Compression, Matrix Range and Completely Positive Map

Transcription

1 Compression, Matrix Range and Completely Positive Map Iowa State University Iowa-Nebraska Functional Analysis Seminar November 5, 2016

2 Definitions and notations H, K : Hilbert space. If dim H = n <, H = C n. B(H, K) : bounded linear operators from H to K. If dim H = m and dim K = n, B(H, K) = M n, m. B(H) = B(H, H) and M n = M n, n. A B(H) is said to be self-adjoint if A = A. B(H) sa will denote the space of self-adjoint operators in B(H). Every A B(H) has a self-adjoint decomposition A = A 1 + ia 2, A 1, A 2 B(H) sa. If dim H = n, B(H) sa = H n, the set of n n Hermitian matrices. S R n, C n is said to be convex if for all x, y S, the line segment {tx + (1 t)y : 0 t 1} S.

3 Compression of linear operators Suppose A B(H) and K is a norm closed subspace of H. Let P K B(H, K) be the orthogonal projection of H onto K. Then B = P K A K B(K) is called a compression of A to K and A is a dilation of B to H. Let A M n and 1 m n. Then B M m is a compression of A if and only if there exists V M n,m such that V V = I m and B = V AV. For A B(H) and m 1, let W m (A) = {B M m : B is a compression of A} For m = 1, W 1 (A) = { Ax, x : x H, x, x = 1} is the numerical range of A, usually denoted by W (A). By the Toeplitz- Hausdorff Theorem, W (A) is convex. For m > 1, W m (A) is usually not convex.

4 Compression of linear operators Theorem 1 (Sz.-Nagy and Foias) A B(K) is a contraction ( A 1) if and only if A has a unitary dilation U B(H) such that A k = P K U k K for all k 1. Given A B(H), the numerical radius of A is given by w(a) = sup{ z : z W (A)}. w(a) is a norm on B(H) and satisfies w(a) A 2w(A). Theorem 2 (Sz.-Nagy and Foias) A B(K) satisfies w(a) 1 if and only if there is a unitary U B(H) such that A k = 2 P K U k K for all k 1.

5 Compression of linear operators Theorem 3 (Ando, Arveson) ( A B(K) satisfies ) w(a) 1 if and only if A is a compression of 0 2IH for some H. 0 0 (( )) (( )) 0 2IH 0 2 Note: W = W = {z C : z 1} (( )) 0 2 w(a) 1 W (A) W 0 0 Let A = A 1 + ia 2 be the self-adjoint decomposition of A B(K). Then W (A) = W (A 1, A 2 ) = {( A 1 x, x, A 2 x, x ) : x K, x, x = 1} R 2 Given A 1,, A p B(K) sa, define the joint numerical range W (A 1,, A p ) = {( A 1 x, x,,, A p x, x ) : x K, x, x = 1} R p

6 Completely positive map A B(H) is said to be positive (A 0) if Ax, x 0 for all x H. An operator system S of a C -algebra A, is a norm-closed self-adjoint (S = S ) subspace S of A containing 1 A. A linear map Φ : S B is positive on S if A 0 Φ(A) 0 Φ k : M k (S) M k (B), Φ k ((A ij )) = (Φ(A ij )) Φ is k-positive if Φ k is positive. Φ is completely positive if Φ is k-positive for all k 1. Theorem 4 (Arveson s Extension Theorem) Let A be a unital C -algebra and S be an operator system of A. Then every completely positive map from S to a C -algebra B can be extended to a completely positive map from A to B.

7 Numerical range and positivity Suppose W (A 1, A 2 ) W (B 1, B 2 ). If c 0 I + c 1 B 1 + c 2 B 2 0, then we have (c 0 I + c 1 B 1 + c 2 B 2 ) x, x 0 for all x H with x, x = 1 c 0 + (c 1, c 2 ) (b 1, b 2 ) 0 for all (b 1, b 2 ) W (B 1, B 2 ) c 0 + (c 1, c 2 ) (a 1, a 2 ) 0 for all (a 1, a 2 ) W (A 1, A 2 ) (c 0 I + c 1 A 1 + c 2 A 2 ) x, x 0 for all x K with x, x = 1 Therefore, the map Φ(c 0 I + c 1 B 1 + c 2 B 2 ) = (c 0 I + c 1 A 1 + c 2 A 2 ) (1) is positive. Remark: If A = A 1 + ia 2 and B = B 1 + ib 2. Then (1) is equivalent to Φ(c 0 I + c 1 B + c 2 B ) = (c 0 I + c 1 A + c 2 A ).

8 Dilation and extension of completely positive map Theorem 5 (Stinespring s dilation theorem) Let A be a unital C -algebra, and let Φ : A B(H) be a linear map. Then Φ is completely positive if and only if there exist a Hilbert space K, a unital C -homomorphism π : A B(K), and a bounded operator V B(H, K) such that Φ(T ) = V π(t )V for all T A. Note: Φ is unital if and only if V V = I H. Therefore, A is a compression of B I H for some H if and only if A = Φ(B) for some unital completely positive map Φ.

9 Reformulation of Theorem 3 Let B = ( ) = B 1 + ib 2, B 1 = ( ) ( 0 i, B 2 = i 0 ). Theorem 3a Suppose A 1, A 2 B(H) sa. Then W (A 1, A 2 ) W (B 1, B 2 ) if and only if the map is completely positive. Φ(c 0 I 2 + c 1 B 1 + c 2 B 2 ) = c 0 I H + c 1 A 1 + c 2 A 2 Theorem 3b Suppose A B(H). Then the map Φ(c 0 I 2 + c 1 B + c 2 B ) = c 0 I H + c 1 A + c 2 A is positive on span(i 2, B, B ) if and only if it is completely positive.

10 Another Proof of Theorem 3b Theorem 6 (Choi) Let S 2 be the space of 2 2 complex symmetric matrices. Then every positive map Φ : S 2 B(H) is completely positive. Choi proves the above theorem for finite dimensional H. The infinite dimensional case can be proven from the finite dimensional case. Theorem 3c Let B M 2 and A B(H). Then the map Φ(c 0 I 2 + c 1 B + c 2 B ) = c 0 I K + c 1 A + c 2 A is positive on span(i 2, B, B ) if and only if it is completely positive. Proof. Every B M 2 is unitarily similar to a symmetric matrix S. Let S = span(i 2, S, S ). If S = S 2, the result follows from Theorem 6. If S S 2, then S = C or C 2 and the result follows.

11 Extension ( ) 0 2 Recall = B ib 2 = ( 1 0 Let B 3 = 0 1 ( Conjecture 1 (Extension of Theorem 3a) Suppose A 1, A 2, A 3 B(H) sa. and only if the map ) ( 0 i + i i 0 ). ). B 1, B 2, B 3 are known as the Pauley matrices. Then W (A 1, A 2, A 3 ) W (B 1, B 2, B 3 ) if Φ(c 0 I 2 + c 1 B 1 + c 2 B 2 + c 3 B 3 ) = c 0 I H + c 1 A 1 + c 2 A 2 + c 3 A 3 is completely positive. 1) The conjecture fails for dim H = 2. Just take A i = B t i. 2) If dim H 2, W (A 1, A 2, A 3 ) is convex but W (B 1, B 2, B 3 ) = {w R 3 : w = 1}. If W (A 1, A 2, A 3 ) W (B 1, B 2, B 3 ), then W (A 1, A 2, A 3 ) is a singleton. Therefore, all A i are scalar and the conjecture holds.

12 Extension Let B 1, B 2, B 3 be the Pauley matrices. Set ˆB i = B i B t i for i = 1, 2, 3. Then W (ˆB 1, ˆB 2, ˆB 3 ) = {w R 3 : w 1} is convex. Conjecture 2 (Extension of Theorem 3b) Suppose A 1, A 2, A 3 B(H) sa. Let Φ(c 0 I 2 + c 1 ˆB1 + c 2 ˆB2 + c 3 ˆB3 ) = c 0 I K + c 1 A 1 + c 2 A 2 + c 3 A 3 Then Φ is positive on span(i 2, ˆB 1, ˆB 2, ˆB 3 ) if and only if Φ is completely positive. The conjecture holds if dim H = n 3. Let Ψ : M 2 M n be given by Ψ(X) = Φ(X X t ). If Φ is positive and n 3, then Ψ is decomposable. There exist completely positive Ψ 1, Ψ 2 : M 2 M n such that Ψ(X) = Ψ 1 (X) + Ψ 2 (X) t. Then the result follows. Question Does the result hold for all n?

13 Extension Theorem 7 (Choi and Li) Suppose B M 2 or B = [b] B 1 M 3. Then for all A B(H), W (A) W (B) if and only if the map is completely positive. Questions: Φ(c 0 I 2 + c 1 B + c 2 B ) = c 0 I H + c 1 A + c 2 A 1) If B M 3 satisfies the conclusion in Theorem 7, must B be unitarily similar to [b] B 1? 2) For which subset S of C can we find B M n such that W (B) = S and satisfies the conclusion in Theorem 7? 3) Does there exists B M n such that W (B) = the square with vertices {1, 1, i, i} and satisfies the conclusion in Theorem 7? B = diag (1, 1, i, i) does not work.

14 k-positive maps Suppose S is an operator system of M m and k 1. For a fixed H, let Clearly, P k (S, H) = {Φ : S B(H) is k-positive}, and CP(S, H) = {Φ : S B(H) is completely posotive}. CP(S, H) P k (S, H) P 2 (S, H) P 1 (S, H) The previous results shows that we have CP(S, H) = P 1 (S, H) for 1) S = span(i, B, B ) with B M 2 or B = [b] B 1 M 3 and any H. 2) S = span(i, B 1, B 2, B 3 ) and dim H 3. Question: When will CP(S, H) = P k (S, H)?

15 Matrix range Let A B(H). For each m 1, Arveson defines the matrix range W n (A) = {Φ(A) : Φ is a unital completely positiive map from B(H) to M n } Theorem 8 (Arveson) 1) W n (A) is C convex. That is, given X 1,..., X k W n (A) and Z 1,..., Z k M n such that k i=1 Z i Z i = I n, we have k i=1 Z i X i Z i W n (A). W n (A) is the closure of the smallest C convex set containing W n (A). 2) Let A be a normal operator and let n 1. Then W n (A) is the closure of { r i=1 λ ih i : r 1, H i 0, λ i sp(t ) and r i=1 H i = I n } 3) For some irreducible operators, the sequence {W n (A)} n=1 is a complete invariant for unitary similarity.

16 Choi s representation theorem Theorem 9 (Choi) Suppose Φ : M n M m is a linear map. Then the following conditions are equivalent: (a) Φ is completely positive. (b) Φ is k-positive for k = min(m, n). (c) The Choi matrix C(Φ) = (Φ(E ij )) is positive semidefinite. (d) There exist V 1,..., V r M n, m such that Φ(A) = r j=1 V j AV j. (2) Furthermore, suppose (d) holds. Then we have (1) The map Φ is unital (Φ(I n ) = I m ) if and only if r j=1 V j V j = I m. (2) The map Φ is trace preserving ( tr (Φ(A)) = tr (A)) if and only if r j=1 V jv j = I n. The minimum of r in (2) is called the rank of Φ.

17 Joint matrix range Given n, m 1, let CP(n, m) be the set of unital completely positive maps from M n to M m. For 1 r mn, let CP r (n, m) be the set of Φ CP(n, m) of rank r. Clearly, CP 1 (n, m) CP 2 (n, m) CP mn (n, m) = CP(n, m) Let A = (A 1, A 2,..., A p ) H p n. For each m 1 and 1 r mn, define We have W r m(a) = {(Φ(A 1 ),..., Φ(A p )) : Φ CP r (n, m)} Wm(A) 1 Wm(A) 2 Wm mn (A) = W m (A) Toeplitz-Haudorff Theorem: W 1 1 (A 1, A 2 ) = W 1 (A 1, A 2 ). Question: When will W r m(a) = W m (A)? Note: W r m(a 1, A 2,..., A p ) = W 1 m(a 1 I r, A 2 I r,..., A p I r ).

18 Joint matrix range Theorem 10 Suppose A 1, A 2,..., A p H n. Let 1 r mn 1. Then W r m(a 1,..., A p ) = W m (A 1,..., A p ) (*) if m 2 (p + 1) 1 < (r + 1) 2 δ mn,r+1. For example, if p = k 2 1 and n > k, then W mk 1 m (A 1,..., A p ) = W m (A 1,..., A p ) for all A 1,..., A p H n. In this case, one can show that r = mk 1 is the smallest number for (*) to hold. Putting m = r = 1, we have W1 1(A 1,..., A p ) = W 1 (A 1,..., A p ) if p < 2 2 δ n,2 = 4 δ n,2. Therefore, W (A 1, A 2, A 3 ) is convex if n 3.

19 Joint matrix range Recall that for A = (A 1, A 2,..., A p ) H p n, Wm(A) 1 Wm(A) 2 Wm mn (A) = W m (A) Let S = span(i n, A 1, A 2,..., A p ) and H = C m. we have Define P k (A) = {(Φ(A 1 ),..., Φ(A p )) : Φ P k (S, H)} W m (A) P k (A) P 2 (A) P 1 (A) Note: P k (A) = W m (A) P k (S, M m ) = CP(S, M m ) For n 3, p = 3 we have W 1 1 (A) = P 1(A)

20 Compression of linear operators Recall that for A B(H) and m dim H, W m (A) = {B M m : B is a compression of A} Theorem 11 (Fan and Pall) Suppose A H n has eigenvalues a 1 a 2 a n and 1 m n. Then W m (A) consists of all B H m with eigenvalues b 1 b 2 b m satisfying the following inequalities: a i b i a n m+i for all 1 i m In particular, B W n 1 (A) if and only if a 1 b 1 a 2 b n 1 a n Suppose A M n is normal with eigenvalues a 1, a 2,..., a n. If a 1, a 2,..., a n C are collinear, then there exist c C and θ R such that e iθ A + ci n H n and we have W m (e iθ A + ci n ) = e iθ W m (A) + ci m

21 Compression of normal matrix Theorem 12 (Fan and Pall) Let A M n and B M n 1 be normal matrices with eigenvalues a 1, a 2,..., a n and b 1, b 2,..., b n 1, respectively. Suppose a 1, a 2,..., a q are each distinct from b 1, b 2,..., b q 1, while a i = b i 1 for q + 1 i n. Then B is a compression of A if and only if a 1, a 2,..., a q are collinear and every segment on this line limited by two adjacent a i s contains one b j, 1 j q 1. If no three a i s are collinear, then up to permutation of indices, we must have a i = b i for i = 1,..., n 2 and b n 1 a n 1 a n.

22 Compression of normal matrix Suppose A M n is normal with non-collinear eigenvalues a 1, a 2,..., a n. Let D m (A) = {diag (B) : B W m (A)} C m. Theorem 13 Suppose A M n is normal with non-collinear eigenvalues a 1, a 2,..., a n. Then the following conditions are equivalent: 1) D m (A) is convex. 2) W m (A) is convex. 3) W m (A) is C -convex. ( W 1 m(a) = W m (A) = W m (A)) 4) Every vertex of W (A) has multiplicity m.

23 Common compression of matrices Given A M n, B M m and 1 k n, m. A and B is said to have a common k-dimensional compression if there exist U M m,k and V M m,k such that U U = I k = V V and U AU = V BV. For m = k n, this is equivalent to the compression of Hermitian matrices studied by Fan and Pall. Extension of the result of Fan and Pall Theorem 14 Suppose and A H n and B H m have eigenvalues a 1 a 2 a n and b 1 b 2 b m, respectively, and 1 k n, m. Then B and C have a common k-dimensional compression if and only if the following inequalities hold: a i b m k+i and b i a n k+i for all 1 i k

24 Common compression of 3 3 normal matrices Theorem 15 Suppose A and B are 3 3 normal matrices with non collinear eigenvalues a 1, a 2, a 3 and b 1, b 2, b 3 respectively. Then 1) A and B have a common 2-dimensional compression C, with degenerate W (C) ( C is normal), if and only if either (a) W (A) and W (B) have a vertex in common and the corresponding opposite sides intersect, or (b) one vertex v of W (A) lies on an edge s of W (B) and the vertex v in W (B) opposite to s lies on the edge s in W (A) opposite to v.

25 Common compression 2) A and B have a common 2-dimensional compression C, with non-degenerate W (C) ( C is not normal), if and only if the following conditions are satisfied: (a) W (A) W (B) is an m-sided polygon P with m 3. (b) Every edge of W (A) and W (B) intersects a side of P at more than one point. (c) For m = 6, the diagonals of P are concurrent. m = 3 m = 4 m = 5 m = 6