Generators of quantum Markov Semigroups Matt Ziemke University of South Carolina Virginia Operator Theory and Complex Analysis Meeting (VOTCAM) November 7th, 2015 Matt Ziemke Generators of QMS
Paper (with G. Androulakis), ”Generators of quantum Markov semigroups”, J. Math. Phys. (2015). Matt Ziemke Generators of QMS
Outline 1. Definitions 2. Background 3. Known Results 4. Some Results 5. Examples Matt Ziemke Generators of QMS
Definitions The σ -weak topology Let A be a von Neumann algebra. Then A has a predual A ∗ and the σ -weak topology on A is the σ ( A , A ∗ ) topology, that is, the weak ∗ topology when A is viewed as the dual of A ∗ . Note 1: Every von Neumann algebra (when viewed as a Banach space) has a predual (Sakai). Note 2: We are mostly interested in the case when A = B ( H ), where H is a separable Hilbert space. In this case B ( H ) ∗ = S 1 ( H ). Matt Ziemke Generators of QMS
Definitions cont. Completely positive operators Let H be a Hilbert space and let T : B ( H ) → B ( H ) be a bounded linear operator. Let B ( H ) ⊗ M n be the ∗ -algebra of n × n matrices with coefficients in B ( H ). We say the operator T is completely positive if for any n ∈ N , any positive element [ A ij ] 1 ≤ i , j ≤ n ∈ B ( H ) ⊗ M n , and any h 1 , h 2 , . . . h n ∈ H we have n � � h i , T ( A ij ) h j � ≥ 0 . i , j =1 Matt Ziemke Generators of QMS
Definitions cont. Quantum dynamical semigroup Let A be a von Neumann algebra. A quantum dynamical semigroup (QDS) is a one-parameter family ( T t ) t ≥ 0 of σ -weakly continuous, completely positive, linear operators on A such that (i) T 0 = 1 (ii) T t + s = T t T s (iii) for a fixed A ∈ A , the map t �→ T t ( A ) is σ -weakly continuous. Further, if T t (1) = 1 for all t ≥ 0 then we say the quantum dynamical semigroup is Markovian or we simply refer to it as a quantum Markov semigroup (QMS) . If the map t �→ T t is norm continuous then we say the semigroup is uniformly continuous . Matt Ziemke Generators of QMS
Generator of a QMS Given a QDS ( T t ) t ≥ 0 , we say that an element A ∈ A belongs to the domain of the infinitesimal generator L of ( T t )) t ≥ 0 , denoted by D ( L ), if 1 lim t ( T t A − A ) t → 0 converges in the σ -weak topology and, in this case, define the infinitesimal generator to be the generally unbounded operator L such that 1 L ( A ) = σ -weak- lim t ( T t A − A ) A ∈ D ( L ) . , t → 0 If ( T t ) t ≥ 0 is uniformly continuous then the generator L is bounded and given by 1 L = lim t ( T t − 1) t → 0 where the limit is taken in the norm topology. In this case T t = e tL . Matt Ziemke Generators of QMS
Background Lindblad (‘76) If ( T t ) t ≥ 0 is a uniformly continuous QMS on B ( H ) then there exists G ∈ B ( H ) and a completely positive map φ : B ( H ) → B ( H ) such that the infinitesimal generator L of ( T t ) t ≥ 0 is given by L ( A ) = φ ( A ) + GA + AG ∗ for all A ∈ B ( H ). Note 1: Lindblad proved this for a uniformly continuous QMS on a hyperfinite factor A of B ( H ) (which includes the case A = B ( H ) by Topping (‘71)). Note 2: Christensen and Evans proved this for uniformly continuous QMS on arbitrary von Neumann algebras in ‘79. Matt Ziemke Generators of QMS
Background cont. Stinespring (‘55) Let B be a C ∗ -subalgebra of the algebra of all bounded operators on a Hilbert space H and let A be a C ∗ -algebra with unit. A linear map T : A → B is completely positive if and only if it has the form T ( A ) = V ∗ π ( A ) V where ( π, K ) is a unital ∗ -representation of A on some Hilbert space K , and V is a bounded operator from H to K . Matt Ziemke Generators of QMS
Background cont. Lindblad + Stinespring Let L be the generator of a uniformly continuous QMS on B ( H ). Then there exists an operator G ∈ B ( H ), a unital ∗ -representation of B ( H ) on some Hilbert space K , and a V ∈ B ( H , K ) such that L ( A ) = V ∗ π ( A ) V + GA + AG ∗ for all A ∈ B ( H ). Note: Due to a result of Kraus (‘70), there exists a sequence ( V j ) j ≥ 1 ⊆ B ( K , H ) such that ∞ V ∗ π ( A ) V = � V ∗ j AV j j =1 where the series � ∞ j =1 V ∗ j AV j converge strongly. Matt Ziemke Generators of QMS
Background cont. Question Does the generator of a general QMS (that is, one which is not uniformly continuous) have a similar form? Note 1: Many important examples of QMS are not uniformly continuous (for example, the QMS associated to the noncommutative heat equation). Note 2: The QMS ( T t ) t ≥ 0 is uniformly continuous if and only if L is bounded. Matt Ziemke Generators of QMS
Known results Davies (‘79) Let T t : S 1 ( H ) → S 1 ( H ) be a semigroup which satisfies: T ∗ t ( C ( H )) ⊆ C ( H ) for all t ≥ 0, There exists e ∈ H\{ 0 } such that T t ( | e �� e | ) = | e �� e | ,and the map [0 , ∞ ) ∋ t �→ T t ( A ) ∈ B ( H ) is SOT-continuous for all A ∈ B ( H ). Then there exists a dense linear subspace D of H and linear operators G : D → H and L n : D → H such that the infinitesimal generator L of ( T t ) t ≥ 0 is given by ∞ � L n AL ∗ n + GA + AG ∗ L ( A ) = n =1 for all A ∈ ( G − 1) − 1 S 1 ( H )( G ∗ − 1) − 1 . Matt Ziemke Generators of QMS
Known results cont. Holevo (‘95) Let ( T t ) t ≥ 0 be a QMS on B ( H ). Assume that there exists a dense linear subspace D of H such that � x , T t A − A � lim y t t → 0 exists for all A ∈ B ( H ) and all x , y ∈ D . Then there exists a linear operator G : D → H , a separable Hilbert space H 0 , and a linear operator L : D → H ⊗ H 0 such that � x , L ( A ) y � = �L x , ( A ⊗ 1 0 )( L y ) � H⊗H 0 + � Gx , Ay � + � x , AGy � for all A ∈ B ( H ) and all x , y ∈ D . Matt Ziemke Generators of QMS
Results Notation If H is a Hilbert space and D is a linear subspace of H , let S ( D ) denote the set of sesquilinear forms on D × D . Definition Let D be a linear subspace of H and A be a linear subspace of B ( H ). A linear map φ : A → S ( D ) is called D-completely positive if for any k ∈ N , and any positive operator A = ( A i , j ) 1 ≤ i , j ≤ k ∈ A ⊗ M k ( C ) and for all x 1 , . . . , x k ∈ D , k � φ ( A i , j )( x i , x j ) ≥ 0 . i , j =1 Matt Ziemke Generators of QMS
Definition Let ( T t ) t ≥ 0 be a QDS, L be its generator and Dom ( L ) its domain. Then A = { A ∈ Dom ( L ) : A ∗ A , AA ∗ ∈ Dom ( L ) } is the domain algebra and is equal to the largest ∗ -subalgebra of Dom ( L ) by Arveson (‘02). Matt Ziemke Generators of QMS
Results cont. Androulakis, Z. (‘15) Let L be the infinitesimal generator of a QMS on B ( H ) and let A be its domain algebra. Assume that there exists e ∈ H such that | e �� e | : ∈ Dom ( L ). Let D e = { x ∈ H : | x �� e | ∈ A} . Then there exists a linear map G : D e → H and a D e -completely positive map φ : A → S ( D e ) such that � x , L ( A ) � = φ ( A )( x , y ) + � x , GAy � + � GA ∗ x , y � for all A ∈ A and x , y ∈ D e . Matt Ziemke Generators of QMS
Results cont. Androulakis, Z. (‘15) Let A be a unital ∗ -subalgebra of B ( H ), D be a linear subspace of H , and φ : A → S ( D ) be a D -completely positive map. Then there exists a Hilbert space K , a ∗ -representation π : A → B ( K ) and a linear map V : D → K such that φ ( A )( x , y ) = � Vx , π ( A ) Vy � K for all x , y ∈ D . Matt Ziemke Generators of QMS
Results cont. Corollary Let L be the infinitesimal generator of a QMS on B ( H ) and let A be its domain algebra. Assume that there exists e ∈ H such that | e �� e | ∈ Dom ( L ). Let D e = { x ∈ H : | x �� e | ∈ A} . Then there exists a Hilbert space K , a ∗ -representation π : A → B ( K ), and a linear map V : D → K such that � x , L ( A ) y � = � Vx , π ( A ) Vy � K + � x , GAy � + � GA ∗ x , y � for all A ∈ A and x , y ∈ D e . Note: Can take G to be G ( x ) = L ( | x �� e | ) e − 1 2 � e , L ( | e �� e | ) e � x . Matt Ziemke Generators of QMS
Example 1 (Parthasarathy (‘92)). Let ( B t ) t ≥ 0 be standard Brownian motion, V be a selfadjoint operator on H and define T t : B ( H ) → B ( H ) by T t ( A ) = E [ e iB t V Ae − iB t V ] . Then ( T t ) t ≥ 0 is a QMS. dx , e ( t ) = e − t 2 / 2 then D e is dense in L 2 ( R ) If H = L 2 ( R ), V = i d and � x , L ( A ) y � = � Vx , AVy � + � x , − 1 2 V 2 Ay � + �− 1 2 V 2 A ∗ x . y � for all x , y ∈ U e and A ∈ A . Matt Ziemke Generators of QMS
Example 2 (Fagnola (‘00), Arveson, (‘02)). Let H = L 2 [0 , ∞ ), and let U t : H → H be defined by � g ( s − t ) if s ≥ t U t ( g )( s ) = 0 otherwise Define E t : L 2 [0 , ∞ ) → L 2 [0 , t ) the natural projection. Define ω : B ( H ) → C by ω ( A ) = � f , Af � where f ∈ H is defined by f ( s ) = e − s ( s ∈ [0 , ∞ )). Define T t : B ( H ) → B ( H ) by T t ( A ) = ω ( A ) E t + U t AU ∗ t . Then ( T t ) t ≥ 0 is a QMS. Matt Ziemke Generators of QMS
Example 2 cont. Fix e ∈ L 2 [0 , ∞ ) such that D e ∈ L 2 [0 , ∞ ) (where D is the differentiation operator), and � e , f � = 0. Then D e ⊆ { x ∈ L 2 [0 , ∞ ) : � x , f � = 0 } hence D e is not dense in H . Also A is not SOT dense in B ( H ). We have � x , L ( A ) y � = ω ( A ) x (0) y (0) + � x , D Ay � + � D A ∗ x , y � for all x , y ∈ D e and A ∈ A . Matt Ziemke Generators of QMS
Thank you! Matt Ziemke Generators of QMS
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