Characterizations are given for the positive and completely positive maps on n n


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1 Special Classes of Positive and Completely Positive Maps ChiKwong Li and Hugo J. Woerdemany Department of Mathematics The College of William and Mary Williamsburg, Virginia ABSTRACT Characterizations are given for the positive and completely positive maps on n n complex matrices that leave invariant the diagonal entries or the kth elementary symmetric function of the diagonal entries, < k n. In addition, it was shown that such a positive map is always decomposable if n 3, and this fails to hold if n > 3. The real case is also considered.. INTRODUCTION Let M n denote the algebra of n n complex matrices, where is hermitian conjugation. A (complex) linear map : M n! M n is positive if maps the set of positive denite matrices into itself; is m?positive if I m is positive on M n M m, where I m is the identity operator on M m ; and is completely positive if is mpositive for every m. The structure of completely positive maps on M n is quite wellunderstood (e.g., see [C, PH, BHH, LoS]). Recently, there has been interest in studying completely positive maps satisfying some special properties such as leaving invariant the trace function, or leaving invariant the identity map, (e.g., see [BPS, LS] and their references). In particular, it has been shown that the structure of such maps is quite complicated. For k n, let S k (x ; : : : ; x n ) denote the kth elementary symmetric function of the numbers x ; : : : ; x n. The purpose of this note is to characterize those positive and completely positive maps : M n! M n that leave invariant the diagonal entries of matrices, i.e., (X) ii = X ii for all X and all i n; and those positive and completely positive maps : M n! M m, m n, that leave invariant the kth elementary symmetric function S k of the diagonal entries, i.e., S k ((X) ; (X) ; : : : ; (X) mm ) = S k (X ; : : : ; X nn ) for all X, where < k m. It turns out that all the trouble is restricted to the case k =. For k >, the structure of those maps is easy to describe. We also obtain similar results for the real case. After the rst version of this paper was submitted, we learned that our Theorems and 3 had been obtained independently by Kye, and would appear in [K]. A sizable part y Partially supported by NSF grant DMS 9594
2 of [K] was devoted to showing that a positive linear map on M n that leaves invariant the diagonal entries must be decomposable if n = 3. In [KK], it was shown that this conclusion fails if n 4. We found a dierent characterization for the decomposability of such positive maps (cf. Theorem 5 (c) ) that led to shorter proofs of the results in [K] and [KK]. We include these results in Section 4. Since the proofs of our Theorems and 3 are rather short, we include them in our discussion for the sake of completeness even though they have been proved in [K]. In our discussion, we shall use fe ; E ; : : : ; E nn g to denote the standard basis of M n. For a matrix A M n, we write A ( A > ) if A is positive semidenite (positive denite).. RESULTS ON COMPLETELY POSITIVE MAPS Recall that a A M n is a correlation matrix if A with all diagonal entries equal to one. It is wellknown that if A M n is a correlation matrix, then the map A : M n! M n of the form X 7! A X is completely positive; here A X denotes the Schur or Hadamard product. Clearly, for such a completely positive map, A (X) and X have the same diagonal entries. In Theorem we show that every completely positive map that leaves invariant the diagonal entries is of this form. As a result, the set of such completely positive maps is linearly isomorphic to the convex set of correlation matrices. For the structure of the set of correlation matrices, one may see [CV, GP, Lo, LiT]. Theorem Suppose : M n! M n is a completely positive map. The following conditions are equivalent. (a) satises (X) ii = X ii for all X and all i n. (b) (E ii ) = E ii for all i n. (c) There is a correlation matrix A M n such that is of the form X 7! A X. Proof. (a) ) (b): If : M n! M n is completely positive and leaves the diagonal entries invariant, then (E ii ) with diagonal entries equal to those of E ii. It follows that (E ii ) = E ii. (b) ) (c): It is known (e.g., see [C] and [PH]) that : M n! M n is a completely positive map if and only if the n n block matrix ((E ij )) i;jn is positive semidenite. It follows that for any i 6= j, (E ij ) = a ij E ij for some a ij C with a ij = a ji. Set a ii = for i = ; : : : ; n, and let A = (a ij ). Then is of the form X 7! A X. Since J = P i;jn E ij, we have A = A J = (J). (c) ) (a): Clear. The set of completely positive maps that leave the trace function invariant have been studied by several authors (see [BPS, LS] and their references). It was shown that the
3 structure of this set is rather complicated. Notice that the trace function is just the rst elementary symmetric function of the diagonal entries. In Theorem we show that for k >, the structure of those completely positive maps that leave invariant the kth elementary symmetric function of diagonal entries is much simpler. To simplify our notation, we shall use S k (X) to denote the kth elemenatry symmetric function of the diagonal entries of the square matrix X. Theorem Let m; n; k be positive integers such that < k m n. A completely positive map : M n! M m satises S k ( (X)) = S k (X) for all X M n if and only if m = n and is of the form X 7! P t (A X)P, where A M n is a correlation matrix, and (i) if < k < n then P M n is a permutation matrix, (ii) if k = n then P M n is a product of a permutation matrix and a diagonal matrix P n a j= je jj for some positive numbers a ; : : : ; a n satisfying n j= a j =. Proof. Suppose : M n! M m is a completely positive map leaving the kth elementary symmetric function of the diagonal entries invariant. Then S k ( (te jj ) + (D)) = S k ( (te jj +D)) = S k (te jj +D) is a polynomial in t of degree one for any positive diagonal matrix D. It follows that (E jj ) = a j E rj ;r j for some r j m. Suppose the function j 7! r j is not injective, say both (E pp ) and (E qq ) are multiples of E ss. Then there exists K f; : : : ; ng with k elements including p and q such that = S k ( P jk E jj), but ( P jk E jj) has less than k diagonal entries and hence S k ( ( P jk E jj)) =. As a result, we have m = n. Furthermore, there is a matrix P M n such that ( P n j= a= j E jj )P is a permutation and the linear operator ~ dened by ~ (X) = P t XP is a completely positive map satisfying ~ (E jj ) = E jj for all j = ; : : : ; n. Applying Theorem to ~, we conclude that ~ is of the form X 7! A X for some correlation matrix A. For any j < < j k n, we have = S k ( P k t= E j t ;j t ) = S k ( ~ ( P k E t= j t ;j t )) = k t=a jt. If k = n, we have n a j= j = ; if k < n, we have a j = for all j = ; : : : ; n. Thus is of the asserted form. The converse of the theorem is clear. 3. RESULTS ON POSITIVE MAPS In this section, we characterize those positive maps that leave invariant the diagonal entries or the kth elementary function of the diagonal entries for a xed k >. As can be seen, the structures of these maps are just slightly more complicated than those in the previous section. We shall denote by A;B the map on M n dened by A;B (X) = I X + A X + B X t for given hermitian matrices A and B with zero diagonals. 3
4 Theorem 3 Suppose : M n! M n is a positive map. The following conditions are equivalent. (a) satises (X) ii = X ii for all X and all i n. (b) (E ii ) = E ii for all i n. (c) = A;B for some hermitian matrices A; B M n with zero diagonals and such that I + D AD + DBD for all diagonal unitary matrices D. Proof. (a) ) (b): Use the same arguments in the proof of Theorem. (b) ) (c): For any r < s n, (E rr + E rs + E sr + E ss ) with diagonal equal to that of E rr + E ss. Thus (E rs + E sr ) = x rs E rs + x sr E sr with x rs = x sr. Similarly, we have (ie rs? ie sr ) = y rs E rs + y sr E sr with y rs = y sr. Set x rr = y rr = for r n, and set X = (x rs ); Y = (y rs ) M n. Then for any hermitian matrix H = R + is, where R is real symmetric and S is real skewsymmetric, we have (H) = (I + X) R + i(y S). Let Z = (H + ih ) + i(g + ig ) M n, where H ; G are real symmetric and H and G are real skewsymmetric. We have (Z) = (I + X) (H + ig ) + Y (ih? G ) = I Z +AZ +B Z t, where A = (X +Y )= and B = (X?Y )=. Since is a positive map, if Z = vv with v = (v ; : : : ; v n ) t where jv j = = jv n j =, and if D = P n j= v je jj, then (Z) = I + D AD + DBD. (c) ) (a): Let be dened as in (c). Clearly, (X) ii = X ii for all X and all i n. It remains to show that is a positive map. Now, for any x = (x ; : : : ; x n ) t C n with x j 6= for all j n, we have (xx ) = X(I +D AD+DBD )X for X = P n j= jx jje jj and some diagonal unitary matrix D. Hence (xx ). By continuity, we have (xx ) for all x C n, and hence maps the cone of positive semidenite matrices into itself. If X is positive denite, then X = ti + ~ X for some t > and some ~ X. It follows that (X) = (ti + ~ X) = ti + ( ~ X) >. Remark Suppose A and B are hermitian matrices with zero diagonals. One easily checks that the following conditions are equivalent. (i) I + D AD + DBD for all diagonal unitary matrices D. (ii) I + A + DBD for all diagonal unitary matrices D. (iii) I + D AD + B for all diagonal unitary matrices D. We chose to use condition (i) in Theorem 3 (c) because of its symmetry with respect to A and B. It would be nice to have an explicit condition on hermitian matrices A and B with zero diagonals such that the conditions (i) { (iii) hold. When n =, this happens if and only if ja j + jb j. The problem seems rather dicult for n 3. As pointed out by Dr. D. Farenick, one can deduce Theorem from Theorem 3. 4
5 By Theorem 3 and arguments similar to those in the proof of Theorem, we have the following result characterizing those positive maps that leave invariant the kth elementary symmetric function of the diagonal entries for k >. Theorem 4 Let m; n; k be positive integers such that < k m n. A positive map : M n! M m satises S k ( (X)) = S k (X) for all X M n if and only if m = n and is of the form X 7! P t ( A;B (X))P for some hermitian matrices A; B M n with zero diagonals, and some P satisfying Theorem (i) or (ii). 4. DECOMPOSABILITY A positive linear map : M n! M n is said to be decomposable if it is the sum of a completely positive map and a completely copositive map. Recall that is completely copositive if the mapping A 7! (A t ) is completely positive. For positive maps that leave invariant the diagonal entries, we have the following equivalent conditions for decomposability. Theorem 5 Let A;B : M n! M n be positive, i.e., satisfy the condition in Theorem 3 (c). The following are equivalent. (a) A;B is decomposable. (b) There exist diagonal matrices D A and D B such that I = D A + D B ; D A + A ; and D B + B : () (c) For all X; Y with I X = I Y we have tr (X + AX + BY ) : () Proof. For the equivalence of (a) and (b) see [K, Theorem.3]. For (b) ) (c) note that tr (D A + A)X ; tr (D B + B)Y implies (), where we use the identities I = D A + D B and I X = I Y. (c) ) (b): First note that it suces to nd diagonal matrices D = P n j= d je jj and E = P n j= e je jj so that I? D? E ; D + A ; E + B : (3) Indeed, if (3) holds we may put D A = D and D B = I? D to obtain (). 5
6 @ A Let A = B A ; Ai = E ii A ; An+i =?E E ii A ; i = ; :::; n:?e ii Then (3) is equivalent to A + nx d i A i + nx e i A n+i : (4) By a standard result on linear matrix inequalities (see, e.g., [BEFB, Section..]), there exist d i and e i such that (4) holds if and only if for every positive semidenite Z M 3n tr (ZA i ) = ; i = ; :::; n; implies tr (ZA ) : (5) Now, suppose X; Y with I X = I Y. Then Z := X X Y is positive semidenite and tr (ZA i ) = ; i = ; :::; n, and thus it follows from (5) that () must hold. By Remark, the positivity requirement on A;B may be reformulated as tr (X + AX + BD XD) (6) for all X and all diagonal unitary matrices D. Since I (D XD) = I X for any diagonal unitary matrix D, we see that (6) is a particular case of (). This, of course, is no surprise as decomposability implies positivity. It is natural to ask to what extent the converse is also true, i.e., when does positivity of A;B imply decomposability? We next show that this converse holds for dimensions 3 and lower, but in general fails to hold for dimensions 4 and higher. Theorem 6 Every positive map of the form A;B, where A; B M n are hermitian matrices with zero diagonals, on M n is decomposable if and only if n 3. Proof. For n = the statement is trivial. For n = one can choose D A = ja ji and D B = I? D A to be the decomposition. Let n = 3. Suppose (6) holds for all diagonal unitary matrices D. We need to show that () holds for all X; Y with I X = I Y. It suces to consider the case when X and Y have positive diagonals. The general case will then follow from continuity. For such X and Y, let E = (I X) =. Then E? XE? and E? Y E? are correlation matrices. Since (e.g., see [LiT]) all 3 3 correlation matrices are convex combinations of rank one correlation matrices, we have E? XE? = P r a ix i and E? Y E? = P s b iy i, where 6
7 a a r > and b b s > are two sets of convex coecients, and X i, Y i are rank one correlation matrices. Suppose a r b s. Write E? XE? as E? XE? = Xr? a i X i + (a r? b s )X r + b s X r : Apply these arguments to the matrices Xr? a i X i + (a r? b s )X r and Xs? b i Y i to remove Y s? if a r? b s b s? or X r if a r? b s < b s?. After nitely many repetitions of this process, we can write E? XE? = c i ~ Xi and E? Y E? = where c i form a set of convex coecients, and ~ X i, ~ Y i are rank one correlation matrices. Since rank one correlation matrices are similar via diagonal unitary matrices, we have ~Y i = D i ~ X i D i for some diagonal unitary D i. It then follows that and Hence Y = E( tr (X + AX + BY ) = X = E( c i ~ Xi )E = c i D i ~ X i D i )E = c i (E ~ X i E) c i D i (E ~ X i E)D i : c i ~ Yi ; c i tr ((E ~ Xi E) + A(E ~ Xi E) + BD i (E ~ Xi E)D i ) : It remains to provide for each n 4 an example of a positive map A;B that is not decomposable. It suces to provide such A;B for n = 4 as one may embed this example in higher dimensions by taking A and B. Introduce the matrices M = p p p p p i p p?i p?i 4 +i 4 CA ; N = 6 7 CA :
8 We claim that there exists an > such that I + M + DND I > (7) for all diagonal unitary matrices D M 4. Subsequently we shall show that for (A; B) = (M; N)=(? ), A;B is positive but not decomposable. To prove our claim, observe that I + M and I + N, and ( B Ker I + M) = span?ca ; p i p CA )?? ; Ker ( I + N) = span ( CA ) : Suppose that no > exists such that (7) holds. Then, by the compactness of the set of 4 4 diagonal unitary matrices, there exists a diagonal unitary matrix D such that Ker ( I + M) \ Ker ( I + DND ) 6= : In particular, there exist ; C such that f(?;?; p ; ) t + (?; i; ; p ) t g= p = D(;?;?; ) t : It follows that jj = ; jj = ; j + j = p ; j? ij = p : Hence = j + j? jj? jj = Re( ); = j? ij? jj? jj = Re(i ); and thus = j j = fre( )g + Re(i )g = ; which is a contradiction. In conclusion, there exists an > such that (7) holds. Now, let (A; B) = (M; N)=(? ). By Theorem 3(c) and (7) we obtain that A;B is positive. On the other hand, if X = i+i p? p? p i?i p? p i p? p? p? p? i p?? CA CA ; Y = ; 8??????
9 then X, Y, I X = I Y and tr (X + AX + BY ) =?4 < : By Theorem 5(c) we see that A;B is not decomposable. Theorem 6 was also obtained in [K] and [KK] using dierent methods. By Theorems 4 and 6, we have the following corollary. Corollary 7 Let k be a given integer with < k n. Every positive map on M n that preserves the kth elementary symmetric function of the diagonal entries is decomposable if and only if n RESULTS ON REAL MATRICES There has also been interest in studying completely positive maps and positive maps on M n (IR), the algebra of n n real matrices. As in the complex case, we say that a (real) linear transformation : M n (IR)! M m (IR) is positive if maps (real symmetric) positive denite matrices to positive denite matrices, and is completely positive if the block matrix ((E ij )) i;jn is positive semidenite. Using these denitions, one easily obtains the real analogs of Theorem and Theorem. However, the results on positive maps are slightly dierent due to the fact that a positive map imposes no condition on the space of skewsymmetric matrices. Theorem 8 Suppose : M n (IR)! M n (IR) is a positive map. The following conditions are equivalent. (a) satises (X) ii = X ii for all X and all i n. (b) (E ii ) = E ii for all i n. (c) There is a correlation matrix A M n (IR) and a linear transformation ~ : fx M n (IR) : X =?X t g! fx M n (IR) : X ii = for i = ; : : : ; ng such that is of the form X 7! A X + ~ (X? X t ). Theorem 9 Let m; n; k be positive integers such that < k m n. A positive map : M n (IR)! M m (IR) satises S k ( (X)) = S k (X) for all X M n if and only if m = n and is of the form X 7! P t ((X))P, where satises condition (c) of Theorem 8, and P satises Theorem (i) or (ii). 9
10 ACKNOWLEDGEMENT The rst author would like to thank Dr. D. Farenick for some helpful discussion and useful references. This research was motivated by a question of Dr. R. Mathias raised at the Fourth International Linear Algebra Society Conference. Thanks are also due to Dr. S.H. Kye for supplying preprints of [K] and [KK]. REFERENCES [BHH] G.P. Barker, R.D. Hill and R.D. Haertel, On the completely positive an positivesemidenitepreserving cones, Linear Algebra Appl. 56:9 (984). [BPS] R. Bhat, V. Pati and V.S. Sunder, On some convex sets and their extreme points, Math. Ann. 96: (993). [BEFB] S. Boyd, L. El Ghaoui, E. Feron and V. Balakrishnan, Linear Matrix Inequalities in Systems and Control Theory, SIAM, Philadelphia, 994. [C] M.D. Choi, Completely positive linear maps on complex matrices, Linear Algebra Appl. :859 (975). [CV] J.P.R. Christensen and J. Vesterstrom, A note on extreme positive denite matrices, Math. Ann. 44:6568 (979). [GPW] R. Grone, S. Pierce and W. Watkins, Extremal correlation matrices, Linear Algebra Appl. 34:6379 (99). [KK] H.J. Kim and S.H. Kye, Indecomposable extreme positive linear maps in matrix algebras, Bulletin London Math. Soc. 6: (994). [K] S.H. Kye, Positive linear maps between matrix algebras which x diagonals, Linear Algebra Appl. 6:3956 (995). [LS] L.J. Landau and R.F. Streater, On Birkho's theorem for doubly stochastic completely positive maps of matrix algebras, Linear Algebra Appl. 93:77 (993). [LiT] C.K. Li and B.S. Tam, A note on extreme correlation matrices, SIAM J. Matrix Analysis Appl. 5:9398 (994). [Lo] R. Loewy, Extreme points of a convex subset of the cone of positive semidenite matrices, Math. Ann. 53:73 (98). [LoS] R. Loewy and H. Schneider, Positive operators on the ndimensional ice cream cone, J. Math. Anal. Appl. 49: (967). [PH] J.A. Poluikis and R.D. Hill, Completely positive and Hermitianpreserving linear transformations, Linear Algebra Appl. 35: (98).
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