. The Choi matrix CL is the operator matrix with LE ij 1 i,j n. Clearly, there is a one-one correspondence between a linear map L and the Choi matrix CL. One can use the Choi matrix to determine whether the map L is k-positive. Proposition 2.2 Let L : M n BK and 1 k n. The following are equivalent. a L is k-positive. b x CL x 0 for all x = k p=1 y p z p with y p z p C n K. c I n P CLI n P is positive semi-definite for all rank-k orthogonal projection P BK. A linear map L : BH BK is called an elementary operator[17] if it has the form LX = A j XB j for some A 1,..., A k, B 1,..., B k BH, K. If H and K are finite dimensional, then every linear map is elementary. Since we are interested in positive linear map, we focus on linear maps which map self-adjoint operators to self-adjoint operators. It is not hard to show that every elementary operator sending the set of self-adjoint operators into itself must have the form q LX = C j XC j D j XD j. 1 Hou [17] gave a condition for an elementary operator in the above form to be k-positive. readily extends Proposition 2.1 to the following. One Proposition 2.3 Suppose L : BH BK has the form 1 with C 1,..., C p, D 1,..., D q BH, K. Then the following are equivalent. a L is k-positive, i.e., I k L is positive. b p I k C r P I k C r q s=1 I k D s P I k D s M k BK + for all rank one orthogonal projection P M k BH. c for all orthonormal subset { x 1,..., x k } H, k i, E ij L x i x j is positive semidefinite, equivalently, i, E ij C r x i x j C r q s=1 i, E ij D s x i x j D s. d for all x C k H, there is an q p matrix T x with operator norm T x 1 such that I k D 1 I k C 1 I k D 2. x = T I k C 2 x I K. x. I k D q I k C p 3

Proof. The equivalence of b and c follows from Proposition 2.1 and the special form of L. For the equivalence of a, b and d, see [17]. 3 A new criterion and refinements of some previous results Recall that, for a linear operator A BH and a positive integer k dim H, the k-numerical range of A is defined by W k A = x j A x j : { x 1,..., x k } is an orthonormal set in H. Denote by x 1 Γ k H = x =. : { x 1,..., x k } is an orthonormal set in H. 2 x k Also for simplicity, denote Γ n,k = Γ k C n. Then the k-numerical range of A can also be reformulated as W k A = { x I k A x : x Γ k H}. If dim H = n <, and A is Hermitian with eigenvalues a 1 a n, then W k A = a n j+1,. For the details of k-numerical ranges, see [2]. The following proposition gives the relation between k-numerical ranges and k-positivity of elementary operators. Theorem 3.1 Suppose L : M n BK has the form X with C 1,..., C p, D 1,..., D q BH, K. C r XC r a If L is k-positive, then W k p C rc r q s=1 D sd s [0,. a j q D s XD s 3 b If for all unit vectors u = u 1,..., u p t C p and v = v 1,..., v q t C q, min W k r u r C r r u r C r s=1 max W k s v s D s s v s D s, 4 then L is k-positive. Here, min S and max S denote the minimum and maximum value of a subset S of real numbers. 4

Proof. If L is k-positive, then I k L x x is positive semi-definite for every x Γ n,k, where Γ n,k is defined in 2. Then part a is trivially followed by taking trace and the definition of k-numerical range. For b, suppose 4 holds for all unit vectors u = u 1,..., u p t C p and v = v 1,..., v q t C q. For x Γ n,k, let C 1 x 1 C p x 1 C x =..... C 1 x k C p x k and Dx = D 1 x 1 D q x 1..... D 1 x k D q x k We will show that C x C x D x D x is positive semi-definite, or equivalently, for all unit vector y K k, y C x 2 y D x 2. Denote by σ 1 A σ 2 A σ min{m,n} A the singular values of A M m,n. Note that there is x Γ n,k so that C x has the smallest p-th singular value σ p C x among all choices of x Γ n,k. Then y C x σ p C x σ p C x for all unit vector y K k. Moreover, there is a unit vector ũ = ũ 1,..., ũ p t C p such that σ C x 2 = C x ũ 2 = r ũrc r x 1 r ũrc r x k t 2 = x j r min W k r ũ r C r r ũ r C r r ũ r C r x j ũ r C r.. Similarly, we can choose ˆx Γ n,k so that Dˆx has the largest maximum singular value σ 1 Dˆx among all choice of x Γ n,k. Then y D x σ 1 D x ˆσ 1 Dˆx. Moreover, there is a unit vector ˆv = ˆv 1,..., ˆv q t C q such that max W k s v s D s s v s D s = ˆx j ˆv s D s ˆv s D s ˆx j s s s ˆv sd s ˆx 1 s ˆv sd s ˆx k t 2 = Dˆx ˆv 2 = σ 1 Dˆx 2. By our assumption, we have σ p C x σ 1 Dˆx, and hence y C x σ p C x σ 1 Dˆx y D x. The desired conclusion follows. 5

Remark Note that in the above proof, if there is x Γ n,k such that C x = 0, then min {W k u r C r } u r C r : u = u 1,..., u p t C p, u u = 1 = 0. r r On the other hand, if min {W k r u r C r r } u r C r : u = u 1,..., u p t C p, u u = 1 > 0, then C x has rank kp for all x Γ n,k. In the following, we give a numerical range criterion for k-positivity of maps φ : M n M m defined by LX = L 1 X L 2 X with L 1 X = γ j F j XF j and L 2 X = where F 1,..., F mn M m,n satisfy the following: j=p+1 γ j F j XF j, γ 1,..., γ mn 0, C1 {F j } mn M m,n is an orthonormal set using the inner product X, Y = tr XY. The positivity of such maps were also considered in [7, 8] using the norm X k = σ j X 2 where σ 1 X σ 2 X σ min{m,n} X are the singular values of X M m,n. The norm X k is known as the 2, k-spectral norm[22]. The authors of [7] mistakenly referred this as the Ky Fan k-norm X k = k σ jx of the matrix X; see [13, p.445]. Consider the following condition on {F j } mn M mn. C2 for all orthonormal basis { x 1, x 2,..., x n } of C n, P k = x i x j F k x i x j F k : 1 k mn i, is a set of mutually orthogonal rank one matrices. Notice that P k is indeed the Choi matrix of the map A F k AF k. In [7], the authors showed that under the assumption C2, if 1 > mn j=p+1 F j 2 k and E ij L 1 E ij i, mn j=p+1 γ j F j 2 k 1 mn j=p+1 F j 2 k 1/2, I n I m j=p+1 then φ is k-positive. In [8], the authors stated the result using the assumption C1 instead of C2 without explaining their relations. In addition, there are some typos in the papers that further obscured the P j, 6

results. In fact, it can be deduced from [31, Proposition 4.14] the special case for m = n was also treated in [29, Lemma 1] that the conditions C1 and C2 are equivalent. In the following, we use results in the previous sections to refine and improve the results in [7, 8] using the concept of the k-numerical range. Proposition 3.2 Suppose that {F j : 1 j mn} is an orthonormal basis of M m,n and L : M n M m has the form LX = γ j F j XF j j=p+1 γ j F j XF j, γ 1,..., γ mn 0. Assume that 1 k min{m, n} and ξ k = 1 max W k mn j=p+1 F j F j > 0. a If then L is k-positive. γ i ξ 1 k max W k j=p+1 γ j F j F j, i = 1,..., p, b If p = mn 1 and γ i < ξ 1 k γ mn max W k F mnf mn = ξ 1 k γ mn F mn 2 k for all i = 1,..., mn 1, then L is not k-positive. Proof. a Let w k = max W k r=p+1 γ rf r F r. Suppose γ i ξ 1 k w k for each i = 1,..., p. We show that I k L x x is positive semidefinite for all x Γ n,k, where Γ n,k is defined in 2. The conclusion will then follow from Proposition 2.3. x 1 We may extend x Γ n,k to x Γ n,n with x =. such that { x 1,..., x n } is x n an orthonormal basis for C n. Since the conditions C1 and C2 are equivalent, the matrices {P 1,..., P mn } defined in C2 form a set of mutually orthogonal set of rank one matrices. Thus, mn P r = I mn and hence F r x i x j F r 1 i,j n = I mn. Focusing on the leading mk mk principal submatrix, we have F r x i x j F r 1 i,j k = I mk r=p+1 F r x i x j F r 1 i,j k. 5 Note that tr r=p+1 γ r F r x i x j F r 1 i,j k = x j r=p+1 γ r F r F r x j w k. 7

Thus, r=p+1 γ r F r x i x j F r 1 i,j k w k I mk. 6 Applying this argument to the special case when γ p+1 = = γ mn, we see that r=p+1 F r x i x j F r 1 i,j k max W k j=p+1 By 5 and the fact that ξ k = 1 max W k j=p+1 F j F j, we have Because w k ξ 1 k ξ k I mk F r x i x j F r 1 i,j k. γ i for each i = 1,..., p, we have w k I mk w k ξ 1 k F r x i x j F r 1 i,j k By 6 and 7, we have the desired operator inequality r=p+1 γ r F r x i x j F r 1 i,j k Fj F j I mk. γ r F r x i x j F r 1 i,j k. 7 γ r F r x i x j F r 1 i,j k. b Suppose that the hypothesis of b holds. We can choose x 1,..., x k in C n so that tr F mn x i x j F mn 1 i,j k = max W k F mnf mn = F mn 2 k, i.e., the rank one matrix γ mn F mn x i x j F mn 1 i,j k has a nonzero eigenvalue γ mn F mn 2 k. Now, mn 1 Thus, the matrix γ r F r x i x j F r 1 i,j k < ξ 1 k γ mn mn 1 mn 1 F r x i x j F r 1 i,j k γ mn I mk. γ r F r x i x j F r 1 i,j k γ mn F mn x i x j F mn 1 i,j k has a negative eigenvalue. The result follows from Proposition 2.3. Part b of the Proposition 3.2 was proved in [7, 8]. Using the fact that max W k r=p+1 γ r F r F r r=p+1 γ r F r 2 k for all nonnegative numbers γ p+1,..., γ mn, we can deduce the main result in [7, 8]. 8

Corollary 3.3 Suppose {F j : 1 j mn} is an orthonormal basis of M m,n and a map L : M n M m has the form mn LX = γ j F j XF j γ j F j XF j, γ 1,..., γ mn 0. j=p+1 Assume that 1 k min{m, n} and ξ k = 1 mn j=p+1 F j 2 k > 0. If then L is k-positive. γ i ξ 1 k j=p+1 γ j F j 2 k for all i = 1,..., p, In [8], the authors mistakenly claimed that if mn j=p+1 F j 2 k+1 < 1 and γ i < 1 ξ k+1 j=p+1 γ j F j 2 k+1 for all i = 1,..., p, then L is not k + 1-positive, see [8, Theorem 1]. The case when p = mn 1 was proved in [7, Theorem 1] and also Proposition 3.2. However, the following example shows that the claim in general does not hold when p < mn 1. Also, this example demonstrates that Proposition 3.2 is stronger than Corollary 3.3 the result in [7, 8]. Example 3.4 Let m = n = 8. Suppose {F 1,..., F 64 } is an orthonormal basis for M 8 such that F 63 = 1 1 1 4 I 4 and F 1 1 64 = 1 1 1 4 I 4. 1 1 Define L : M 8 M 8 by 62 64 LX = γ j F j XF j F j XF j. Then F 63 2 2 + F 64 2 2 = 1/2 + 1/2 = 1 and W 64 2 j=63 F j F j = W 2 I 8 /4 = {1/2}. So, Corollary 3.3 is not applicable. By Proposition 3.2, the map L is 2-positive if γ i 1 for i = 1,..., 62. 4 D-type linear maps In this section, we consider linear maps L : M n M n of the form A = a ij diag a kk d k1,..., a kk d kn A 8 k=1 for an n n nonnegative matrix D = d ij. This type of maps will be called D-type linear maps. The question of when a D-type map is positive was studied intensively by many authors and applied in quantum information theory to detect entangled states and construct entanglement witnesses [18, 26]. For example, if D = n 1I n + E 12 + + E n 1,n + E n,1, we get a positive map A = a ij n 1 diag a 11, a 22, a 33,..., a nn + diag a nn, a 11, a 22,..., a n 1,n 1 A, which is not completely positive. This can be viewed as a generalization of the Choi map [5]. j=63 k=1 9

4.1 Criteria for k-positivity For γ 0, define L γ : M n M n by L γ A = γtr AI n A. This is a well-known example in literature. It is known that for all k {1,..., n}, L γ is k-positive if and only if γ k, for example, see [31, p.119], which can also be proved easily by Proposition 2.1. Notice that the map L γ is a D-type map with all entries of D being γ. In the following, we present a necessary and sufficient criteria of D-type linear map to be k-positive. Proposition 4.1 Suppose L : M n M n is a D-type map 8 for an n n nonnegative matrix D = d ij. The following conditions are equivalent. a L is k-positive. b d jj > 0 for all j = 1,..., n, and for all k n matrix U with columns u 1,..., u n C k satisfying tr U U = 1, we have u j Udiag d 1j,..., d nj U [ 1] u j 1, where X [ 1] is the Moore-Penrose generalized inverse of X. Proof. for all k n matrix U with column u 1,..., u n C k satisfying tr U U = 1, consider the unit vector u = u j ê j C nk, which can also be expressed as u = e j û j, where { e 1,..., e k } and { ê 1,..., ê n } are the standard basis of C k and C n, respectively, and û j is the j-th column of U T. Let F ij = L û i û j + û i û j = û j diag d 1l,..., d nl û i ê l ê l, l=1 which is a diagonal matrix with the l-th diagonal entry equal to û j diag d 1l,..., d nl û i. Proposition 2.1, L is k-positive if and only if the nk nk matrix I k L u u = e i e j F ij û i û j = e i e j F ij u u i, i, By is positive semi-definite. Notice that the above matrix is permutationally similar to F ij û i û j e i e j = F ij e i e j û û, i, i, 10

where û = n ê j u j. Recall that û j is the j-th column of U T. Direct computation shows that F ij e i e j = û j diag d 1l,..., d nl û i ê l ê l e i e j i, = = = i, l=1 ê l ê l l=1 l=1 l=1 û i diag d 1l,..., d nl û j e i e j i, ê l ê l U T diag d 1l,..., d nl U T T ê l ê l Udiag d 1l,..., d nl U = D 1 D n, where D l = U diag d 1l,..., d nl U for l = 1,..., n. Therefore, the condition is equivalent to: c û lies in the range of D 1 D n and D 1 D n 1/2 [ 1] û 1. Note that for all choice of U satisfying tr U U = 1 the corresponding û j always lies in the range of D j for j = 1,..., n if and only if d jj > 0 for all j = 1,..., n; the norm inequality in c is the same as the inequality stated in b. Therefore, condition a is equivalent to condition c, which is equivalent to condition b. Notice that the inequality in Proposition 4.1b is equivalent to tr Udiag d 1j,..., d nj U [ 1] UU 1. Also, Proposition 4.1 is particularly useful when k = 1. Corollary 4.2 Let L : M n M m be a D-type map of the form 8 with D = d ij. For u = u 1, u 2,..., u n t C n, let f j u = n d ij u i 2. Then, L is positive if and only if any one of the following equivalent conditions hold 1 d ii > 0 for all i = 1,..., n and u j 2 u j 0 f j u 1 for every unit vector u = u 1, u 2,..., u n t C n. 2 d ii > 0 for all i = 1,..., n and n u j 2 f j u 1 for every vector u = u 1, u 2,..., u n t C n with u i 0 for all i = 1,..., n. Proof. For k = 1, 1 is equivalent to condition b in Proposition 4.1. 2 is equivalent to 1 because u j 2 f j u is continuous and homogeneous in u. 4.2 Construction of D-type positive maps In this section, we discuss how to construct D-type positive linear maps using the results in previous sections. Recall that a permutation π of i 1,..., i l is an l-cycle if πi j = i j+1 for j = 1,..., l 1 and πi l = i 1. Note that every permutation π of 1,..., n has a disjoint cycle decomposition π = π 1 π 2 π r, that is, there exists a set {F s } r s=1 of disjoint cycles of π with r s=1 F s = {1, 2,..., n} such that π s = π Fs and πi = π s i whenever i F s. We have the following. 11 T

Proposition 4.3 Suppose π is a permutation of 1, 2,..., n with disjoint cycle decomposition π 1 π r such that the maximum length of π i is equal to l > 1 and P π = δ iπj is the permutation matrix associated with π. For t 0, let Φ t,π : M n M n be the D-type map of the form 8 with D = n ti n + tp π. Then Φ t,π is positive if and only if t n l. Proof. It is easily checked that for 0 t 1, the function gr 1, r 2,..., r s = s 1 s t + tr i 1 for all r i > 0 and r 1 r 2 r s = 1, 9 and the function g attains the maximum 1 when r 1 = = r s = 1. Suppose 0 t n l. We are going to use condition 2 in Corollary 4.2 to show that Φ t,π is positive. for all vector u = u 1, u 2,..., u n t C n, with u i 0 for all i = 1,..., n, we have f i u = n t u i 2 + t u πi 2. So, by Corollary 4.2, Φ t,π is positive if fu 1, u 2,..., u n = u i 2 n t u i 2 + t u πi 2 1 10 for all vector u = u 1, u 2,..., u n t with nonzero entries. Suppose π is a product of r disjoint cycles, that is, π = π 1 π 2 π r. Let F j be the set of indices corresponding to the cycle π j and l j denote the number of elements in F j for j = 1,..., r. Then l = max{l 1,..., l r } and j l j = n. for all vector u = u 1, u 2,..., u n t C n, with u i 0 for all i = 1,..., n, we have i F j u πj i 2 u i 2 = 1. It follows that fu 1, u 2,..., u n = = r u i 2 r n t u i 2 + t u πi 2 = l j n 1 i F j l j l j n t + l j n t u π j i 2 u i 2 u i 2 n t u i 2 + t u i F πj j i 2 r l j n 1 = 1, whenever 0 l j n t 1 for all 1 j r by 9, or equivalently, 0 t n l j n l. Therefore, 10 holds. Conversely, suppose t > n l. Let π = π 1π 2 π r be a decomposition of π into disjoint cycles. Without loss of generality, we may assume that l 1 = l l j for all j = 2,..., r, and π 1 is a cycle on 1, 2,..., l. Let u i = ɛ i 2, where 0 < ɛ < 1 n lt for i = 1,..., l and u i = 1 for l + 1 i n. Then we have fu 1, u 2,..., u n = l 1 1 n t + tɛ + 1 n t + t l 1 n t + tɛ + n l n l 1 > n t + t 1 n + n l n lt = l 1 n n + n l = l l n n + n l n 12 ɛ l 1 + = 1, i=l+1 1 n t + t

which implies Φ t,π is not positive. Next, we consider a general map Λ D of the form 8. Proposition 4.4 Let Λ D : M n M n have the form 8 for a nonnegative matrix D = d ij with all row sum and column sum equal to n. Then Λ D is positive if d ii n 1 for all i = 1,..., n. Moreover, the following conditions are equivalent. a Λ D is completely positive. b Λ D is 2-positive. c D = ni n. Proof. Suppose d ii n 1 for all i = 1,..., n. Then D = n 1I + S for a doubly stochastic matrix S, which is a convex combination of permutation matrices for example, see [13, Theorem 8.7.1, pp. 527]. We may represent S as S = m p i P πi for some permutations π 1, π 2,..., π m of {1, 2,..., n} and positive scalars p i with m p i = 1. Let S i = n 1I n + P πi and Λ Si be the linear map of the form as in 8. By Proposition 4.3, Λ Si is a positive map. Thus, Λ D is a convex combination of positive maps, and is therefore positive. Next, we prove the three equivalent conditions. The implication. a b is clear. For c a, it is well known and easy to check, say, by considering the Choi matrix, that Λ D is completely positive if D = ni n. It remains to prove b c. Suppose D ni n. Then d ii < n for some i. Without loss of generality, we assume that i = 1. Let U = u 1 u n = 1 1 0 0 M n 0 1 1 2,n and define D j := Udiag d 1j, d 2j,..., d nj U = 1 n d1j 0 0 n d 1j, j = 1,..., n. As n > d 11, û 1 D [ 1] 1 û 1 = 1 d 11 > 1 n and û j D [ 1] j û j = 1 n d 1j 1 n for j = 2,..., n. Hence, and Λ D is not 2-positive by Proposition 4.1. û j D [ 1] j û j > 1, In [26], the positive map Λ D with D = n 1I n +P for a permutation matrix P was considered, and the special case when P is a length n-cycle was discussed in details. By Propositions 4.3 and 4.4, we have the following corollary. Corollary 4.5 Let Λ D : M n M n be a D-type map of the form 8 with D = n 1I n + P for a permutation matrix P. Then Λ D is positive. Moreover, the following are equivalent. a Λ D is completely positive. b Λ D is 2-positive. c D = ni n. 13

The condition d ii n 1 for each i is not necessary for Λ D in Proposition 4.4 to be positive as seen below. 1.35 1 0.65 Example 4.6 Let D = 0.65 1.35 1. Here, d ii < 2 = 3 1. Direct computation shows 1 0.65 1.35 that 3 u j 2 f j u 1 for all u 1, u 2, u 3 C 3. Therefore, Λ D is positive by Corollary 4.2. Example 4.7 In Proposition 4.3, let 0 t 1 and D = d ij = n ti n + ts, where S = s 1 s 2 s n s n s 1 s n 1...... with s i 0 i = 1, 2,..., n and n s i = 1. Define Λ D : M n M n by s 2 s 3 s 1 where f 1 a 12 a 1n a 21 f 2 a 2n Λ D a ij =......, a n1 a n2 f n f 1 = n t 1 + ts 1 a 11 + ts n a 22 + ts n 1 a 33 + + ts 2 a nn, f 2 = ts 2 a 11 + n t 1 + ts 1 a 22 + ts n a 33 + + ts 3 a nn,. f n = ts n a 11 + ts n 1 a 22 + ts n 2 a 33 + + n t 1 + ts 1 a nn. By Proposition 4.3, the map Λ D is positive. Finally, we give an example which illustrates how to apply Proposition 4.3 to construct positive elementary operators for all dimension. Example 4.8 Let H and K be Hilbert spaces of dimension at least n, and let { e i } n and { ê j } n be any orthonormal sets of H and K, respectively. for all permutation π id of {1, 2,..., n}, let lπ = l n. Let Φ t,π : BH BK be defined by Φ t,π A = n t E ii AE ii + t E i,πi AE i,πi E ii A E ii for all A BH, where E ji = ê j e i. Then Φ t,π is positive if and only if 0 t n l. In fact, for the case dim H = dim K = n, Φ t,π is a D-type map of the form 8 with D = n ti + tp π as discussed in Proposition 4.3. 14

4.3 Decomposability of D-type positive maps Decomposability of positive linear maps is a topic of particular importance in quantum information theory since it is related to the PPT states that is, the states with positive partial transpose. In this section, we will give a new class of decomposable positive linear maps. The following result is well known see [15]. Proposition 4.9 Suppose L : M n M m has the form 8. Then L is decomposable if and only if the Choi matrix CL is a sum of two matrices C 1 and C 2 such that C 1 and the partial transpose of C 2 are positive semi-definite. In [26], it was shown that the linear maps Φ k = Φ 1,π with πi = i + k mod n in Proposition 4.3 are indecomposable whenever either n is odd or k n 2. It was asked in [26] that whether or not Φ n 2 is decomposable when n is even. In this section, we will answer this question by showing that Φ n 2 is decomposable. In fact, this is a special case of the following proposition as π 2 = id. Proposition 4.10 Let π be a permutation of {1, 2,..., n}. If π 2 = id, then the positive linear map Φ 1,π in Proposition 4.3 is decomposable. Proof. For simplicity, denote Φ = Φ 1,π. Let F be the set of fixed points of π. Since ΦE ii = n 2E ii + E πi,πi and ΦE ij = E ij, the Choi matrix of Φ is CΦ = n 2E ii E ii + E ij E ij E πi,πi E ii i j = n 1E ii E ii + n 2E ii E ii i F i F Let and i j;πi j E ij E ij + i F E πi,πi E ii i F C 1 = n 1E ii E ii + i F i F E i,πi E i,πi. n 2E ii E ii E ij E ij i j;πi j C 2 = E πi,πi E ii E i,πi E i,πi. i F i F Since π 2 = id, the cardinal number of F c must be even. Thus we have C 2 = E πi,πi E ii + E ii E πi,πi E i,πi E i,πi E πi,i E πi,i. As i<πi C T 2 2 = i<πi E πi,πi E ii + E ii E πi,πi E i,πi E πi,i E πi,i E i,πi 0, we see that C 2 is PPT. Observe that C 1 = A 0, where A = aij M n is a Hermitian matrix satisfying a ii = n 2 or n 1, a ij = 0 or 1 so that n a ij = 0. It is easily seen from the strictly diagonal dominance theorem [13, Theorem 6.1.10, pp. 349] that A is positive semi-definite. So C 1 0, and by Proposition 4.9, Φ is decomposable. 15

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