Invariant Projections for Covariant Quantum Markov Semigroups

Journal of Stochastic Analysis

Volume 1 Number 4 Article 3

December 2020

Franco Fagnola Politecnico di Milano, [email protected]

Emanuela Sasso University of Genova, 16100 Genova, Italy, [email protected]

Veronica Umanità University of Genova, 16100 Genova, Italy, [email protected]

Follow this and additional works at: https://digitalcommons.lsu.edu/josa

Part of the Analysis Commons, and the Other Mathematics Commons

Recommended Citation Fagnola, Franco; Sasso, Emanuela; and Umanità, Veronica (2020) "Invariant Projections for Covariant Quantum Markov Semigroups," Journal of Stochastic Analysis: Vol. 1 : No. 4 , Article 3. DOI: 10.31390/josa.1.4.03 Available at: https://digitalcommons.lsu.edu/josa/vol1/iss4/3 Journal of Stochastic Analysis Vol. 1, No. 4 (2020) Article 3 (14 pages) digitalcommons.lsu.edu/josa DOI: 10.31390/josa.1.4.03

INVARIANT PROJECTIONS FOR COVARIANT QUANTUM MARKOV SEMIGROUPS

FRANCO FAGNOLA, EMANUELA SASSO*, AND VERONICA UMANITA`

Dedicated to Professor Leonard Gross on the occasion of his 88th birthday

Abstract. In this paper we investigate consequences of covariance of a uniformly Quantum Markov Semigroup, under a group action, on the structure of its minimal invariant projections. We obtain that, under suitable hypotheses, minimal invariant projections correspond to irreducible sub-representations in which the initial covariant representation is decomposed. We apply this results in the study circulant Quantum Markov Semigroups.

1. Introduction Semigroups of completely positive unital maps describe irreversible dynamics of a open quantum system. They often arise from scaling limits of a quantum system interacting with an external environment and covariance properties reflect the symmetries of the total Hamiltonian such as spatial isotropy or translation invariance. A semigroup ( t)t 0 on (h) is called covariant with respect to the action g π(g) of a groupT on ≥ (h)ifB → B (π(g)∗xπ(g)) = π(g)∗ (x)π(g), Tt Tt for all x (h), g G and t 0. It is clear that the generator satisfies the corresponding∈B covariance∈ property≥ and it is natural to ask if thisL property influences the structure of . A.S. Holevo investigated the structure of the infinitesimal generator ([14, 15,L 16, 17]). In particular, he proved that the covariance property imposesL strong restrictions on . Indeed, in the uniformly continuous case, if we consider a (covariant) (Gorini-Kossakowski-Sudharshan-Lindblad)L [18] GKSL representation of ,(H, (Lk)k), we have that H commutes with the family of operators π(g)(g GL) of the representation π. A key step in the∈ study of a QMS is determining whether it is irreducible and, if not, its irreducible sub-semigroups. These are in one to one correspondence with common invariant subspaces of operators k Lk∗Lk +2iH and Lk (see [9, Theorem III.1]). Finding all common invariant subspaces of a set of operators, however, is typically a difficult problem (see [21])! therefore it is worth investigating the

Received 2019-12-10; Accepted 2020-6-24; Communicated by A. Stan et al. 2010 Mathematics Subject Classification. 82C10; 47D06; 46L55. Key words and phrases. Quantum Markov semi-groups, covariance, atomic algebras, irreducibility. *Correspondingauthor. 1 2 F. FAGNOLA, E. SASSO, AND V. UMANITA` implications of covariance in order to gain insight on the structure of QMSs by the interplay of representation theory and invariant subspace problems. Other objects are strongly connected to the structure of the infinitesimal generator. The most important for the study of the asymptotic behavior of the semigroup are the Decoherence-free sub algebra ( ) and the set of the fixed points ( ). In [13], the authors analyzed the structureN T of the decoherence free-subalgebraF T ( ) of a uniformly continuous covariant semigroup with respect to a represen- tationN T π of a compact group G on h. In particular, they obtained that, when π is d irreducible, ( ) is isomorphic to ( (k) 1l m) for suitable Hilbert spaces k and m, and a integerN T d related to the numberB ⊗ of connected components of G.This is the easy case, because if π is irreducible, the Hilbert space h has to be finite dimensional and so ( ) is an atomic algebra. In order to extend this result when π is reducible, theyN hadT to add the additional hypothesis that ( ) is atomic. As we showed in [7], the reason is that the atomicity of the decoherence-freeN T subalgebra of a uniformly-continuous QMS forces the Lindblad operators to have a block-diagonal structure, inherited by the same atomic decomposition of ( )in type I factors. N T In this paper, we want to compare the covariance property with the set of the fixed points. The latter, in general, is not an algebra, but when there exists a faithful normal invariant state, it is a subalgebra of ( ). Moreover, we have that ( ) is the image of a conditional expectation and,N asT a consequence, it is always atomic.F T So the aim of this work is to show what consequences the covariance property has for the invariant projections of a uniformly continuous QMS. In Section 2 we introduce preliminary definitions and results about atomic decoherence-free subalgebras, the set of the fixed points and covariant QMSs. In Section 3, we investigate the GKSL representation of a covariant QMS, showing that, if is covariant, there exists a covariant GKSL representation, that is “compatible”T with respect to the representation of G. But it can happen that for a covariant QMS there exists a GKLS representations that it is not covariant. Then in Section 4, we focus on the relationship between the representation π and the minimal projections pi in the center of ( ), that appear in its atomic decomposition. When the representation is reducible,F T we obtain that the projections, (qj )j, on the invariant spaces for the representation π are fixed points and, when we have that every invariant projections commutes with π(g) for every g G, ∈ we have some important consequences. For example, that the qj ’s are the unique minimal invariant projections for and every minimal projection in the center of T ( ) can be written as sum of qj. In Section 5, we show that the circulant QMSs satisfiesF T this condition and in this context we have a description of ( )(= ( )) F T N T as direct sum of the family qj .

2. Preliminary Results Let h be a complex separable Hilbert space. A Quantum Markov Semigroup on the algebra (h) of all bounded operators on h is a weakly*-continuous semi- B group =( t)t 0 of completely positive, identity preserving normal maps. Such T T ≥ INVARIANT PROJECTIONS 3 semigroup is uniformly continuous if it satisﬁes

lim t 1l =0. t 0+ %T − % → t In this case its generator is a linear and bounded operator such that t = e L in the uniform topology. Moreover,L we can express in a GKSL representationT [18] L 1 (x)=i[H, x] (L∗L x 2L∗xL + xL∗L ) L − 2 k k − k k k k k 1 "≥ for some H = H∗ and (Lk)k operators in (h) for which the series k Lk∗Lk is strongly convergent. In particular, we considerB here special representations of , ! L by means of operators (Lk)k such that 1l,L1,L2,...are linearly independent. More precisely, we have the following characterization (see [20], Proposition 30.14 and the discussion below the proof of Theorem 30.16): Theorem 2.1. Let be the generator of a uniformly continuous QMS on (h). L B Then there exist a bounded selfadjoint operator H and a sequence (Lk)k 1 of ele- ments in (h) such that: ≥ B (1) k 1 Lk∗Lk is strongly convergent, ≥ 2 (2) if k 0 ck < and c01l + k 1 ckLk =0for scalars (ck)k 0 then ! ≥ | | ∞ ≥ ≥ ck =0for every k 0, ! 1 ≥ ! (3) (x)=i[H, x] 2 k 1 (Lk∗Lkx 2Lk∗xLk + xLk∗Lk) for all x (h). L − ≥ − ∈B Moreover, if H$, (Lk$ )k 1 is! another family of bounded operators in (h) with H$ selfadjoint, then it satisfies≥ conditions (1)-(3) if and only if the lengthsB of sequences (Lk)k 1, (Lk$ )k 1 are equal and there exists a sequence (αk)k 1 C with 2 ≥ ≥ ≥ ⊆ αk < and β R such that k | | ∞ ∈ 1 ! H$ = H + β1l + (S S∗), L$ = u L + α 1l (2.1) 2i − k kj j k j " for some unitary matrix U =(ukj)kj, where S := kj αkukj Lj . We conclude this section recalling some basic definitions! about two important subspaces of (h) associated with . They are strongly connected to the phenom- enon of decoherenceB (see e.g. [4,T 6, 1, 19]) and the asymptotic properties of the semigroup ([5, 8, 11, 12]). The decoherence-free (DF) subalgebra of is denoted by ( ) and is defined by T N T

( )= x (h): t(x∗x)= t(x)∗ t(x), t(xx∗)= t(x) t(x)∗ t 0 . N T { ∈B T T T T T T ∀ ≥ }(2.2) Since is uniformly continuous, ( ) is the biggest subalgebra of (h) on which T N T itH B itH every operator t is a -automorphism, and we have t(x)=e xe− x ( ) ([10]). T ∗ T ∀ ∈ N MoreoverT we denote by ( ) the set of ﬁxed points of ,i.e. F T T ( )= x (h):T (x)=x t 0 = x (h): (x)=0 . (2.3) F T { ∈B t ∀ ≥ } { ∈B L } When ( ) is atomic (i.e. for every non-zero projection p ( )there exists a non-zeroN T minimal projection q ( ) such that q p), we∈ obtainNT some ∈N T ≤ 4 F. FAGNOLA, E. SASSO, AND V. UMANITA` additional information on the structure of the semigroup (see [7]). In general, ( ) could be no atomic, while if there exists a faithful invariant state, ( )is alwaysN T atomic. F T

3. Structure of the Generator of a Covariant QMS The following deﬁnition establishes the main property of that we study in this paper. T Deﬁnition 3.1. Let G be a compact group and π : g π(g) a continuous unitary representation of G on h. The uniformly continuous QMS,→ on (h) is said to be covariant with respect to the representation π if T B

(π(g)∗xπ(g)) = π(g)∗ (x)π(g) (3.1) Tt Tt for all x (h), g G and t 0. ∈B ∈ ≥ This property can be expressed equivalently in terms of the generator as

(π(g)∗xπ(g)) = π(g)∗ (x)π(g) (3.2) L L for all x (h) and g G. The structure∈B of the∈ generator of a covariant uniformly continuous QMS was fully characterized by Holevo in [14], Section 2, in the case of amenable locally compact groups. When G is compact, the result can be restated as follows. Theorem 3.2. Let G be a compact group, π : g π(g) acontinuousunitary representation of G on h.If is a uniformly continuous,→ QMS, then it is covariant if and only if there exists a GKSLT representation of (called covariant) given by operators H, L : k 1 satisfying: L { k ≥ } (1) L∗π(g)∗xπ(g)L = π(g)∗L∗xL π(g) for all x (h) and g G, k k k k k k ∈B ∈ (2) H π(g):g G $. ! ∈{ ∈ } ! Moreover, the condition in Item 1 is equivalent to

π(g)∗Lj π(g)= v(g)jkLk (3.3) "k for all g G, where V (g)=(v(g) ) is a unitary matrix. In particular, for all ∈ jk jk g G, the operators H,π(g)∗Lkπ(g):k 1 give another GKSL representation of∈ . { ≥ } L Theorem 3.2 directly implies the following corollary that we will widely use in the reminder of the paper. Corollary 3.3. Let H, (L ) be in a covariant GKSL representation of . Then k k L H and k Lk∗Lk intertwine the representation π. Proof. By! Theorem 3.2 we already know that π(g)H = Hπ(g) for all g G.To conclude the proof is enough to note that by item 1 of the same theorem∈ and the unitarity of π we have

π(g)∗ Lk∗Lk π(g)= Lk∗(π(g)∗π(g))Lk = Lk∗Lk. # $ "k "k "k Therefore π(g)( L∗L )=( L∗L ) π(g) for all g G. k k k k k k ∈ ! ! ! INVARIANT PROJECTIONS 5

Theorem 3.2 gives the existence of a covariant GKSL representation: but what happens if we consider another GKSL representation given by operators H$, (Lk$ )k? The following result shows that in general it is not covariant. Proposition 3.4. Let be a uniformly continuous covariant QMS w.r.t. a unitary T representation π of a compact group G, and let H, Lk : k 1 be in a covariant GKSL representation of the generator . { ≥ } L Then another GKSL representation H$,Lk$ : k 1 is covariant if and only if it is connected to the former by equation{ (2.1) with≥ }

αk =(UV(g)U ∗α)k, (3.4) where V (g)=(v(g)ij )ij is the unitary matrix of equation (3.3) for the operators (Lk)k 0. ≥ Proof. We begin by proving that equation (3.4) ensures the existence of a unitary matrix V $(g) such that equation (3.3) holds also for the operators (Lk$ ). Since H, Lk : k 1 and H$,Lk$ : k 1 give two GKSL representations of , by equation{ (2.1)≥ we} have { ≥ } L

π(g)∗Lk$ π(g)= ukhπ(g)∗Lhπ(g)+αk1l "h = (UV(g))khLh + αk1l "h = (UV(g)U ∗) (L$ α 1l ) + α 1l kh h − h k "h

= (UV(g)U ∗)khLh$ + αk (UV(g)U ∗)khαh 1l . # − $ "h "h If condition (3.4) holds, then (Lk$ )k clearly satisﬁes the covariance condition (3.3) with respect to the unitary matrix V $(g)=UV(g)U ∗. On the other hand, relation π(g)∗Lk$ π(g)= h w(g)khLh$ for a unitary matrix W (g)=(w(g)kh)kh implies !

(UV(g)U ∗ W (g))khLh$ + αk (UV(g)U ∗)khαh 1l = 0 . − # − $ "h "h Since 1l ,L$ : k 1 are linearly independent and { k ≥ } 2 2 2 (UV(g)U ∗ W (g)) (UV(g)U ∗) + (W (g)) =2< , | − kh| ≤ | kh| | kh| ∞ k k k " 2" " 2 2 αk (UV(g)U ∗)khαh αk + (UV(g)U ∗)khαh < . % − % ≤ | | % % ∞ k % h % k k % h % " % " % " " %" % We immediately% have UV(g)U ∗% = W (g) and αk = % (UV(g)U ∗)khαh,% i.e. equa- % % % h % tion (3.4) is fulﬁlled. This condition is also suﬃcient for [H$,π(g)] = 0 to hold for all g G. First of all we note that ! ∈ 1 1 [H$,π(g)] = [H + β1l + (S S∗), π(g)] = [S S∗, π(g)] 2i − 2i − 6 F. FAGNOLA, E. SASSO, AND V. UMANITA`

1 for all g G and β R. Moreover, since [S∗,π(g)] = [S,π(g− )]∗, it is enough to prove∈ that [S,π(g∈)] = 0 for all g G to conclude the− proof. Indeed we have ∈

Sπ(g)= αiuij Ljπ(g) i,j " = π(g) αi(UV (g))ij Lj i,j " = π(g) αh(UV (g)∗U ∗UV(g))hjLj "h,j = π(g)S for all g G and therefore [H$,π(g)] = 0 for all g G. ∈ ∈ ! We conclude the section proving that ( ) and ( )behavenicelywithre- spect to the covariance property. N T F T Proposition 3.5. Let be a covariant QMS w.r.t. a unitary representation π of a compact group G. ThenT

π(g)∗ ( )π(g)= ( ) and π(g)∗ ( )π(g)= ( ) N T N T F T F T for all g G. ∈ Proof. Let g G and x be in ( ). We have to prove that y := π(g)∗xπ(g) belongs to (∈ ). The covarianceN ofT gives N T T (y∗y)= (π(g)∗x∗xπ(g)) = π(g)∗ (x∗x)π(g) T Tt Tt =(π(g)∗ (x∗) π(g)) (π(g)∗ (x)π(g)) = (π(g)∗x∗π(g)) (π(g)∗xπ(g)) Tt Tt Tt Tt = (y∗) (y). Tt Tt In the same way we can show the equality t(yy∗)= t(y) t(y∗), i.e. y ( ). For the set of ﬁxed points ( ) the proofT is very similar.T T ∈N T F T !

Whenever the representation π is irreducible it is possible to further specify the structure of the generator . Indeed, in this case, CorollaryL 3.3 and Schur’s Lemma give

L∗Lk C1l and H C1l , k ∈ ∈ "k so that the following result immediately follows. Proposition 3.6. Let G be a compact group, π : g π(g) an irreducible unitary ,→ representation of G on a ﬁnite dimensional Hilbert space h. Let also H, (Lk)k be operators in a covariant GKSL representation of . Then can be written as L L (x)= L∗xL $x (3.5) L k k − "k where $ is a real positive constant such that k Lk∗Lk = $1l . ! INVARIANT PROJECTIONS 7

4. Structure of Invariant Projections for a Covariant QMS In this section, assuming the existence of a faithful normal invariant state for the covariant QMS , we clarify the relationships between the structure of ( ) and operators π(g).T F T First of all we recall the following facts about the set of ﬁxed points of an arbitrary QMS (not necessarily covariant). a) A projection p belongs to ( ) if and only if it commutes with operators H and (L ) in any GKSLF representationT of , k k L b)Ifp in as invariant projection, then we have t(pxp)=p t(x)p for every x (h) and t 0. This means that the algebraT p (hT) p = (p(h)) is preserved∈B by the≥ semigroup, and so we obtain by restrictionB a QMSB p on (p(h)), i.e. T B p(x)= (x)= (pxp)=p (x)p x (p(h)). Tt Tt Tt Tt ∀ ∈B Moreover we have ( p)=p ( )p. c)If has a faithfulF normalT invariantF T state, then ( ) is the image of a normalT conditional expectation and so it is an atomicF T algebra. There- fore there exist two countable sequences (ki)i I and (mi)i I of separable Hilbert spaces such that ∈ ∈

( )= i I ( (ki) 1l m ) F T ⊕ ∈ B ⊗ i in accordance to the decomposition of h given by h = i I (ki mi). Moreover, denoting by p the orthogonal projection onto⊕ ∈k m⊗ for i I, i i ⊗ i ∈ the collection (pi)i I is a family of mutually orthogonal minimal projec- ∈ tions in ( ( )) such that i I pi =1l. Z F T ∈ On the other hand, if is covariant (but non necessarily with a faithful invari- ! ant state), since π can beT decomposed into the direct sum of ﬁnite-dimensional irreducible sub-representations acting on orthogonal subspaces, there exists a collection (Vi)j J of pairwise orthogonal, ﬁnite-dimensional and π-invariant subspaces of h such that:∈

(1) h = j J Vj , ⊕ ∈ (2) the restriction πj of π to every Vj is an irreducible unitary representation, (3) each orthogonal projection q onto V belongs to ( ) and j j F T π (g)=q π(g)q = π(g)q = q π(g) g G, (4.1) j j j j j ∀ ∈ (4) the restriction j of to (V ) is covariant with respect to π . T T B j j For more details we refer the reader to [13, Theorem 8]. We will prove that the family (pi)i I determines the structure of non zero central invariant projections. ∈ Proposition 4.1. Assume with a faithful normal invariant state. Every non zero invariant projection q inT ( ( )) can be written as Z F T q = pi i I "∈ 0 for some non empty subset I0 of I. 8 F. FAGNOLA, E. SASSO, AND V. UMANITA`

Proof. Since both pi and q belong to ( ( )) for all i I, each piq is a projection in ( ( )). Therefore, set I = i ZI F: pTq =0 and∈ taken i I , the inequality Z F T 0 { ∈ i . } ∈ 0 piq = piqpi pi implies piq = pi by the minimality of pi in the center of ( ). So ≤ F T q = piq = piq = pi, i i I i I " "∈ 0 "∈ 0 concluding the proof. ! When the semigroup satisﬁes the additional condition

( ) π(g):g G $ (4.2) F T ⊆{ ∈ } we can provide a more accurate description of invariant central projections and the general structure of ( ). F T Remark 4.2. In general condition (4.2) does not imply that the semigroup is covariant with respect to the representation π, but it forces an invariant projec-T tion p for the semigroup to be invariant with respect the conjugation with π,i.e. π(g)∗pπ(g)=p for every g G. ∈ Proposition 4.3. Let π be a unitary representation of a compact group G on h, and let be a uniformly continuous (not necessarily covariant) QMS on (h) possessingT a faithful normal invariant state and satisfying ( ) π(g):Bg F T ⊆{ ∈ G $.Ifp is an invariant projection, its range p(h) is a π-invariant subspace and the} restriction π : G (p(h)) of π to p(h) is the sub-representation given by p →B π (g)=pπ(g)=π(g)p = pπ(g)p g G. (4.3) p ∀ ∈ Moreover, if is covariant with respect to π, p is covariant with respect to π . T T p Proof. Since p is an invariant projection, by assumption it commutes with every π(g) and so p(h)isπ-invariant and equation (4.3) immediately follows. p If is π-covariant, the covariance of with respect to πp is clear since preserves (Tp(h)). T T B ! In addition, given an invariant projection p, equation (4.2) allows us to prove the equivalence between the irreducibility of the representation πp and that one of the semigroup p. Recall that aT QMS is irreducible if there does not exist non-trivial projections T q such that t(q) q for all t 0 (see [9] Deﬁnition II.2) In particular, if possesses a faithfulT ≥ invariant state,≥ since a projection q satisfying (q) q,byT Tt ≥ tr(ρ( t(q) q)) = 0, turns out to be invariant, irreducibility is equivalent to the non-existenceT − of non-trivial invariant projections. Proposition 4.4. Assume covariant with respect to the representation π of G T with a faithful normal invariant state ρ and ( ) π(g):g G $. Given an invariant projection p, then the following factsF areT equivalent⊆{ ∈ } (1) p is minimal in ( ); (2) p is irreducible;F T T (3) ( p)=Cp; F T (4) πp is irreducible. INVARIANT PROJECTIONS 9

Proof. 1. 2. Assume p minimal in ( ) and take q (p(h)) an invariant ⇒ p F Tp ∈B p projection for .Thenq p and q = t (q)= t(q) by deﬁnition of .This clearly meansT that q belongs≤ to ( ),T and so eitherT q = 0 or q = Tp by the minimality of p in this algebra. ThereforeF T p is irreducible. T 2. 3. Denote by ρ the faithful normal invariant state of .Sincetr(ρp) =0 ⇒ 1 T . (otherwise p =0bytheﬁdelity),thenpρp(tr (pρ))− is a faithful normal invariant state for p. Therefore, the irreducibility of p and the atomicity of ( p) force T T F T to have ( p)=Cp. F T 3. 4. Assume πp reducible. Then by Peter-Weyl Theorem there exists a non trivial⇒ projection q ( p) (p(h)) (see (4.1) here) such that π is irreducible ∈F T ⊆B q and p = q. This contradicts the assumption ( p)=Cp. . F T 4. 1. Assume πp irreducible and let q be a non zero projection in ( )such that q⇒ p. Hence, by Proposition 4.3, q(h)isaπ-invariant subspace ofFp(Th), and so it coincides≤ with p(h), i.e. q = p. This proves the minimality of p in ( ). F T ! Finally we can characterize the structure of ( ) showing that every invariant F T projection p coincides with some qj . Theorem 4.5. Let be a covariant uniformly continuous QMS with a faithful normal invariant stateT and satisfying

( ) π(g):g G $. F T ⊆{ ∈ } If p is a non zero minimal projection in ( ), then there exists a unique index F T j J such that p = qj . In particular, the qj’s are the unique minimal invariant projections∈ for . T Proof. First of all we claim that each qj is a minimal projection in ( ). Indeed, if q is a non zero projection in ( ) such that q q (so that q F T(V )), since F T ≤ j ∈B j the restriction j of to (V )isπ -covariant and π is irreducible, statement 2 T T B j j j in Proposition 4.4 implies p =1lVj = qj , so that qj is minimal. Now we consider pj := qj p, the orthogonal projection onto qj(h) p(h). Since we clearly have p q and∧ p p and both q and p are in ( ),∩ we get j ≤ j j ≤ j F T t(pj ) qj and t(pj)& p for all t 0. Consequently t(pj) qj p = pj for all tT 0, so≤ that p Tbelongs≤ to ( ) thanks≥ to existence ofT a faithful≤ ∧ invariant state. ≥ j & F T & Since& qj is minimal& in ( ), this implies either pj = 0 or p&j = qj . If the ﬁrst& case happens for every j JF,T the relation & ∈ p(h)= j J qj(h) p(h)= j J qj(&h) p(h) &= j J pj (h) ∪ ∈ ∩ ∪ ∈ ∩ ∪ ∈ gives the contradiction p = 0. Therefore, there exists at least one j J such that ' ( ' ( ∈ pj = qj . Finally, since the projections qj are mutually orthogonal,& we can have p = q for a unique j J. j j ∈ ! Theorem& 4.6. Assume covariant with a faithful normal invariant state and T & ( ) π(g):g G $. Then, for all i I there exists a subset J J such that F T ⊆{ ∈ } ∈ i ⊆ pi = qj . j J "∈ i Moreover, for diﬀerent i, k I, we have Ji Jk = , i.e. every qj belongs to a unique block (k m ). ∈ ∩ ∅ B ij ⊗ ij 10 F. FAGNOLA, E. SASSO, AND V. UMANITA`

Proof. Given i I deﬁne Ji = j J : piqj =0 . Then, since each qj belongs to ( ) and p is∈ in the center of{ this∈ algebra, p. q }= q p q is an invariant non zero F T i i j j i j projection in (Vj ) for all j Ji. Now, by the commutation between ( ) and every π(g)impliesB ∈ F T

π (g)∗p q π (g)=π(g)∗p q π(g)=p q g G, j i j j i j i j ∀ ∈ i.e. piqj intertwines the representation πj . Therefore, by Schur’s Lemma the equality p q = q follows for all j J .So i j j ∈ i pi = piqj = piqj = qj j J j J j J "∈ "∈ i "∈ i

Finally, if there exists l Ji Jk for some i, k I with i = k,thenql pi and q p , giving the contradiction∈ ∩ q = 0. ∈ . ≤ l ≤ k l ! As the last result of this section, we analyze the action of the conjugation with the representation π on the minimal projections pi in the center of ( ), that appear in the atomic decomposition of the algebra. F T

Proposition 4.7. For all g G there exists a unique permutation σg of I such that ∈ π(g)p π(g)∗ = p i I. (4.4) i σg (i) ∀ ∈ Moreover, if π is irreducible and ( ) is not a factor, there is at least one g G such that σ (i) = i for all i I. F T ∈ g . ∈ Proof. Assume ( ) is not a factor (otherwise equation (4.4) is trivially satisﬁed), so that I has cardinalityF T greater than one. Let i I and g G.Sinceπ(g)is ∈ ∈ unitary and π(g)∗ ( )π(g)= ( ) by Proposition 3.5, there exists a unique projection q ( F)T dependingF onTg, such that i ∈F T pi = π(g)∗qiπ(g). (4.5)

We claim that the minimality of pi implies that one of qi. Indeed, taken a non zero projection qi$ ( ) satisfying qi$ qi,sinceπ(g)∗qi$π(g) belongs to ( ) we have ∈FT ≤ F T π(g)∗q$π(g) π(g)∗q π(g)=p , i ≤ i i so that the minimality of pi gives either π(g)∗qi$π(g) = 0 or π(g)∗qi$π(g)=pi.The ﬁrst equality contradicts the assumption q$ = 0, so that the equation i . π(g)∗qi$π(g)=pi = π(g)∗qiπ(g) holds. This means qi$ = qi,i.e.qi is minimal in ( ), proving the claim. Now, since the atomicity of ( ) gives F T F T q = q 1l i j(i) ⊗ mj(i) j I "∈ for some projection q (k ), we immediately obtain q q 1l for all j(i) ∈B j(i) i ≥ l(i) ⊗ ml(i) l(i) I.Butqi is minimal in ( ), to which also ql(i) 1l ml(i) belongs, and so we have∈ either q 1l = 0 orF qT 1l = q . Now,⊗ if q 1l = 0 for all l(i) ⊗ ml(i) l(i) ⊗ ml(i) i l(i) ⊗ ml(i) l(i) I,wehaveqi = 0 contradicting the assumption pi = 0. Hence, there exists aunique∈ l(i) I satisfying q 1l = q , and we can. put σ(i)=l(i), with ∈ l(i) ⊗ ml(i) i INVARIANT PROJECTIONS 11

σ : I I. The uniqueness of l(i) is a consequence of the the fact that qi i I is a set of→ orthogonal projections. It follows that { } ∈

p = π(g)∗ q 1l π(g) π(g)∗ p π(g), i σ(i) ⊗ mσ(i) ≤ σ(i) being q 1l a projection' in (k ( m ) and'p (the unit of this space. σ(i) ⊗ mσ(i) B σ(i) ⊗ σ(i) σ(i) Since pσ(i) is minimal in ( ) we get pi = π(g)∗pσ(i)π(g), i.e. equality (4.4) holds. Moreover σ is a permutation.F T Indeed σ(i)=σ(j)withi = j implies . pi = π(g)∗pσ(i)π(g)=π(g)∗pσ(j)π(g)=pj , and this is not possible. Finally assume π irreducible. Since ( ) is not a factor, we have that p =1l F T i . for all i I. If for all g G there exists a permutation σg of I such that σg(i)=i for at least∈ one i I,then∈ ∈ p = π(g)∗p π(g) g G, i i ∀ ∈ i.e. each pi intertwines the representation. Therefore, Schur’s Lemma implies pi C1l, contradicting the assumption. ∈ ! The meaning of this result is the following: for all g G the conjugation with ∈ π(g)∗ changes the block (k m )intheblock (k m ), where j = σ (i). B i ⊗ i B j ⊗ j g We can eventually have j = i,i.e. (ki mi) is invariant with respect to the transformation. In particular, if π is irreducibleB ⊗ and ( ) is not a factor, there is F T no invariant blocks with respect to the conjugation with π(g)∗.

5. Application to Circulant Quantum Markov Semigroups J. R. Bolaños-Serv´ın and R. Quezada [3] introduced the class of circulant QMSs whose properties have been further studied in [2]. The name of this family of uniformly continuous QMS depends the GKSL form of their generator involving circulant matrices. We fix a dimension p 2 and we choose as our von Neumann ≥ algebra the algebra of the p p matrices with complex entries, i.e. (h)= p(C). × p B M We indicate with (ek)k p the canonical basis of C . We underline that, since the ∈Z index set is Zp, all the operations among indices should be meant modulus p. Let p k Jc be a circulant matrix associated to a cycle c of order p (i.e. c = id and c = id for k =1,...,p 1). We can express J in terms of the canonical basis as . − c J = e e . c | c(k)56 c(k+1)| k "∈Zp Without loss of generality, we can consider as c,thecyclec =(0, 1, 2, 3,....p 1). − In this case J = Jc (the primary permutation matrix) has an easier formula, since J = k ek ek+1 . Now we can define a circulant generator on p(C) as ∈Zp | 56 | L M p 1 ! − k k (x)= γ(p k)J ∗ xJ x L − − k"=1 p 1 p 1 with γ a vector in C − such that γ(k) > 0 for k =1,...,p 1 and − γ(k) = 1. − k=1 Clearly is written in the GKSL representation by means of operators Lk = L ! 12 F. FAGNOLA, E. SASSO, AND V. UMANITA`

γ(p k) J k for k = 1,...,p 1, and H = 1l. The (uniformly continuous) QMS generated− by is the circulant− semigroup associated with the cycle c. T) L 1 Remark 5.1. Since J ∗J = 1l, the state p− 1l is a faithful invariant state for the circulant QMS .So ( ) is an algebra and T F T k k ( )= ( )= J ,J∗ : k Zp $. N T F T { ∈ } In particular every invariant projection p satisfies k k J pJ ∗ = p k Zp. (5.1) ∀ ∈ From (5.1) we can deduce a necessary condition for a projection to be invariant. Proposition 5.2. Every invariant projection p,withp = p e e ,fora h,j| h56 j | circulant QMS satisfies ph+1,j+1 = ph,j for every h, j Zp. ∈ ! k k Proof. Every invariant projections p has to commute with J and J ∗ for every k Zp, but it is equivalent to require that p commutes with J,i.e.pJ = Jp. Now ∈ pJ = p e e e e h,j| h56 j | | i+156 i| i "h,j " = ph,j eh ej 1 | 56 − | "h,j = p e e , h+1,j+1| h+156 j | "h,j and, analogously, Jp = e e p e e | i+156 i| h,j| h56 j | i " "h,j = p e e . h,j| h+156 j | "h,j This concludes the proof. ! k Define now a representation π : Zp Md(C)bysettingπ(k)f = J f for all p → k Zp and f h = C . Clearly is covariant with respect to π and ( )= ∈ ∈ T F T π(k):k Zp $, i.e. condition (4.2) is trivially satisfied. { Moreover,∈ the} representation π is not irreducible, since, for example, the space span (1,...,1) is invariant for every π(g), with g Zp. Therefore, we can de- { } ∈ compose π into the direct sum of irreducible sub-representations (πj )j J on the ∈ mutually orthogonal subspaces (Vj )j j such that the orthogonal projection qj onto V is minimal in ( ) and ( j )=∈ q ( )q ,being j the restriction of to j F T F T j F T j T T Vj (see Proposition 4.4). In order to determine projections qj , we recall the following result: Proposition 5.3. The permutation matrix J is diagonalized by F , the discrete Fourier transform, i.e. p 1 − i FJF∗ = ω e e , | i56 i| i=0 " 1 p 1 kl where ω is a primitive p-root of the unit and F = − ω e e . √p k,l=0 | k56 j | ! INVARIANT PROJECTIONS 13

p 1 This clearly means that the family of vectors Fe − is an orthonormal basis of { i}i=0 h given by eigenvectors of J related to the eigenvalue ωi. Therefore each projection p 1 p 1 − − 1 ki li 1 (k l)i q := Fe Fe = ω ω e e = ω − e e (5.2) i | i56 i| p | k56 l| p | k56 l| k",l=0 k",l=0 p 1 is a minimal invariant projection for , and the family q − is a collection of T { i}i=0 mutually orthogonal minimal invariant projections such that i qi =1l. Since each q commutes with J k = π(k) for all k =0,...,p 1, its range i ! − Vi := qi(h)isπ-invariant, and so the restriction of π to Vi gives a sub-representation π of π with π (k)=q π(k)q for all i =0,...,p 1. i i i i − Finally, note that πi is irreducible, being Vi an one-dimensional space. Summarizing: p 1 Proposition 5.4. The unitary representation π splits in the direct sum i=0− πi, where each π : G (q (h)) is an irreducible representation of G and the⊕ semi- i →B i group qi , given by restricting to (q (h)),isπ -covariant. T T B i i p 1 We now show that the family q − gives an atomic decomposition of ( ). { i}i=0 N T Lemma 5.5. Every q belongs to ( ( )), the center of ( ). i Z N T N T Proof. Let p be a projection in ( ). Since it commutes with J, p has to commute N T with its spectral projections and so, in particular, with each qi = Fei Fei . Therefore, since ( )= ( ) is generated by its projections, we immediately| 56 | obtain q ( (F ))T for allN iT=0,...,p 1. i ∈Z N T − ! Theorem 5.6. We have ( )= ( )=Cq0 ... Cqp 1. N T F T ⊕ ⊕ − Proof. By Lemma 5.5 projections q ’s belong to ( ( )), so that i Z N T p 1 ( )= − q ( )q , N T ⊕i=0 iN T i being i qi = 1l. Finally, since each qi has a one-dimensional range, we obtain qi ( )qi = Cqi, concluding the proof. ! N T! Acknowledgment. The authors would like to thank the organizers for the invi- tation to the 40th International Conference on Quantum Probability and Inﬁnite Dimensional Analysis held at Ohio State University, Columbus on August 2019. Special thanks go to Professor Aurel Stan for support, invaluable assistance and a lively atmosphere for discussion and exchange of knowledge.

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Franco Fagnola: Department of Mathematics, Politecnico di Milano, 20133 Milano, Italy E-mail address: [email protected]

Emanuela Sasso: Department of Mathematics, University of Genova, 16100 Genova, Italy E-mail address: [email protected]

Veronica Umanita:` Department of Mathematics, University of Genova, 16100 Gen- ova, Italy E-mail address: [email protected]