arXiv:1712.00637v1 [quant-ph] 2 Dec 2017 addtsfrsoigadmnpltn unu information. su quantum such manipulating in leaving and States storing therein). for references candidates the disappea and dissipation 29] attent and 22, Special irreversibility 14, c where QMS. systems subspaces of quantum or features algebras open special in on equilibrium investigation to motivate approach and computing tum fivratsae,a ela eoeec-resbytm an subsystems decoherence-free as determinin well parts, decom dissipative as deco a purely the states, induces and invariant if it noiseless atomic, that, of its is [13] into QMS in system continuous showed the uniformly we a Recently, of QMS. th subalgebra on whole consequences the important of has algebra, action decoherence. Neumann induced von environment a int s insight and as a gain ture states is to invariant us which its allow points, QMS, also fixed a state, invariant of faithful set a the exists and there automorphisms, as act group n hi eeaos a enteojc fsgicn attentio significant of object the been has generators, their and idld[3 h tutr fuiomycniuu unu Marko quantum continuous uniformly of structure Gorini-Kossakowski-Su (QMS) of the papers [23] fundamental Lindblad the from Starting Introduction 1 eainhp ewe h eoeec-realgebra decoherence-free the between Relationships h eoeec-resbler,weecmltl oiiemap positive completely where subalgebra, decoherence-free The h nraigitrs nmteaia oeln fdchrne c decoherence, of modelling mathematical in interest increasing The e frvril ttsadepiil ecieterelati the subalgebra properties. describe point equivalent fixed above explicitly the and and subalgebra states decoherence-free decoherenc reversible w the Moreover, of of equivalent. set are atomicity decoherence the environmental state, and invariant normal ful T eso ht o unu akvSmgop(M)wt fait a with (QMS) Semigroup Markov Quantum a for that, show We ( = T t ) t ≥ 0 r ntepyia emnlg,qatmdnmclsemigroups, dynamical quantum terminology, physical the in or, , .Fagnola F. n h e ffie points fixed of set the and 1 .Sasso E. , Abstract 1 2 .Umanit`a V. , nhpbtenthe between onship o Mswt the with QMSs for hrceiethe characterize e -resubalgebra e-free sae r promising are bspaces 2 usae 2] In [29]. subspaces d n. tutr n the and structure e o spi osub- to paid is ion hrhn[9 and [19] dharshan se[,3 ,13, 7, 3, [2, (see r h tutr of structure the o ned t struc- its Indeed, s bler when ubalgebra h structure the g T hrn quan- oherent semigroups v t herence-free ftesemi- the of niusto ontinues oiinof position h- particular, we provided a full description of invariant states extending known ones in the finite dimensional case [6]. In this paper we push further the analysis of uniformly continuous QMS with atomic decoherence-free subalgebra and a faithful invariant state proving a number new results we briefly list and outline below.
1. Environment induced decoherence ([7, 9, 11]) holds if and only if ( ) is atomic. In this case the decoherence-free subalgebra is generated byN theT set of eigenvectors corresponding to modulus one eigenvalues of the completely positive maps t, namely, in an equivalent way, by the eigenvectors with purely imaginary eigenvalueT of the generator (Theorem 12).
2. The decoherence-free subalgebra and the set of so-called reversible states, i.e. the linear space generated by eigevectors corresponding to modulus 1 eigenval- ues of predual maps are in the natural duality of a von Neumann algebra T∗t with its predual (Theorem 16). Moreover, Theorems 18 and 20 explicitly de- scribe the structure of reversible states.
3. We find a spectral characterization of the decomposition of the fixed point algebra (Theorem 23). Moreover, the exact relationship between ( ) and ( ) (Theorems 23 and 25) is established in an explicit and constructiveF T way N T allowing one to find the structure of each one from the structure of the other.
Loosely speaking one can say that, for QMSs with a faithful invariant state, the same conclusions can be drawn replacing finite dimensionality of the system Hilbert space by atomicity of the decoherence-free subalgebra. Counterexamples (Examples 10 and 17) show that, in general, the above conclusions may fail if for QMSs without faithful normal invariant states. The above results, clarify then the relationships between the atomicity of the decoherence-free subalgebra, environmental decoherence, ergodic decomposition of the trace class operators, and the structure of fixed points. In particular the first result implies that the decomposition induced by deco- herence coincides with the Jacobs-de-Leeuw-Glickeberg (JDG) splitting. Such de- composition was originally introduced for weakly almost periodic semigroups and generalized to QMSs on von Neumann algebras in [24, 21] at all. It is among the most useful tools in the study of the asymptotic behavior of operators semigroups on Banach spaces or von Neumann algebras. Indeed, it provides a decomposition of the space into the direct sum of the space generated by eigenvectors of the semi- group associated with modulus 1 eigenvalues, and the remaining space, called stable, consisting of all vectors whose orbits have 0 as a weak cluster point. Under suitable conditions, we obtain the convergence to 0 for each vector belonging to the stable space.
2 On the other hand, the ergodic decomposition of trace class operators (which is a particular case of the JDS splitting applied to the predual of ), allows one, for T instance, to determine reversible subsytems by spectral calculus. Determining re- versible states, in particular, is an important task in the study of irreversible (Marko- vian) dynamics because these states retain their quantum features that are exploited in quantum computation (see [3, 29] and the references therein). More precisely, reversible (or rotating) and invariant states of a quantum channel (acting on Mn(C) for some n> 1) allow to classify kinds of information that the process can preserve. When the space is finite-dimensional and there exists a faithful invariant state, the structure of these states can be easily found (see e.g. Lemma 6 and Section V in [8], and Theorems 6.12, 6.16 in [31]) through the decomposition of ( ) and the algebra of fixed points ( ) in “block diagonal matrices”, i.e. in theirN T canonical form given by the structureF T theorem for matrix algebras (see Theorem 11.2 in [28]). Since the same decomposition holds for atomic von Neumann algebras, in this pa- per we generalize these results to uniformly continuous QMSs acting on (h) with h B infinite-dimensional. The paper is organized as follows. In Section 2 we collect some notation and known results on the structure of norm-continuous QMS with atomic decoherence- free subalgebra and the structure of their invariant states. In Section 3, after recalling some known results from [11] on the relationship between EID and Jacobs-de Leeuw- Glickeberg decomposition, we prove the main result of this paper: the equivalence between EID and atomicity of the decoherence free subalgebra. The predual algebra of the decoherence-free subalgebra is characterized in Section 4 as the set of reversible states. Finally, in Section 5, we study the structure of the set of fixed points of the semigroup and its relationships with the decomposition of ( ) when this algebra is atomic. N T
2 The structure of the decoherence-free algebra
Let h be a complex separable Hilbert space and let (h) the algebra of all bounded B operators on h with unit 1l. A QMS on (h) is a semigroup =( t)t≥0 of completely positive, identity preserving normal mapsB which is also weaklyT ∗ Tcontinuous. In this paper we assume uniformly continuous i.e. T
lim sup t(x) x =0, t→0+ kxk≤1 kT − k so that there exists a linear bounded operator on (h) such that = etL. The L B Tt operator is the generator of , and it can be represented in the well-known (see L T
3 [26]) Gorini-Kossakowski-Sudarshan-Lindblad (GKSL) form as 1 (x) = i[H, x] (L∗L x 2L∗xL + xL∗L ) , (1) L − 2 ℓ ℓ − ℓ ℓ ℓ ℓ ℓ X≥1 ∗ h ∗ where H = H and (Lℓ)ℓ≥1 are bounded operators on such that the series ℓ≥1 Lℓ Lℓ is strongly convergent and [ , ] denotes the commutator [x, y]= xy yx. The choice · · − P of operators H and (Lℓ)ℓ≥1 is not unique, but this will not create any inconvenience in this paper. More precisely, we have the following characterization (see [26], Propo- sition 30.14 and the discussion below the proof of Theorem 30.16). Theorem 1. Let be the generator of a uniformly continuous QMS on (h). Then L B there exist a bounded selfadjoint operator H and a sequence (Lℓ)ℓ≥1 of elements in (h) such that: B ∗ 1. ℓ≥1 Lℓ Lℓ is strongly convergent,
2 2.P if ℓ≥0 cℓ < and c01l+ ℓ≥1 cℓLℓ = 0 for scalars (cℓ)ℓ≥0 then cℓ = 0 for every ℓ | 0|, ∞ P ≥ P 3. (x) = i[H, x] 1 (L∗L x 2L∗xL + xL∗L ) for all x (h). L − 2 ℓ≥1 ℓ ℓ − ℓ ℓ ℓ ℓ ∈ B We recall that, for anP arbitrary von Neumann algebra , its predual space ∗ is the space of w∗-continuous functionals on (said normalM). It is a well-known factM M that for all ω ∗ there exists ρ I(h), the space of trace-class operators, such that ω(x)=tr(∈ρx M) for all x . In∈ particular, if ω is a positive and ω = 1, it is called state, and ρ is positive∈M with tr (ρ) = 1, i.e. ρ is a density. || || If = (h), every normal state ω has a unique density ρ. Therefore, in this case, we canM identifyB them. Finally, ρ is faithful if tr (ρx) = 0 for a positive x (h) implies ∈ B x = 0 (see [28], Definition 9.4). Given a w∗-continuous operator : , we can define its predual map S M→M : as (ω)= ω . S∗ M∗ →M∗ S∗ ◦ S In particular, for = (h), by considering the predual map of every t, we obtain the predual semigroupM B =( ) satisfying T T∗ T∗t t tr( (ρ)x)=tr(ρ (x)) ρ I(h), x (h). T∗t T∗t ∀ ∈ ∈ B The decoherence-free (DF) subalgebra of is defined by T ( )= x (h): (x∗x)= (x)∗ (x), (xx∗)= (x) (x)∗ t 0 . (2) N T { ∈ B Tt Tt Tt Tt Tt Tt ∀ ≥ } It is a well known fact that ( ) is the biggest von Neumann subalgebra of (h) N T B on which every t acts as a -homomorphism (see e.g. Evans[16] Theorem 3.1). Moreover, the followingT facts hold∗ (see [14] Proposition 2.1).
4 Proposition 2. Let be a QMS on (h) and let ( ) be the set defined by (2). Then T B N T
1. ( ) is invariant with respect to every , N T Tt 2. ( )= δ(n)(L ), δ(n)(L∗): n 0 ′, where δ (x) := [H, x], N T { H k H k ≥ } H 3. (x)= eitH xe−itH for all x ( ), t 0, Tt ∈N T ≥ 4. if possesses a faithful normal invariant state, then ( ) contains the set of fixedT points ( )= L , L∗,H : k 1 ′. N T F T { k k ≥ } In addition, if the QMS is uniformly continuous, its action on ( ) is bijective. N T Theorem 3. If is a uniformly continuous QMS, then ( ) is the biggest von T N T Neumann subalgebra on which every map acts as a -automorphism. Tt ∗ Proof. The restriction of every to ( ) is clearly injective thanks to item 3 of Tt N T Proposition 2. Now, given x ( ) and t> 0, we have to prove that x = t(y) for some y ( ). ∈N T T ∈N T First of all note that, since the QMS is norm continuous, it can be extended to norm continuous group ( t)−∞
As we said in the introduction, we will study the relationships between the struc- ture of ( ) and other problems in the theory of uniformly continuous QMSs in which theN atomicityT of ( ) plays a key role. First of all, as shownN T in [13], the structure of the decoherence-free subalgebra ( ) gives information on the whole QMS. Let us recall these results. Assume that N (T ) is an atomic algebra, that is, there exists an (at most countable) family (p ) N T i i∈I of mutually orthogonal non-zero projections, which are minimal projections in the center of ( ), such that i∈I pi = 1l and each von Neumann algebra pi ( )pi is a type I factor.N T In that case, the subalgebra ( ) can be decomposed as N T P N T ( )= p ( )p . (3) N T ⊕i∈I iN T i
The properties of the projections pi imply their invariance under the semigroup and more generally the -invariance of each factor p ( )p . Moreover, each p ( )p Tt iN T i iN T i is a type I factor acting on the Hilbert space pih; thus, there exist two countable sequences of Hilbert spaces (ki)i∈I , (mi)i∈I , and unitary operators Ui : pih ki mi such that → ⊗
∗ ∗ U p ( )p U = (k ) 1lm , U p (h)p U = (k m ). (4) i iN T i i B i ⊗ i i iB i i B i ⊗ i 5 Therefore, defining U = i∈I Ui, we obtain a unitary operator U : h i∈I (ki mi) such that ⊕ → ⊕ ⊗ ∗ U ( )U = ( (k ) 1lm ) . (5) N T ⊕i∈I B i ⊗ i As a consequence, we find the following result. Theorem 4. ( ) is an atomic algebra if and only if there exist two countable se- quences of separableN T Hilbert spaces (k ) , (m ) such that h = (k m ) (up to a i i∈I i i∈I ⊕i∈I i ⊗ i unitary operator) and ( )= ( (k ) 1lm ) (up to an isometric isomorphism). N T ⊕i∈I B i ⊗ i In this case: 1. for every GSKL representation of by means of operators H, (L ) , we have L ℓ ℓ≥1 (i) L = 1lk M ℓ ⊕i∈I i ⊗ ℓ for a collection (M (i)) of operators in (m ), such that the series M (i)∗M (i) ℓ ℓ≥1 B i ℓ≥1 ℓ ℓ strongly convergent for all i I, and ∈ P (i) H = K 1lm + 1lk M ⊕i∈I i ⊗ i i ⊗ 0 for self-adjoint operators K (k ) and M (i) (m ), i I, i ∈ B i 0 ∈ B i ∈ m (i) (i) m i 2. defining on ( i) the GKSL generator associated with operators M0 , Mℓ ): ℓ(i) 1 , weB have L { ≥ } m (x y ) = eitKi x e−itKi i (y ) Tt i ⊗ i i ⊗ T i mi for all t 0, xi (ki) and yi (mi), where is the QMS generated by m ≥ ∈ B ∈ B T i , L m 3. if there exists a faithful normal invariant state, then the QMS i is irre- T ducible, possesses a unique invariant state τmi which is also faithful, and we mi mi have ( ) = ( ) = C1lmi . Moreover, for all i I, Ki has pure point spectrum.F T N T ∈ Proof. The proof of the necessary condition is given in Theorems 3.2 and 4.1, Propo- sition 4.3 and Lemma 4.2 in [13]. Conversely, given two countable sequences of Hilbert spaces (k ) , (m ) , such that h = (k m ) up to a unitary operator, and i i i i ⊕i∈I i ⊗ i ( ) = ( (k ) 1lm ) (up to the corresponding isometric isomorphism), set p N T ⊕i∈I B i ⊗ i i the orthogonal projection onto ki mi. Then we obtain a family (pi)i∈I of mutu- ally orthogonal non-zero projections,⊗ which are minimal projections in the center of ( ), such that p = 1land each von Neumann algebra p ( )p = (k ) 1lm N T i∈I i iN T i B i ⊗ i is a type I factor. P
6 Now, we recall a characterization of the atomicity of a von Neumann algebra in terms of the existence of a normal conditional expectation, i.e. a weakly*-continuous norm one projection (see [30] Theorem 5). To this end we will use that, given x h and k complex separable Hilbert spaces and σ a normal state on k, there always exists∈ (see e.g. Exercise 16.10 in [26]) a normal completely positive linear map : (h k) (h) satisfying Eσ B ⊗ → B tr( (X)η)=tr(X(η σ)) X (h k), η I(h). (6) Eσ ⊗ ∀ ∈ B ⊗ ∈
Since σ is positive and σ σ = σ, it is a normal conditional expectation called σ-conditionalE expectationE. ◦E E Theorem 5 (Tomiyama). Let be a von Neumann algebra acting on the Hilbert space h. Then is atomic if andM only if is the image of a normal conditional M M expectation : (h) . E B →M Proof. If there exists a normal conditional expectation : (h) onto , then is atomic by Proposition 3 and Lemma 5 in [20], beingE B(h) atomic→M and semifinite.M M B On the other hand, if is atomic, let (p ) be a sequence of orthogonal projections M i i in the center of such that pi pi is a type I factor; moreover, assume pi pi k Mk m M M ≃ ( i) 1lmi with ( i)i and ( i)i be two sequences of complex and separable Hilbert Bspaces⊗ such that h (k m ). Set (σ ) a sequence of normal states on (m ) , ≃ ⊕i i ⊗ i i i i i define the normal conditional expectations π : (k m ) (k ) 1lm as π (x)= i B i ⊗ i → B i ⊗ i i (x) 1lm for all x (k m ). So, Eσi ⊗ i ∈ B i ⊗ i π(x) := π(x ) for x =(x ) ( (k m )) (h) ii ij ij ∈ B ⊕i i ⊗ i ≃ B i X gives a normal conditional expectation onto . M Corollary 6. Let be an atomic von Neumann algebra acting on h and let M N ⊆ be a von Neumann subalgebra. If there exists a normal conditional expectation M: onto , then is atomic. E M→N N N Proof. By Tomiyama’s Theorem we know that, since is atomic, it is the image of a normal conditional expectation : (h) (seeM also the proof of Theorem 5 in [30]). Therefore, the map : F(h)B →Mis a normal conditional expectation onto E◦F B →N . Indeed, since = Ran is contained in = Ran , for x (h) we have N N E M F ∈ B ( )( )x = 2( (x))=( )(x), E◦F E◦F E F E◦F i.e. is a projection. Therefore, 1. On the other hand, since and E◦Fare norm one operators, we clearlykE ◦Fk obtain ≥ = 1. The normality ofE F kE ◦Fk is evident, and so we can conclude that the algebra is atomic by Tomiyama Theorem.E◦F N
7 Remark 7. Theorem 5 is a simplified version of Theorem 5 in [30], given in terms of atomicity of the subalgebra . Moreover, Corollary 6 generalizes to atomic algebras M one implication of the same theorem. In particular, we give easier proofs of these results. In the following we assume the existence of a faithful normal invariant state; note that, in general, this condition is not necessary for the decoherence-free subalgebra to be atomic. This is always the case for any QMS acting on a finite dimensional algebra. However, we show the following example that will be useful later. 3 Example 8. Let h = C with the canonical orthonormal basis (ei)i=1,2,3 and (h)= M (C). We consider the operator on M (C) given by B 3 L 3 1 (x) = iω [ e e , x] ( e e x 2 e e x e e + x e e ) L | 1ih 1| − 2 | 3ih 3| − | 3ih 2| | 2ih 3| | 3ih 3| for all x M (C), with ω R, ω = 0. Clearly is written in the GKSL form with ∈ 3 ∈ 6 L H = ω e e and L = e e , and so it generates a uniformly continuous QMS | 1ih 1| | 2ih 3| =( t)t≥0 on M3(C). TAn easyT computation shows that any invariant functional has the form a e e + b e e | 1ih 1| | 2ih 2| for some a, b C, and so the semigroup has no faithful invariant states. Since [H, L] =∈ 0, by item 2 of Proposition 2 we have ( ) = L, L∗ ′, so that an element x (h) belongs to ( ) if and only if N T { } ∈ B N T
xe2 = x33e2 xe e = e x∗e 2 3 2 3 | ih | | ih | , i.e. xe3 = x22e3 . xe e = e x∗e 3 2 3 2 | ih | | ih | x31 = x21 =0 Therefore we get ( )= x e e + x ( e e + e e ) x , x C , N T { 11| 1ih 1| 22 | 2ih 2| | 3ih 3| | 11 22 ∈ } i.e. ( ) is isometrically isomorphic to the atomic algebra C Cp, where p denotes N T ⊕ the identity matrix in M2(C).
3 Atomicity of ( ) and decoherence N T In this section, we explore the relationships between the atomicity of ( ) and the property of environmental decoherence, under the assumption of theN existenceT of a faithful normal invariant state ρ. Following [11], we say that there is environment induced decoherence (EID) on the open system described by if there exists a t- invariant and -invariant weakly∗ closed subspace of (h)T such that: T ∗ M2 B 8 (EID1) (h)= ( ) with = 0 , B N T ⊕M2 M2 6 { } (EID2) w∗ lim (x) = 0 for all x . − t→∞ Tt ∈M2 Unfortunately, if EID holds and h is infinite-dimensional, it is not clear if the space is uniquely determined. However, is always contained in the -invariant and M2 M2 T -invariant closed subspace ∗ ∗ 0 = x (h): w lim t(x)=0 . M ∈ B − t→∞ T n o In [13] we showed that, if ( ) is atomic, then EID holds (see Theorem 5.1) and, in N T particular, ( ) is the image of a normal conditional expectation : (h) ( ) compatibleN withT the faithful state ρ (i.e. ρ = ρ) and such that E B →N T ◦E Ker = = x (h) : tr(ρxy)=0 y ( ) . (7) E M2 { ∈ B ∀ ∈N T } (see Theorem 19 in [11]). In the following we will show that, if ( ) is atomic, the N T decomposition unique, i.e. the only way to realize it, is taking 2 given by (7) (see Theorem 12 and Remark 13.2). M Moreover, in this case, we will study the relationships of such a decomposition with another famous asymptotic splitting of (h), called the Jacobs-de Leeuw-Glickberg splitting: this comparison is very naturalB since the decomposition (h)= ( ) B N T ⊕M2 is clearly related too to the asymptotic properties of the semigroup. We recall that, since there exists ρ faithful invariant, the Jacobs-de Leeuw-Glickberg splitting holds (see e.g. Corollary 3.3 and Proposition 3.3 in [21]) and is given by (h)= M M with B r ⊕ s w∗ itλ Mr := span x (h): t(x)= e x for some λ R, t 0 (8) { ∈ B T ∗ ∈ ∀ ≥ } w M := x (h):0 (x) . (9) s { ∈ B ∈ {Tt }t≥0}
Moreover, in this case Mr is a von Neumann algebra. The relationship between the decomposition induced by decoherence and the Jacobs-de Leeuw-Glickberg splitting is given by the following result (see Proposition 31 in [11]). Proposition 9. If there exists a faithful normal invariant state ρ, then the following conditions are equivalent:
1. EID holds with 2 = 0 and the induced decomposition coincides with the Jacobs-de Leeuw-GlickbergM M splitting,
2. ( ) M = 0 , N T ∩ s { } 3. ( )= M . N T r 9 Moreover, if one of the previous conditions holds, then ( ) is the image of a normal conditional expectation compatible with ρ and such thatN TKer = = M . E E M0 s Clearly, if Mr is not an algebra, it does not make sense to pose the problem to understand if it coincides with ( ). In particular, this could happen when has N T T no faithful invariant states, as the following example shows. Example 10. Let us consider the uniformly continuous QMS on M (C) defined T 3 in Example 8. We have already seen that does not posses faithful invariant states, and T ( )= x e e + x ( e e + e e ) x , x C . N T { 11| 1ih 1| 22 | 2ih 2| | 3ih 3| | 11 22 ∈ } We want now to find the space Mr, generated by eigenvectors of corresponding to purely imaginary eigenvalues. Easy computations show that we haveL (x) = iλx for some λ R if and only if L ∈ 3 3 iλ x e e = iω (x e e x e e ) ij| iih j| 1j| 1ih j| − j1| jih 1| i,j=1 j=1 X X 1 3 3 x e e 2x e e + x e e , − 2 3j | 3ih j| − 22| 3ih 3| i3| iih 3| j=1 i=1 ! X X i.e., in the case when λ = 0, xij = 0 for i = j and x22 = x33, and, in the case λ = 0, if and only if the following identities hold 6 6
x =0, x =0, x (ω λ)=0, 11 22 12 − x 1 + i(ω λ) =0 x (ω + λ)=0, x ( 1 + iλ)=0, 13 − 2 − 21 23 2 1 1 x31 2 + i(ω + λ) =0, x32( 2 + iλ)=0, x33(1 + iλ)= x22. Since ω and λ belong to R this is equivalent to have either x = x e e and λ = ω, 12| 1ih 2| or x = x e e and λ = ω. Therefore, we can conclude that 21| 2ih 1| −
x11 x12 0 M r = x21 x22 0 : x11, x22, x12, x21 C . ∈ 0 0 x22 In particular, M is not an algebra and it is strictly bigger that ( ). r NT Remarks 11. 1. Note that if Mr is contained in ( ), then it is a -algebra. N T iλt ∗ iµt Indeed, if Mr ( ), taken x, y Mr such that t(x)= e x and t(y)= e y for some λ,µ R⊆Nand anyT t, by property∈ 3 in PropositionT 2 we have T ∈ (x∗y)= (x)∗ (y)= eit(µ−λ)x∗y t 0. Tt Tt Tt ∀ ≥ 10 ∗ As a consequence x y belongs to Mr. 2. If h is finite-dimensional, then also the opposite implication is true. itλ Indeed, if Mr is a -algebra, given x (h) such that t(x) = e x, λ R, we ∗ ∗ ∗ ∈ B T ∗ ∗∈ have t(x ) t(x) = x x. Then, by the Schwarz inequality, t(x x) x x for all t 0.T Set T r := . Since in this case is a strongly continuousT semigroup,≥ by ≥ Tt Tt|Mr T definition of Mr and by Corollary 2.9, Chapter V, of [15], the strong operator closure r M r −1 of t : t 0 is a compact topological group of operators in ( r). Hence, ( t ) {T ≥ } r r −1 B ∗ T M is the limit of some net ( tα )α and so ( t ) is a positive operator. Since x x r, for all α we have r (x∗xT) x∗x, and soT ( r)−1(x∗x) x∗x. On the other hand,∈ Ttα ≥ Tt ≥ ( r)−1( r(x∗x)) = x∗x ( r)−1(x∗x). Tt Tt ≥ Tt r −1 ∗ ∗ ∗ ∗ ∗ Therefore, ( t ) (x x) = x x and this implies t(x x) = x x = t(x) t(x). Sim- T ∗ ∗ T ∗ T T ilarly we can prove the equality t(xx ) = xx = t(x) t(x) , and so x belongs to ( ). T T T N T Now we are able to prove one of the central results of this paper. Theorem 12. Assume that there exists a faithful normal invariant state ρ. Then ( ) is atomic if and only if EID holds with ( )= M and = . N T N T r M2 M0 Proof. If ( ) is atomic, then EID holds by Theorem 5.1 in [13]. It remains to prove N TM k that ( )= r and 2 = 0. The atomicity implies ( )= i∈I ( ( i) 1lmi ) N T M M N T ⊕ B M ⊗ up to a unitary isomorphism. Let x = i∈I (xi 1lmi ) be in ( ) s, with ∗ ⊗ N T ∩ xi (ki) for every i I, and assume w limα t (x) = 0. Given ui, vi ki and τi ∈ B ∈ −P T α ∈ an arbitrary state on mi, by Theorem 4
tr(( u v τ ) (x)) = v , eitαKi x e−itαKi u . (10) | iih i| ⊗ i Ttα h i i ii Choosing u and v such that K u = λ u and K v = µ v , λ ,µ R, equation (10) i i i i i i i i i i i i ∈ becomes tr(( u v τ ) (x)) = eitα(µi−λi) v , x u , | iih i| ⊗ i Ttα h i i ii so that v , x u = 0, i.e. x = 0 because the eigenvectors of K from an orthonormal h i i ii i i basis of ki (see item 3 in Theorem 4). This proves the equality ( ) Ms = 0 . So we can conclude thanks to item 3 of Proposition 9. N T ∩ { } Conversely, if EID holds with ( )= Mr and 2 = 0, by Proposition 9 there exists a normal conditional expectationN T : (h) M ( )M onto ( ), compatible to E B →N T N T ρ. Therefore, ( ) is atomic thanks to Theorem 5. N T Remarks 13. As a consequence of Theorem 12 and Proposition 9 the following facts hold:
1. ( ) is atomic if and only if ( ) Ms = 0 , if and only if ( ) = M , i.eN T( ) is generated by eigenvectorsN T of∩ corresponding{ } to purely imaginaryN T r N T L 11 eigenvalues. Moreover, in this case we also have ( ) = 0 , being M : this N T ∩ M0 { } M0 ⊆ s means that, assuming ( ) atomic and the existence of a faithful invariant state, the situation is similarN toT the finite-dimensional case, i.e ( ) does not contain operators going to 0 under the action of the semigroup. N T 2. if ( ) is atomic and ( )= C1l, the semigroup satisfies the following prop- erties givenN T by non-commutativeF T Perron-Frobenius Theorem (see e.g. Propositions 6.1 and 6.2 in [5], Theorem 2.5 in [4]):
- the peripheral point spectrum σp( t) T of each t is a subgroup of the circle group T, T ∩ T - given t 0, each peripheral eigenvalue α of is simple and we have σ ( ) T = ≥ Tt p Tt ∩ α(σ ( ) T), p Tt ∩ - the restriction of ρ to ( ) is a trace. N T As a consequence, the peripheral point spectrum of each t is the cyclic group of all h-roots of unit for some h N. T ∈ 3. If ( ) is atomic, the decomposition induced by decoherence is uniquely deter- mined. ThisN T fact follows from Proposition 5 in [11], since we have ( ) M = 0 . N T ∩ 0 { } 4. Note that Theorem 12 does not exclude the possibility to have a QMS displaying decoherence with ( ) a non-atomic type I algebra. Clearly, in this case,T N T we will get ( ) ) M or ( . N T r M2 M0 Remark 14. In [25] the authors prove that EID holds when the semigroup commutes with the modular group associated with a faithful normal invariant state. However, our result in Theorem 12 is stronger since we find the equivalence between EID and the atomicity of ( ), which is a weaker assumption of the commutation with the modular group. InN fact,T it can be shown (see [27], section 3) that commutation with the modular group implies atomicity of ( ). Moreover, it is not difficult to find an N T example of a QMS on (h), with h finite dimensional which does not commute with the modular group. ItsB decoherence-free subalgebra, as any finite dimensional von Neumann algebra, will be atomic.
4 Structure of reversible states
In this section, assuming ( ) atomic and the existence of a faithful invariant state ρ, we study the structureN of reversibleT states, i.e. states belonging to the vector space ( ):= span σ I(h): (σ)= eitλσ for some λ R, t 0 (11) R T∗ { ∈ T∗t ∈ ∀ ≥ } = span σ I(h): (σ) = iλ σ for some λ R . (12) { ∈ L∗ ∈ }
12 In particular we will prove that ( ∗) is the predual of the decoherence-free algebra ( ). R T N T To this end, we recall the following result which is a version of the Jacobi- De Leeuw-Glicksberg theorem for strongly continuous semigroup (see Propositions 3.1, 3.2 in [24] and Theorem 2.8 in [15]). Theorem 15. If there exists a normal density ρ I(h) satisfying ∈ tr(ρ ( (x)∗ (x))) tr(x∗x) x (h), t 0, (13) Tt Tt ≤ ∀ ∈ B ≥ then we can decompose I(h) as w I(h)= ( ) σ I(h):0 (σ) . (14) R T∗ ⊕{ ∈ ∈ {T∗t }t≥0} Since each faithful invariant state clearly fulfills (13), we obtain the splitting given by equation (14). On the other hand, denoting by ⊥A the vector space σ I(h) : tr(σx) = 0 x A for all subset A of (h), the atomicity of ( {) ensures∈ the following ∀ ∈ } B N T facts:
(F1). (h)= ( ) 0 with ( )= Mr = Ran and 0 = Ms = Ker , where B : (hN) T ⊕M( ) is a conditionalN T expectationE compatibleM with theE faithful E B → N T state ρ (see Theorem 12 and Proposition 9); (F2). I(h)= ⊥ ⊥ ( ) with M0 ⊕ N T ⊥ = Ran ( ) , ⊥ ( ) = Ker . M0 E∗ ≃N T ∗ N T E∗ ≃M2∗ Moreover each acts as a surjective isometry on ⊥ , and lim (σ)=0 T∗t M0 t T∗t for all σ ⊥ ( ) (see Theorem 10 in [11]). ∈ N T As a consequence, every state ω ( )∗ is represented by a unique density σ in ⊥ , and, in this case, we write∈ Nω T= ω to mean that ω(x) = tr(σx) for all M0 σ x ( ). Therefore, if we denote by =( t)t≥0 the restriction of to ( ), we have∈N T S S T N T ( ω )(x)= ω ( (x)) = tr σ eitH xe−itH = tr (e−itH σeitH )x S∗t σ σ Tt E∗ for all x = (x) ( ), concluding that ∗tωσ is represented by the density (e−itH σeitH )E ⊥ ∈ N. InT a equivalent way, weS have E∗ ∈ M0 (σ)= (e−itH σeitH ) σ ⊥ . (15) T∗t E∗ ∀ ∈ M0 Theorem 16. If ( ) is atomic and there exists a faithful invariant state, then N T ( )= ⊥ = σ I(h): (σ)= (e−itH σ eitH ) t 0 ( ) , R T∗ M0 { ∈ T∗t E∗ ∀ ≥ }≃N T ∗ for every Hamiltonian H in a GKSL representation of the generator of . T 13 ⊥ −itH itH Proof. The inclusion 0 σ I(h): ∗t(σ) = ∗(e σ e ) t 0 follows M ⊆ { ∈ T E ∀ ≥ −}itH itH from the previous discussion. On the other hand, if we have ∗t(σ)= ∗(e σe ) T⊥ E for all t 0, taking t = 0 we get σ = ∗(σ), i.e. σ belongs to 0. Now,≥ given σ ( ) such that E(σ)= eitλσ for all t 0,Mλ R, we have ∈ R T∗ T∗t ≥ ∈ −itλ −itλ tr(σx) = lim tr(σx) = lim tr ∗t(σ)e x = lim e tr(σ t(x))=0 t→∞ t→∞ T t T ⊥ for all x 0, so that σ belongs to 0. This proves that ( ∗) is contained in ⊥ ∈ M M R T 0. InM order to prove the opposite inclusion it is enough to show that ⊥ ( ) contains w N T σ I(h):0 (σ) , since we have { ∈ ∈ {T∗t }t≥0} w I(h)= ( ) σ I(h):0 (σ) = ⊥ ⊥ ( ) R T∗ ⊕{ ∈ ∈ {T∗t }t≥0} M0 ⊕ N T by equation (14), item (F2) and Theorem 12. w So, let σ I(h) such that 0 (σ) ; given (t ) with w lim (σ) = 0 and ∈ ∈ {T∗t }t≥0 α − α T∗tα x M such that (x)= eitλx for some λ R, we have ∈ r Tt ∈ −itλ −itλ tr(σx) = lim tr σe t (x) = lim e tr( ∗t (σ)x)=0. α T α α T α This means that σ belongs to ⊥ ( ) by Theorem 12. N T In general, when there does not exist a faithful invariant state ( ) could be R T∗ different from ( ) , as we can see in Example 17. N T ∗ Example 17. Let us consider a generic Quantum Markov Semigroup with C3, more precisely the uniformly continuous QMS generated by
(x)= G∗x + L∗ xL + xG L 3j 3j j=1,2 X where γ33 G = + iκ3 e3 e3 , L3j = √γ3j ej e3 for j =1, 2, − 2 | ih | | ih | with κ3 R, γ3j > 0 for j =1, 2, and γ33 = γ31 γ32. We know that, the restriction of to∈ the diagonal matrices is the generator− of− a continuous time Markov chain (XL) with values in 1, 2, 3 . For more details see [1, 10]. t t { } Since 1 and 2 are absorbing states for (Xt)t, and 3 is a transient state, by Proposition 2 in [12] we know that any invariant state of is supported on span e , e . In T { 1 2} particular, this implies there is no faithful invariant state. Moreover, Theorem 8 in [12] gives ( )= C1l, since the absorbing states are acces- sible from 3. As a consequence, (N )T = C1l. N T ∗ On the other hand, since 1 is absorbing, the state e1 e1 is invariant, and so it belongs in particular to ( ). Therefore, we have (| ih) =| ( ) . R T∗ R T∗ 6 N T ∗ 14 We can now give the structure of reversible states when ( ) is a type I factor. N T Theorem 18. Let be a QMS on (k m) with a faithful invariant state ρ and T B ⊗ ( ) = (k) 1lm and let τm be the unique invariant state of the partially traced N T B m ⊗ semigroup defined in Theorem 4 item 2. Then a state η belongs to ( ∗) ( )∗ if and onlyT if R T ≃N T η = σ τm (16) ⊗ for some state σ on (k). B Proof. Let (ej)j≥1 be an orthonormal basis of eigenvectors of K so that Kej = κjej for some κ R. Given a state η, we can write j ∈ η = e e η | jih k| ⊗ jk j,k X≥1 with ηjk trace class operator on m, so that
m (η)= ei(κk−κj )t e e (η ) T∗t | jih k| ⊗ T∗t jk j,k X≥1 by Theorem 4. Therefore, by the linear independence of operators ej ek , we have (η)= eitλη for some λ R if and only if | ih | T∗t ∈ m (η )= eit(λ−κk+κj )η j, k, T∗t jk jk ∀ m i.e. if and only if each ηjk belongs to ( ∗ ). R T m Now, by item 3 of Theorem 4 we know that has a unique (faithful) invariant m m T m state τm and ( )= ( ); hence, Theorem 16 and Proposition 21 give ( ∗ ) m N Tm F Tm R T ≃ ( ) = ( ) = ( ) = span τm . As a consequence, we can conclude that N T ∗ F T ∗ F T∗ { } η belongs to ( ) if and only if η = tr(η ) τm for all j, k 1, i.e. if and only if N T ∗ jk jk ≥
η = (tr(η ) e e ) τm = σ τm jk | jih k| ⊗ ⊗ j,k X with σ := tr(η ) e e . j,k jk | jih k| This is enough to prove the statement since ( ∗) is the vector space generated by eigenstatesP of corresponding to modulo 1R eigenvalues.T T∗t If ( ) is not a type I factor, but it is atomic, we can obtain a similar result usingN thatT reversible states are “block-diagonal”, as the following proposition shows. Proposition 19. Assume ( ) atomic. Let be a QMS with a faithful invariant N T T state and let ( ) as in (3) with (pi)i∈I minimal projections in the center of ( ). Then p σp =0N Tfor all i = j and for all reversible state σ ( ). N T i j 6 ∈ R T∗ 15 itλ ⊥ Proof. Let σ ( ∗) such that ∗t(σ)= e σ for some λ R. Since ( ∗)= 0, by (F1) we have∈ R trT (σx) = tr(σxT ) for all (h) x = x ∈+ x with xR T ( )M and 1 B ∋ 1 2 1 ∈ N T x . Moreover, the property (p )= p for all k I and t 0 gives 2 ∈M0 Tt k k ∈ ≥ (p σp )= p (σ)p = eitλ p σp i, j I, T∗t i j iT∗t j i j ∀ ∈ so that each p σp belongs to ( ) ( ) . Therefore, since tr (p σp x) = i j R T∗ ≃ N T ∗ i j tr(σpjxpi) = 0 for all x ( ) = i∈I pi ( )pi, i = j, we obtain that piσpj = 0 for all i = j. ∈ N T ⊕ N T 6 6 As a consequence, by Theorem 4 we have the following characterization of re- versible states Theorem 20. Assume ( ) atomic and suppose there exists a faithful - invariant state. Let (p ) , (k ) N, (Tm ) be as in Theorem 4. A state η belongsT to ( ) if i i∈I i i∈I i i∈I R T∗ and only if it can be written in the form
η = tr(ηp ) σ τm i i ⊗ i i∈I X where, for every i I, ∈ m 1. τm is the unique i -invariant state which is also faithful, i T 2. σi is a density on ki. We have thus derived the general form of reversible states starting from the struc- ture of the atomic decoherence-free algebra ( ). This is a well known fact for a N T completely positive and unitary map (i.e. a channel) on a finite-dimensional space (see e.g. Theorem 6.16 in [31] and section V in [8]), but the proof of this result is not generalizable to the infinite dimensional case since it is based on a spectral decomposition of the channel based on eigenvectors.
5 Relationships with the structure of fixed points
In this section we investigate the structure of the set ( ) of fixed points of the semigroup and its relationships with the decomposition ofF T( ) given in the previous N T section. First of all we prove the atomicity of ( ) and relate this algebra with the space of invariant states. Really, the reader canF findT the proof of these results in [17]. We report them for sake of completeness. Proposition 21. If there exists a faithful normal invariant state, then ( ) is an F T atomic algebra and ( )∗ is isomorphic to the space ( ∗) of normal invariant functionals. F T F T
16 Proof. Since there exists a faithful invariant state, by Theorem 2.1 in [17] and in [18] ( ) it is the image of a normal conditional expectation : (h) ( ) given by F T E B →F T ∞ t ∗ −λt ∗ 1 (x)= w lim λ e t(x) dt = w lim s(x) ds. (17) E − λ→0 T − t→+∞ t T Z0 Z0 Hence, ( ) is atomic by Theorem 5, the range of the predual operator coincides F T E∗ with ⊥Ker and it is isomorphic to ( ) through the map E F T ∗ Ran = ⊥Ker σ σ ( ) . E∗ E ∋ 7→ ◦E∈F T ∗ Moreover, we clearly have Ran = ( ), (see also Corollary 2.2 in [17]). E∗ F T∗ Therefore, assuming the existence of a faithful invariant state, we can find a countable set J, and two sequences (sj)j∈J , (fj)j∈J of separable Hilbert spaces such that
h (s f ) (unitary equivalence) (18) ≃ ⊕j∈J j ⊗ j ( ) (s ) 1lf , ( -isomorphism isometric) (19) F T ≃ ⊕j∈J B j ⊗ j ∗ f where 1lfj denote the identity operator on j. Even if the decomposition (19) is given up to an isometric isomorphism, for sake of simplicity we will identify h with (s f ) and ( ) with (s ) 1lf . ⊕j∈J j ⊗ j F T ⊕j∈J B j ⊗ j Now we can state for ( ) a similar result to Theorem 4. Note that, it has F T already been proved for a quantum channel on a matrix algebra in [8] Lemma 6, and in [31] Theorems 6.12 and 6.14. Here, we extend this in the infinite-dimensional framework.
Theorem 22. Assume there exists a faithful normal invariant state. Let (sj)j∈J and (fj)j∈J be two countable sequences of Hilbert spaces such that (18) and (19) hold. Then we have the following facts: 1. for every GKSL representation (1) of the generator by means of operators L Lℓ,H, we have
(j) L = 1ls N ℓ 1, ℓ ⊕j∈J j ⊗ ℓ ∀ ≥ (j) H = λ 1ls 1lf + 1ls N , ⊕j∈J j j ⊗ j j ⊗ 0 (j) f (j) ∗ (j) where Nℓ are operators on j such that the series ℓ(Nℓ ) Nℓ are strongly convergent for all j J, (λ ) is a sequence of real numbers, and every M (j) ∈ j j∈J P 0 is a self-adjoint operator on fj;
17 f f j s f j 2. t(x y) = x t (y) for all x ( j) and y ( j), for j J, where T ⊗ ⊗ T ∈ Bf ∈ B ∈ T is the QMS on (f ) generated by j , whose GKSL representation is given by B j L N (j), N (j) : ℓ 1 ; { ℓ 0 ≥ } f 3. every j is irreducible and possesses a unique (faithful) normal invariant state T τfj ;
4. every invariant state η has the form η = σ τf with σ an arbitrary j∈J j ⊗ j j positive trace-class operator on sj such that tr(σj)=1. P j∈J Proof. Since (19) holds, like in the proof of TheoremsP 3.1, 3.2 in [13] there exist (j) (j) operators (N ) on (f ) such that L = 1ls N for all ℓ 1 and j J. ℓ ℓ B j ℓ ⊕j∈J j ⊗ ℓ ≥ ∈ s f Now, if pj is the orthogonal projection onto j j, we have H = l,m plHlmpm ∗ ⊗ with Hlm : hs hf hs hf and H = Hml for all l, m J. Since every m ⊗ m → l ⊗ l lm ∈ P x = (x 1lf ) ( ) commutes with H we get 0 = [x, H], i.e. ⊕j∈J j ⊗ j ∈F T
0= p [x 1lf ,H ]p + p (x 1lf )H H (x 1lf ) p , j j ⊗ j jj j j j ⊗ j jm − jm m ⊗ m m j j6=m X X which implies
[x 1lf ,H ]=0 j J, (x 1lf )H = H (x 1lf ) j = m. j ⊗ j jj ∀ ∈ j ⊗ j jm jm m ⊗ m ∀ 6 (j) The first condition is equivalent to have Hjj = λj1lsj 1lfj + 1lsj N0 for some (j) f ⊗ ⊗ N0 ( j) and λj R; the second one gives Hjm = 0 for all j = m, and so we obtain∈ item B 1. ∈ 6 Item 2 trivially follows. The proof of items 3 and 4 are similar to the ones of Theorem 4.1 and 4.3, respectively, in [13]. We want now to understand the relationships between decompositions (5) and (19) making use of the notations introduced in Theorems 4 and 22. In particular, in Theorem 23, we find a spectral characterization of the decomposition of the fixed point algebra, up to an isometric isomorphism. Indeed, in this representation, the spaces sj undergoing trivial evolutions are the eigenspaces of suitable Hamiltonians Ki corresponding to their different eigenvalues. First of all we introduce the following notation: for every i I denote by ∈ σ(K ) := κ(i) : j J (20) i { j ∈ i} with κ(i) = κ(i) for j = l in J , the (pure point) spectrum of the Hamiltonian K j 6 l 6 i i ∈ (k ) for some at most countable set J N. Note that, if has a faithful normal B i i ⊆ T invariant state, then σ(Ki) is exactly the spectrum of Ki thanks to Theorem 4.
18 Without of loss of generality we can choose the family Ji : i I such that Jh Jl = whenever h = l. In this way, set { ∈ } ∩ ∅ 6 J := J , (21) ∪i∈I i for j J there exists a unique i I such that j = j J . ∈ ∈ i ∈ i Theorem 23. Assume ( ) atomic and let ( ) = ( (k ) 1lm ) with (k ) , N T N T ⊕i∈I B i ⊗ i i i (mi)i two countable sequences of Hilbert spaces such that h = i∈I (ki mi). If there exists a faithful normal invariant state, up to an isometric isomorphism⊕ ⊗ we have
( )= (s ) 1lf (22) F T ⊕j∈J B j ⊗ j with J defined in (21), and
(i) s = s := Ker K κ 1lk , f = f := m j J , i I. (23) j ji i − j i j ji i ∀ i ∈ i ∈ K κ(i)q q Proof. By considering the spectral decomposition i = j∈Ji j ji with ( ji)j∈Ji mutually orthogonal projections such that P (i) q k = Ker K κ 1lk =: s ji i i − j i ji q m and j∈Ji ji = 1l i , we immediately obtain P k m =( s ) m = (s f ) i ⊗ i ⊕j∈Ji ji ⊗ i ⊕j∈Ji ji ⊗ ji by setting f := m for all j J . Therefore, by definition of J, since every j J ji i ∈ i ∈ belongs to a unique Ji, we have
h = (k m )= (s f )= (s f ) . ⊕i∈I i ⊗ i ⊕i∈I ⊕j∈Ji ji ⊗ ji ⊕j∈J j ⊗ j In order to conclude the proof we have to show equality (22). Given x ( ) ∈F T ⊆ ( ) (see item 4 in Proposition 2), we can write x = i∈I (xi 1lmi ) with (xi)i∈I N(kT), and so, by Theorem 4 we have ⊕ ⊗ ⊆ B i itK −itK x = (x 1lm )= ( (x 1lm )) = (e i x e i 1lm ). ⊕i∈I i ⊗ i Tt ⊕i∈I i ⊗ i ⊕i∈I i ⊗ i Consequently, x = eitKi x e−itKi for all i I, i.e. every x commutes with K , and i i ∈ i i then with each projection qji with j Ji. This means that each xi = j∈Ji qjixiqji k ∈ k s ⊕ belongs to the algebra j∈Ji qji ( i)qji = j∈Ji (qji i) = j∈Ji ( j), so that x is in s ⊕ B ⊕ B ⊕ B j∈J ( j ) 1lfj . ⊕ B ⊗ (i) On the other hand, given i I and j J , for u, v s = Ker(K κ 1lk ) we get ∈ ∈ i ∈ j i − j i itK itK ( u v 1lm )= e i u e i v 1lm = u v 1lm t 0, Tt | ih | ⊗ i | ih | ⊗ i | ih | ⊗ i ∀ ≥ 19 (i) where u and v are eigenvectors of Ki associated with the same eigenvalue κj . Since (i) Ker(K κ 1lk ) is generated by elements of the form u v , and the net ( (z)) i − j i | ih | Tt t is uniformly bounded for all z, we obtain the inclusion (s ) 1lm ( ) for all B j ⊗ i ⊆ F T j J , i I, i.e. (s ) 1lf ( ) for all j J = J . ∈ i ∈ B j ⊗ j ⊆F T ∈ ∪i∈I i Theorem below shows how we can derive an “atomic decomposition ”of ( ) from one of ( )’s. We want now to analyze the opposite procedure. F T N T s s f Assuming ( )= j∈J ( j) 1lfj with ( j)j, ( j)j two countable sequences of Hilbert spacesF suchT that⊕ h =B ⊗(s f ), and using notations of Theorem 22, we j∈j j j set an equivalence relation on ⊕J in the⊗ following way: Definition 24. Given j, k J, we say that j is in relation with k (and write j k) if there exist a complex separable∈ Hilbert space m and unitary isomorphisms ∼
V : f m, V : f m (24) j j → k k → (j) ∗ (j) ∗ (k) ∗ (k) ∗ such that operators VjNl Vj ,VjN0 Vj : l 1 and VkNl Vk ,VkN0 Vk : l 1 give the same Lindbladian{ operator on (m)≥. } { ≥ } B We obtain in this way an equivalence relation which induces a partition of J, J = n∈I In, for some finite or countable set I N, where each In is an equivalence class∪ with respect to . ⊆ ∼ Theorem 25. Assume that there exists a faithful normal invariant state and ( ) s s f N T atomic. Let ( ) = j∈J ( j) 1lfj with ( j)j, ( j)j two countable sequences of Hilbert spacesF suchT that⊕h = B (s⊗ f ), and let I : n I be the set of equivalence ⊕j∈j j ⊗ j { n ∈ } classes of J with respect to the relation . Then ( ) is isometrically isomorphic ∼ N T to the direct sum ( (k ) 1lm ) with ⊕n∈I B n ⊗ n k := s , m := V f j I , (25) n ⊕j∈In j n j j ∀ ∈ n where Vj’s are the unitary isomorphisms given in (24). Proof. Given n I, by definition of m we can define a unitary operator by setting ∈ n U : (s f ) ( s ) m = k m n ⊕j∈In j ⊗ j → ⊕j∈In j ⊗ n n ⊗ n (j) j I (uj zj) u Vjzl ⊕ ∈ n ⊗ 7→ j∈In j ⊗ (j) P where u in hs denotes the vector j ⊕i∈In i
(j) 0 if i = j uj := i∈In vi, vi := 6 . ⊕ u if i = j j 20 Now, since h = (s f )= ( (s f )) ⊕j∈j j ⊗ j ⊕n∈I ⊕j∈In j ⊗ j by the equality J = n∈I In, by setting U := n∈I Un we get a unitary operator U : h (k m ∪) such that ⊕ → ⊕n∈I n ⊗ n ∗ U ( )U = (( (s )) 1lm ) . F T ⊕n∈I ⊕j∈In B j ⊗ n In order to conclude the proof we have to show that
∗ U ( )U = ( ( s ) 1lm ) . N T ⊕n∈I B ⊕j∈In j ⊗ n
To this end recall that, by Theorem 22, the operators (Lℓ)ℓ, H in a GKSL represen- tation of the generator can be written as L (j) (j) L = 1ls N ℓ 1, H = λ 1ls 1lf + 1ls N ℓ ⊕j∈J j ⊗ ℓ ∀ ≥ ⊕j∈J j j ⊗ j j ⊗ 0 (j) s (j) (j) ∗ f with (Nℓ )ℓ≥1 ( j), N0 = (N0 ) ( j) and λj R for all j J; moreover, ⊆ B ∈ B (n) ∈ (n) ∈(n) ∗ m by definition of In, we can choose operators (Ml )l and M0 = (M0 ) in ( n) (n) (n) B mn such that Ml , M0 : l 1 is a GKSL representation of the the generator { m ≥ } j j L of a QMS n on (m ) equivalent to V N ( )V ∗,V N ( )V ∗ : l 1 for all j I . T B n { j l j j 0 j ≥ } ∈ n Therefore, the operators
′ ∗ (n) L := 1ls V M V ℓ ⊕n∈I ⊕j∈In j ⊗ j ℓ j ′ ∗ (n) H := λ 1ls 1lf + 1ls V M V , ⊕n∈I ⊕j∈In j j ⊗ j ⊕j∈In j ⊗ j 0 j clearly give the same GKSL representation of L , H : ℓ 1 . Moreover we have { ℓ ≥ } ′ ∗ ∗ (n) ∗ (n) UL U = U 1ls V M V U = 1lk M ℓ ⊕n∈I n ⊕j∈In j ⊗ j ℓ j n ⊕n∈I n ⊗ ℓ ′ ∗ ∗ (n) ∗ UH U = U λ 1ls 1lf + 1ls V M V U ⊕n∈I n ⊕j∈In j j ⊗ j ⊕j∈In j ⊗ j 0 j n (n) = K 1lm + 1lk M ⊕n∈I n ⊗ n n ⊗ 0 ∗ with K := λ 1ls = K (k ), so that n ⊕j∈In j j n ∈ B n
m ′ ∗ m ′ ∗ m (n) Uδ (L )U = δ ′ ∗ (UL U )= 1lk δ (n) (M ) H′ ℓ UH U ℓ n∈I n M ℓ ⊕ ⊗ 0 m ′∗ ∗ m ′∗ ∗ m (n)∗ Uδ (L )U = δ ′ ∗ (UL U )= 1lk δ (n) (M ) H′ ℓ UH U ℓ n∈I n M ℓ ⊕ ⊗ 0
21 ∗ m ′ ∗ m ′∗ ∗ ′ for all m 0. Therefore, since U ( )U = UδH′(Lℓ)U , UδH′(Lℓ )U : m 0 by item 2 of≥ Proposition 2, we obtainN thatT an operator{ x ( (k m ))≥ belongs} ∈ B ⊕n∈I n ⊗ n to U ( )U ∗ if and only if it commutes with N T m (n) m (n)∗ 1lk δ (n) (M ) , 1lk δ (n) (M ) n∈I n M ℓ n∈I n M ℓ ⊕ ⊗ 0 ⊕ ⊗ 0 for every m 0. ≥ k m Now, let qn := 1lkn 1lmn the orthogonal projection onto n n. We clearly have ∗ ⊗ ⊗h k m qn U ( )U and n∈I qn = 1l. Therefore, the algebra qn ( )qn = ( n n) is ∈ F T ∗ ∗ B B ⊗ preserved by the action of every map t := U t(U U)U , and so we can consider the P T T · (n) restriction of to this algebra, getting a QMS on (kn mn) denoted by and satisfying (T (n))= q U ( )U ∗q ,e where ( (n)B) is the⊗ decoherence-freeT algebra N T n N T n N T of (n). Notee that, since each q commutes with every UL′ U ∗, UL′∗U ∗ and UH′U ∗, T n ℓ ℓ a GKSL representation of the generator (n) of (n) is given by operators L T ′ ∗ (n) ′ ∗ (n) q UH U q = K 1lm + 1lk M q UL U q = 1lk M , n n n ⊗ n n ⊗ 0 n ℓ n n ⊗ ℓ so that
m (x y)= eitKn xe−itKn n (y) x (k ), y (m ) Tt ⊗ ⊗ Tt ∀ ∈ B n ∈ B n and e (n) m (n) m (n)∗ ′ ( )= 1lk δ (n) (M ), 1lk δ (n) (M ): ℓ 1, m 0 n M ℓ n M ℓ N T { ⊗ 0 ⊗ 0 ≥ ≥ } m = (k ) ( n ). (26) B n ⊗N T Since U ( )U ∗ = q U ( )U ∗q N T n N T m n,m∈I M and equation (26) holds, to conclude the proof we have to show that
∗ m q U ( )U q = 0 n = m, and ( n )= C1lm . n N T m { } ∀ 6 N T n So, let x U ( )U ∗ and consider n, m I with n = m. Since the net ( (q xq )) ∈ N T ∈ 6 Tt n m t≥0 is bounded in norm and the unit ball is weakly* compact, there exists a weak* ∗ cluster point y such that y = w limα tα (qnxqm). Therefore, for σ ( ∗) with (σ)= eitλσ for some λ R, we− have T ∈ R T T∗t ∈ itαλ tr(σy) = lim tr(σ t (qnxqm)) = lim e tr(σqnxqm) . α T α α
Now, if λ = 0 this implies tr (σqnxqm)=0=tr(σy); otherwise, since σ is an invariant state, we automatically6 have q σq = 0 for n = m by Theorem 22, so that tr (σy) is m n 6 22 0 again. Consequently, tr (σy) = 0 for all σ ( ∗). On the other hand, taking η w ∈ R T such that 0 (η) , up to passing to generalized subsequences we have ∈ {T∗t }t≥0 tr(ηy) = lim tr(η t (qnxqm)) = lim tr( ∗t (η)qnxqm)=0. α T α α T α
We can then conclude that tr (σy) = 0 for all σ I( n (kn mn)) by virtue of equation (14), and so y = 0. This means that q xq ∈belongs⊕ to ⊗ ( ) M and this n m N T ∩ s intersection is 0 by Corollary 12 ( ( ) is atomic). Therefore qnxqm = 0 for n = m and so { } N T 6 U ( )U ∗ = q ( )q = ( (n)). N T ⊕n∈I nN T n ⊕n∈I N T Moreover, since ( ) is atomic by Corollary 12, the general theory of von Neumann N T (n) mn algebras (see e.g. [28]) says that each algebra ( )= (kn) ( ) is atomic m N T B ⊗N T too, and consequently ( n ) is atomic. Finally, recalling that, by construction, f f mn j N T j t = t for all j In with t an irreducible QMS having a faithful invariant state T T ∈ T m (see Theorem 22), we get ( n )= C1lm thanks to Proposition 4.3 in [13]. N T n Remark 26. Theorem 25 provides in particular a way to pass from the decomposition of GKSL operators H, (Ll)l of according to the splitting of h = j∈j(sj fj) associated with the atomic algebraL ( ), to the other one decomposition⊕ with respect⊗ to the splitting h = (k m )F associatedT with ( ). ⊕n∈I n ⊗ n N T More precisely, if ( )= (s ) 1lf and F T ⊕j∈J B j ⊗ j (j) (j) L = 1ls N ℓ 1, H = λ 1ls 1lf + 1ls N ℓ ⊕j∈J j ⊗ ℓ ∀ ≥ ⊕j∈J j j ⊗ j j ⊗ 0 (j) s (j) f with (Nℓ )ℓ≥1 ( j ), N0 ( j) and λj R for all j J, then we can decompose H, (L ) with respect⊆ B to the splitting∈ B h = ∈ (k m )∈ given by (25) as follows: ℓ ℓ ⊕n∈I n ⊗ n (n) (n) L = 1lk M , H = K 1lm + 1lk M , ℓ ⊕n∈I n ⊗ ℓ ⊕n∈I n ⊗ n n ⊗ 0 (n) (n) (n)∗ m where (Mℓ )ℓ≥1, M0 = M0 are operators in ( n) giving the same GKSL rep- (j) (j) B resentation of (Nl )ℓ≥1, N0 for all j In, and every Kn is the self-adjoint operator on (k ) having (λ ) as eigenvalues∈ and (s ) as corresponding eigenspaces. B n j j∈In j j∈In Acknowledgements. The financial support of MIUR FIRB 2010 project RBFR10COAQ “Quantum Markov Semigroups and their Empirical Estimation” is gratefully acknowl- edged.
References
[1] L. Accardi, F. Fagnola, S. Hachicha, Generic q-Markov semigroups and speed of convergence of q-algortithms, Infin. Dim. Anal. Quantum Prob. Rel. Topics 9, 567 (2006).
23 [2] J. Agredo, F. Fagnola and R. Rebolledo, Decoherence free subspaces of a quantum Markov semigroup, J. Math. Phys. 55, 112201 dx.doi.org/ 10.1063/1.4901009, (2014).
[3] V.V. Albert, B. Bradlyn, M. Fraas, L. Jiang, Geometry and response of Lind- bladians, Phys. Rev. X 6, 041031 (2016).
[4] S. Albeverio and R. Høegh-Krohn, Frobenius theory for positive maps of von Neumann algebras, Commun. Math. Phys., 64, 83–94 (1978).
[5] A. Batkai, U. Groh, D. Kunszenti-Kovacs and M. Schreiber, Decomposition of operator semigroups on W ∗-algebras, Semigroup Forum, 84, no.1, 8–24 (2012).
[6] B. Baumgartner and H. Narnhofer, The structure of state space concerning quantum dynamical semigroups. Rev. Math. Phys. 24, no. 2, 1250001, (2012).
[7] Ph. Blanchard and R. Olkiewicz, Decoherence Induced Transition form Quantum to Classical Dynamics, Rev. Math. Phys. 15, no. 3, 217–243, (2003).
[8] R. Blume-Kohout, H. K. Ng, D. Poulin, L. Viola, Information-preserving struc- tures: a general framework for quantum zero-error information, Phys. Rev. A 82, 062306, (2010).
[9] R. Carbone, E. Sasso and V. Umanit`a, Decoherence for Quantum Markov Semi- groups on matrix spaces, Annales Henri Poincar´e 14, no. 4, 681–697 (2013).
[10] R. Carbone, E. Sasso, V. Umanit`a, On the asymptotic behavior of generic quan- tum Markov semigroups, Infin. Dimens. Anal. Quantum Probab. Relat. Top., 17, no. 1, 1450001, p.18 (2014).
[11] R. Carbone, E. Sasso and V. Umanit`a, Environment induced decoherence for markovian evolutions. J. Math. Phys. 56, 092704 (2015).
[12] R. Carbone, E. Sasso, V. Umanit`a, Some remarks on decoherence for generic quantum Markov semigroup, Infin. Dimens. Anal. Quantum Probab. Relat. Top. 20, 1750012 (2017).
[13] J. Deschamps, F. Fagnola, E. Sasso, V. Umanit`a, Structure of Uniformly Con- tinuous Quantum Markov Semigroups, Rev. in Math. Phys. 28, n.1, 165003 (32 pages), (2016).
[14] A. Dhahri, F. Fagnola and R. Rebolledo, The decoherence-free subalgebra of a quantum Markov semigroup with unbounded generator. Infin. Dimens. Anal. Quantum Probab. Relat. Top. 13, 413–433, (2010).
24 [15] K. J. Engel, R. Nagel, One-parameter semigroups for linear evolution equations, Springer-Verlag, New York, 2000.
[16] D.E. Evans, Irreducible quantum dynamical semigroups, Commun. Math. Phys. 54, 293–297, (1977).
[17] A. Frigerio, Stationary States of Quantum Dynamical Semigroups, Commun. Math. Phys. 63, 269–276, (1978).
[18] A. Frigerio and M. Verri, Long-Time Asymptotic Properties of Dynamical Semi- groups on W ∗-algebras. Math. Z. 180, 275-286, (1982).
[19] V. Gorini, A. Kossakowski, E.C.G. Sudarshan, Completely positive dynamical semigroups of N-level systems. J. Math. Phys. 17, 821–825, (1976).
[20] Hayes, Normal conditional expectations between von Neumann algebras. Preprint.
[21] M. Hellmich, Quantum dynamical semigroups and decoherence, Advances in Math. Phys., 625978 (16 pages), (2011).
[22] D. A. Lidar and K. B. Whaley, in: Decoherence-free Subspaces and Subsystems, Lecture Notes in Physics, 622 (2003), 83–120.
[23] G. Lindblad, On the Generators of Quantum Dynamical Semigroups. Commun. Math. Phys. 48, 119–130, (1976).
[24] B. Kummerer, and R. Nagel, Mean ergodic semigroups on W ∗-algebras. Acta Sci. Math. (Szeged) 41, no. 1-2, 151–159, (1979).
[25] P. Lugiewicz, R. Olkiewicz, Classical properties of infinite quantum open sys- tems, Comm. Math. Phys. 239, no.1, (2003) 241–259.
[26] K. R. Parthasarathy, An introduction to quantum stochastic calculus, Mono- graphs in Mathematics 85, Birkh¨auser-Verlag, Basel 1992.
[27] M. Takesaki, Conditional expectations in von Neumann algebras, J. Funct. Anal. 9, 306–321 (1972).
[28] M. Takesaki, Theory of operator algebras I, Springer-Verlag, New York- Heidelberg, 1979.
[29] F. Ticozzi and L. Viola, Quantum Markovian Subsystems: invariance, attractiv- ity, and control, IEEE Transaction on automatic control, 53, 2048–2063, (2008).
25 [30] J. Tomiyama, On the projection of norm one in W∗-algebras III, Tˆohoku Math. J. 11, 125–129, (1959).
[31] M. M. Wolf, Quantum channel and operations, guided tour, notes, (2012).
1 Department of Mathematics, Politecnico di Milano, Piazza Leonardo da Vinci 32, I - 20133 Milano, Italy E-mail: [email protected] 2 Department of Mathematics, University of Genova, Via Dodecaneso, I - 16146 Genova, Italy E-mail: [email protected] 3 Department of Mathematics, University of Genova, Via Dodecaneso, I - 16146 Genova, Italy E-mail: [email protected]
26