Nilpotent Completely Positive Maps. Nirupama Mallick

Transcription

1 Nilpotent Completely Positive Maps Nirupama Mallick (Joint work with Prof. B. V. Rajarama Bhat) OTOA ISI Bangalore December 16, 2014 Nirupama (ISIBC) OTOA 1 / 15

2 Nilpotent linear maps Nilpotent maps are very important and they appear in various contexts. Nilpotent CP-maps appears when we consider some special classes of unital CP-map called roots of states. Nirupama (ISIBC) OTOA 2 / 15

3 Nilpotent linear maps Nilpotent maps are very important and they appear in various contexts. Nilpotent CP-maps appears when we consider some special classes of unital CP-map called roots of states. V is a n-dimensional vector space, L : V V linear. L is nilpotent of order p 1 if L p = 0 and L p 1 0. Nirupama (ISIBC) OTOA 2 / 15

4 Nilpotent linear maps Nilpotent maps are very important and they appear in various contexts. Nilpotent CP-maps appears when we consider some special classes of unital CP-map called roots of states. V is a n-dimensional vector space, L : V V linear. L is nilpotent of order p 1 if L p = 0 and L p 1 0. Take V i = ker(l i ) for 1 i p. Then {0} V 1 V 2 V p = V. Nirupama (ISIBC) OTOA 2 / 15

5 Nilpotent linear maps Nilpotent maps are very important and they appear in various contexts. Nilpotent CP-maps appears when we consider some special classes of unital CP-map called roots of states. V is a n-dimensional vector space, L : V V linear. L is nilpotent of order p 1 if L p = 0 and L p 1 0. Take V i = ker(l i ) for 1 i p. Then {0} V 1 V 2 V p = V. l i := dim V i /V i 1, V 0 = {0}. l 1 + l l p = n and l 1 l 2 l p. (l 1, l 2,, l p ) is a partition of n. We call (l 1, l 2,, l p ) as nilpotent type of L. Nirupama (ISIBC) OTOA 2 / 15

6 Nilpotent maps and invariant subspaces M V invariant subspace of L. R := L M. Define S : V/M V/M by S(v + M) = L(v) + M. Nirupama (ISIBC) OTOA 3 / 15

7 Nilpotent maps and invariant subspaces M V invariant subspace of L. R := L M. Define S : V/M V/M by S(v + M) = L(v) + M. R, S are nilpotent of order at most p. Suppose R, S are of nilpotent type (r 1,, r p ), (s 1,, s p ) respectively. The tuples are extended by 0 s if necessary. Nirupama (ISIBC) OTOA 3 / 15

8 Nilpotent maps and invariant subspaces M V invariant subspace of L. R := L M. Define S : V/M V/M by S(v + M) = L(v) + M. R, S are nilpotent of order at most p. Suppose R, S are of nilpotent type (r 1,, r p ), (s 1,, s p ) respectively. The tuples are extended by 0 s if necessary. Littlewood-Richardson rules, Horn s inequalities. Nirupama (ISIBC) OTOA 3 / 15

9 Nilpotent maps and invariant subspaces M V invariant subspace of L. R := L M. Define S : V/M V/M by S(v + M) = L(v) + M. R, S are nilpotent of order at most p. Suppose R, S are of nilpotent type (r 1,, r p ), (s 1,, s p ) respectively. The tuples are extended by 0 s if necessary. Littlewood-Richardson rules, Horn s inequalities. Majorization inequalities: l l k (r r k ) + (s s k ) for all 1 k < p and l l p = (r r p ) + (s s p ). Nirupama (ISIBC) OTOA 3 / 15

10 Completely positive maps Suppose H is a finite dimensional Hilbert space with dim(h) = n. Let B(H) be the algebra of all bounded linear maps on H. Nirupama (ISIBC) OTOA 4 / 15

11 Completely positive maps Suppose H is a finite dimensional Hilbert space with dim(h) = n. Let B(H) be the algebra of all bounded linear maps on H. Definition A linear map α : B(H) B(H) is said to be CP-map if k i,j=1 Y i α(x i X j )Y j 0 X i, Y i B(H), k 1. Nirupama (ISIBC) OTOA 4 / 15

12 Completely positive maps Suppose H is a finite dimensional Hilbert space with dim(h) = n. Let B(H) be the algebra of all bounded linear maps on H. Definition A linear map α : B(H) B(H) is said to be CP-map if k i,j=1 Y i α(x i X j )Y j 0 X i, Y i B(H), k 1. What happens if α is nilpotent? We want to define nilpotency type of α. We will show that nilpotency order is not bigger than n Nirupama (ISIBC) OTOA 4 / 15

13 Choi-Kraus decomposition Suppose α : B(H) B(H) is CP-map. Then there exists operators L 1,, L d B(H) such that α(x) = d L i XL i X B(H). ( ) i=1 Nirupama (ISIBC) OTOA 5 / 15

14 Choi-Kraus decomposition Suppose α : B(H) B(H) is CP-map. Then there exists operators L 1,, L d B(H) such that α(x) = d L i XL i X B(H). ( ) i=1 The decomposition is not unique. But if α(x) = d j=1 M j XM j is another decomposition, then span{l i : 1 i d} = span{m j : 1 j d }. Nirupama (ISIBC) OTOA 5 / 15

15 Choi-Kraus decomposition Suppose α : B(H) B(H) is CP-map. Then there exists operators L 1,, L d B(H) such that α(x) = d L i XL i X B(H). ( ) i=1 The decomposition is not unique. But if α(x) = d j=1 M j XM j is another decomposition, then span{l i : 1 i d} = span{m j : 1 j d }. We fix one such decomposition say ( ). Then we can have Nirupama (ISIBC) OTOA 5 / 15

16 Choi-Kraus decomposition Suppose α : B(H) B(H) is CP-map. Then there exists operators L 1,, L d B(H) such that α(x) = d L i XL i X B(H). ( ) i=1 The decomposition is not unique. But if α(x) = d j=1 M j XM j is another decomposition, then span{l i : 1 i d} = span{m j : 1 j d }. We fix one such decomposition say ( ). Then we can have ker(α(1)) = d i=1 ker(l i). Nirupama (ISIBC) OTOA 5 / 15

17 Choi-Kraus decomposition Suppose α : B(H) B(H) is CP-map. Then there exists operators L 1,, L d B(H) such that α(x) = d L i XL i X B(H). ( ) i=1 The decomposition is not unique. But if α(x) = d j=1 M j XM j is another decomposition, then span{l i : 1 i d} = span{m j : 1 j d }. We fix one such decomposition say ( ). Then we can have ker(α(1)) = d i=1 ker(l i). For X 0, α(x) = 0 iff ran(x) d i=1 ker(l i ). Nirupama (ISIBC) OTOA 5 / 15

18 Choi-Kraus decomposition Suppose α : B(H) B(H) is CP-map. Then there exists operators L 1,, L d B(H) such that α(x) = d L i XL i X B(H). ( ) i=1 The decomposition is not unique. But if α(x) = d j=1 M j XM j is another decomposition, then span{l i : 1 i d} = span{m j : 1 j d }. We fix one such decomposition say ( ). Then we can have ker(α(1)) = d i=1 ker(l i). For X 0, α(x) = 0 iff ran(x) d i=1 ker(l i ). {x : α( x x ) = 0} = d i=1 ker(l i). Nirupama (ISIBC) OTOA 5 / 15

19 Choi-Kraus decomposition Suppose α : B(H) B(H) is CP-map. Then there exists operators L 1,, L d B(H) such that α(x) = d L i XL i X B(H). ( ) i=1 The decomposition is not unique. But if α(x) = d j=1 M j XM j is another decomposition, then span{l i : 1 i d} = span{m j : 1 j d }. We fix one such decomposition say ( ). Then we can have ker(α(1)) = d i=1 ker(l i). For X 0, α(x) = 0 iff ran(x) d i=1 ker(l i ). {x : α( x x ) = 0} = d i=1 ker(l i). ran(α(1)) = span{l i u : u H}. Nirupama (ISIBC) OTOA 5 / 15

20 Choi-Kraus decomposition Suppose α : B(H) B(H) is CP-map. Then there exists operators L 1,, L d B(H) such that α(x) = d L i XL i X B(H). ( ) i=1 The decomposition is not unique. But if α(x) = d j=1 M j XM j is another decomposition, then span{l i : 1 i d} = span{m j : 1 j d }. We fix one such decomposition say ( ). Then we can have ker(α(1)) = d i=1 ker(l i). For X 0, α(x) = 0 iff ran(x) d i=1 ker(l i ). {x : α( x x ) = 0} = d i=1 ker(l i). ran(α(1)) = span{l i u : u H}. α p (X) = 0 for all X B(H) iff L i1 L i2...l ip=0 for all i 1, i 2..., i p 1. Nirupama (ISIBC) OTOA 5 / 15

21 CP Nilpotent Type Let α : B(H) B(H) be nilpotent CP-map of order p. Nirupama (ISIBC) OTOA 6 / 15

22 CP Nilpotent Type Let α : B(H) B(H) be nilpotent CP-map of order p. Let H 1 := ker(α(1)) and H k := ker(α k (1)) ker(α k 1 (1)) for 2 k p. Then H = H 1 H 2 H p. Nirupama (ISIBC) OTOA 6 / 15

23 CP Nilpotent Type Let α : B(H) B(H) be nilpotent CP-map of order p. Let H 1 := ker(α(1)) and H k := ker(α k (1)) ker(α k 1 (1)) for 2 k p. Then H = H 1 H 2 H p. Let a i := dim(h i ) for 1 i p. We call (a 1, a 2,, a p ) as the CP nilpotent type of α. Nirupama (ISIBC) OTOA 6 / 15

24 CP Nilpotent Type Let α : B(H) B(H) be nilpotent CP-map of order p. Let H 1 := ker(α(1)) and H k := ker(α k (1)) ker(α k 1 (1)) for 2 k p. Then H = H 1 H 2 H p. Let a i := dim(h i ) for 1 i p. We call (a 1, a 2,, a p ) as the CP nilpotent type of α. Some inequalities: a 1 + a a p = dim(h). a i+1 d.a i for all 1 i (p 1). Nirupama (ISIBC) OTOA 6 / 15

25 CP Nilpotent Type Let α : B(H) B(H) be nilpotent CP-map of order p. Let H 1 := ker(α(1)) and H k := ker(α k (1)) ker(α k 1 (1)) for 2 k p. Then H = H 1 H 2 H p. Let a i := dim(h i ) for 1 i p. We call (a 1, a 2,, a p ) as the CP nilpotent type of α. Some inequalities: a 1 + a a p = dim(h). a i+1 d.a i for all 1 i (p 1). Conversely, given a tuple (a 1, a 2,, a p ) of natural numbers adding up to dim(h) i.e., a 1 + a a p = dim(h) and satisfying a i+1 d.a i, then there exists a nilpotent completely positive map α : B(H) B(H) of order p. Nirupama (ISIBC) OTOA 6 / 15

26 CP Nilpotent Type Let α : B(H) B(H) be nilpotent CP-map of order p. Let H 1 := ker(α(1)) and H k := ker(α k (1)) ker(α k 1 (1)) for 2 k p. Then H = H 1 H 2 H p. Let a i := dim(h i ) for 1 i p. We call (a 1, a 2,, a p ) as the CP nilpotent type of α. Some inequalities: a 1 + a a p = dim(h). a i+1 d.a i for all 1 i (p 1). Conversely, given a tuple (a 1, a 2,, a p ) of natural numbers adding up to dim(h) i.e., a 1 + a a p = dim(h) and satisfying a i+1 d.a i, then there exists a nilpotent completely positive map α : B(H) B(H) of order p. If α : B(H) B(H) is a completely positive map of nilpotent order p, then p dim(h). Nirupama (ISIBC) OTOA 6 / 15

27 Dual nilpotent CP-maps Define α : B(H) B(H) by α (X) = d i=1 L ixl i. Nirupama (ISIBC) OTOA 7 / 15

28 Dual nilpotent CP-maps Define α : B(H) B(H) by α (X) = d i=1 L ixl i. trace(α(x) Y ) =trace(x α (Y )) for all X, Y B(H). Nirupama (ISIBC) OTOA 7 / 15

29 Dual nilpotent CP-maps Define α : B(H) B(H) by α (X) = d i=1 L ixl i. trace(α(x) Y ) =trace(x α (Y )) for all X, Y B(H). α is a nilpotent of order p if and only if α is a nilpotent of order p. Nirupama (ISIBC) OTOA 7 / 15

30 Dual nilpotent CP-maps Define α : B(H) B(H) by α (X) = d i=1 L ixl i. trace(α(x) Y ) =trace(x α (Y )) for all X, Y B(H). α is a nilpotent of order p if and only if α is a nilpotent of order p. Type of a nilpotent operator and the type of its adjoint are same. CP nilpotent type of α and α are may not be same. Nirupama (ISIBC) OTOA 7 / 15

31 Dual nilpotent CP-maps Define α : B(H) B(H) by α (X) = d i=1 L ixl i. trace(α(x) Y ) =trace(x α (Y )) for all X, Y B(H). α is a nilpotent of order p if and only if α is a nilpotent of order p. Type of a nilpotent operator and the type of its adjoint are same. CP nilpotent type of α and α are may not be same. Decompose the Hilbert space H as H = H 1 H 2 H p and also H = H 1 H 2 H p where H 1 = ker(α (1)) and H k = ker((α ) k (1)) ker((α ) k 1 (1)) for all k 2. Nirupama (ISIBC) OTOA 7 / 15

32 Dual nilpotent CP-maps Define α : B(H) B(H) by α (X) = d i=1 L ixl i. trace(α(x) Y ) =trace(x α (Y )) for all X, Y B(H). α is a nilpotent of order p if and only if α is a nilpotent of order p. Type of a nilpotent operator and the type of its adjoint are same. CP nilpotent type of α and α are may not be same. Decompose the Hilbert space H as H = H 1 H 2 H p and also H = H 1 H 2 H p where H 1 = ker(α (1)) and H k = ker((α ) k (1)) ker((α ) k 1 (1)) for all k 2. Let a k := dim(h k ) for 1 k p. Then a p i+1 + a p i a p a 1 + a a i for 1 i p and a 1 + a a p = a 1 + a a p. Nirupama (ISIBC) OTOA 7 / 15

33 Invariant subspaces and majorization α : B(H) B(H) nilpotent CP map. Nirupama (ISIBC) OTOA 8 / 15

34 Invariant subspaces and majorization α : B(H) B(H) nilpotent CP map. Let M be a subspace of H. Take N = M. Then H = M N. ] Consider B(M) B(H) via X. [ X Nirupama (ISIBC) OTOA 8 / 15

35 Invariant subspaces and majorization α : B(H) B(H) nilpotent CP map. Let M be a subspace of H. Take N = M. Then H = M N. ] Consider B(M) B(H) via X. [ X We say that M is invariant under α, if α leaves B(M) invariant. Nirupama (ISIBC) OTOA 8 / 15

36 Invariant subspaces and majorization α : B(H) B(H) nilpotent CP map. Let M be a subspace of H. Take N = M. Then H = M N. ] Consider B(M) B(H) via X. [ X We say that M is invariant under α, if α leaves B(M) invariant. This happens iff every Choi-Kraus coefficient leaves N invariant. Nirupama (ISIBC) OTOA 8 / 15

37 Invariant subspaces and majorization α : B(H) B(H) nilpotent CP map. Let M be a subspace of H. Take N = M. Then H = M N. ] Consider B(M) B(H) via X. [ X We say that M is invariant under α, if α leaves B(M) invariant. This happens iff every Choi-Kraus coefficient leaves N invariant. We get α(x) = L i XL i with L i = B(M N) for some [ ] Bi 0 D i C i operators B i B(M), C i B(N), D i B(M, N), 1 i d. Nirupama (ISIBC) OTOA 8 / 15

38 Invariant subspaces and majorization α : B(H) B(H) nilpotent CP map. Let M be a subspace of H. Take N = M. Then H = M N. ] Consider B(M) B(H) via X. [ X We say that M is invariant under α, if α leaves B(M) invariant. This happens iff every Choi-Kraus coefficient leaves N invariant. We get α(x) = L i XL i with L i = B(M N) for some [ ] Bi 0 D i C i operators B i B(M), C i B(N), D i B(M, N), 1 i d. Define CP-maps β : B(M) B(M) and γ : B(N) B(N) by β(x) = B i XB i and γ(y ) = C i Y C i. Nirupama (ISIBC) OTOA 8 / 15

39 Invariant subspaces and majorization α : B(H) B(H) nilpotent CP map. Let M be a subspace of H. Take N = M. Then H = M N. ] Consider B(M) B(H) via X. [ X We say that M is invariant under α, if α leaves B(M) invariant. This happens iff every Choi-Kraus coefficient leaves N invariant. We get α(x) = L i XL i with L i = B(M N) for some [ ] Bi 0 D i C i operators B i B(M), C i B(N), D i B(M, N), 1 i d. Define CP-maps β : B(M) B(M) and γ : B(N) B(N) by β(x) = Bi XB i and γ(y ) = Ci Y C i. Note that for any i 1, i 2,..., i p, L i1 L i2... L ip = for some operator D i1,i 2,...,i p. [ ] Bi1 B i2... B ip 0 D i1,i 2,...,i p C i1 C i2... C ip Nirupama (ISIBC) OTOA 8 / 15

40 Invariant subspaces and majorization α : B(H) B(H) nilpotent CP map. Let M be a subspace of H. Take N = M. Then H = M N. ] Consider B(M) B(H) via X. [ X We say that M is invariant under α, if α leaves B(M) invariant. This happens iff every Choi-Kraus coefficient leaves N invariant. We get α(x) = L i XL i with L i = B(M N) for some [ ] Bi 0 D i C i operators B i B(M), C i B(N), D i B(M, N), 1 i d. Define CP-maps β : B(M) B(M) and γ : B(N) B(N) by β(x) = Bi XB i and γ(y ) = Ci Y C i. Note that for any i 1, i 2,..., i p, L i1 L i2... L ip = [ ] Bi1 B i2... B ip 0 D i1,i 2,...,i p C i1 C i2... C ip for some operator D i1,i 2,...,i p. Thus if α is nilpotent of order p, then β and γ are nilpotent of order at most p. Nirupama (ISIBC) OTOA 8 / 15

41 Majorization Theorem Suppose (a 1,, a p ), (b 1,, b p ) and (c 1,, c p ) are CP nilpotent type of α, β and γ respectively. Then k a i i=1 k b i + k i=1 i=1 c i for all 1 k < p, and p a i = i=1 p p b i + c i. i=1 i=1 Nirupama (ISIBC) OTOA 9 / 15

42 Majorization Theorem Suppose (a 1,, a p ), (b 1,, b p ) and (c 1,, c p ) are CP nilpotent type of α, β and γ respectively. Then k a i i=1 k b i + k i=1 i=1 c i for all 1 k < p, and p a i = i=1 p p b i + c i. i=1 i=1 Proof: For the inequality part, first consider the case k = 1. Let {u 1, u 2,..., u r } be a basis for ( i ker(b i)) ( i ker(d i)). Let {v 1, v 2,..., v c1 } be a basis for i ker(c i). Nirupama (ISIBC) OTOA 9 / 15

43 Proof continues: ( ) ( ) ( ) ( ) ( ) ( ) u1 u2 ur {,,...,,,,..., } is linearly v 1 v 2 v c1 independent in i ker(l i). Extend this collection to: ( ) ( ) ( ) ( u1 u2 ur 0 {,,...,, a basis of i ker(l i). In particular, a 1 = r + c 1 + s. v 1 ) ( 0, v 2 ) ( ) 0,...,, v c1 ( x1 y 1 ), Now we observe that x 1, x 2,..., x s are vectors in i ker(b i). ( x2 y 2 ),..., We claim that {u 1, u 2,..., u r, x 1, x 2,..., x s } are linearly independent in i ker(b i). We have r + s b 1 and hence a 1 b 1 + c 1. Nirupama (ISIBC) OTOA 10 / 15

44 Proof continues: Suppose j p ju j + j q jx j = 0 for some scalars p j, q j. Fix 1 i d. Then as u j i ker(d i) for all j, j q jx j ker(d i ). Further as ( ) xj is in ker(l i ), we have D i x j + C i y j = 0 or C i y j = D i x j for y j all j. Consequently j C iq j y j = j D iq j x j = 0. Therefore, j q jy j i ker(c i). So there exist scalars t i, 1 i c 1, such that i t iv i = j q jy j. Then, j p j ( uj ) + j t j ( 0 ) + j q j ( xj ) = 0. 0 v j y j Now due to linear independence of these vectors, p j 0, q j 0 and t j 0. Consider α k (1), β k (1) and γ k (1), k j=1 a j k j=1 b j + k j=1 c j for 1 k < p as k j=1 a j =dim ker(α k (1)). Nirupama (ISIBC) OTOA 11 / 15

45 Roots of states Definition Let H be a finite dimensional Hilbert space and let u H be a unit vector in H. Consider the pure state X u, Xu I on B(H). Then a unital completely positive map τ : B(H) B(H) is said to be an n th root of this state if τ n (X) = u, Xu I X B(H). Nirupama (ISIBC) OTOA 12 / 15

46 Roots of states Definition Let H be a finite dimensional Hilbert space and let u H be a unit vector in H. Consider the pure state X u, Xu I on B(H). Then a unital completely positive map τ : B(H) B(H) is said to be an n th root of this state if τ n (X) = u, Xu I X B(H). Theorem Let τ : B(H) B(H) be a unital CP-map such that τ p (X) = u, Xu I where u is a unit vector of H. Set H 0 = {x H : x, u = 0} so that H = Cu H 0. Suppose α : B(H 0 ) B(H 0 ) is the compression of τ to B(H 0 ), then α is nilpotent CP map of order at most p. Nirupama (ISIBC) OTOA 12 / 15

47 Roots of states Theorem Let α : B(H 0 ) B(H 0 ) be a contractive CP-map ( such ) that α p (Y ) = 0 1 for all Y B(H 0 ). Take H = C H 0 and u =. Suppose 0 τ : B(H) B(H) is a map defined by ( ) ( ) X11 X τ( 12 X11 0 ) = X 21 X 22 0 α(x 22 ) + X 11 (I α(i)) for all X = [X ij ] B(C H 0 ). Then τ is a CP-map and τ p (X) = u, Xu I for all X B(H). Nirupama (ISIBC) OTOA 13 / 15

48 References Wing Suet Li; Vladmir Müller: Invariant subspaces of nilpotent operators and LR-sequences. Integral Equations Operator Theory 34 (1999). William Fulton: Eigenvalues, invariant factors, highest weights, and Schubert calculus, Bull. Amer. Math. Soc. (N.S.) 37 (2000). Hari Bercovici, Wing Suet Li; Thomas Smotzer: Continuous versions of the Littlewood-Richardson rule, selfadjoint operators, and invariant subspaces, J. Operator Theory 54 (2005). B. V. Rajarama Bhat: Roots of states. Commun. Stoch. Anal. 6 (2012). B. V. Rajarama Bhat; Nirupama Mallick: Nilpotent Completely Positive Maps, Positivity (November 2013). Nirupama (ISIBC) OTOA 14 / 15

49 THANKS Nirupama (ISIBC) OTOA 15 / 15