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For example, the notions of transience and recurrence have been studied in [5], proper definitions of these notions have been developed in this context and the analogues of transient or recurrent points have been characterized. An interesting feature is that the internal degrees of freedom introduce a source of memory which gives rise to a specific non-Markovian behavior. Recall that, in the classical context ([22]), an exact dichotomy exists for irreducible Markov chains: a point is either recurrent or transient, and the nature of a point can be characterized in terms of the first return time and the number of visits. In contrast, irreducible open quantum walks exhibit three possibities regarding the behavior of return time and number of visits. In this article, we study the recurrence and transience, as well as their characterizations, for continuous-time versions of OQW. In the same way that open quantum walks are quantum extensions of discrete-time Markov chains, there exist natural quantum extensions of continuous-time Markov processes. One can point to two different types of continuous-time evolutions with a structure akin to open quantum walks. The first (see [6]) is a natural extension of classical Brownian motion and is called open quantum brownian motion; it is obtained by considering OQW in the limit where both time and in space are properly rescaled to continuous variables. The other type of such evolution (see [25]) is an analogue of continuous-time Markov chains on a graph, is obtained by rescaling time only, and is called continuous-time open quantum walks (CTOQW). In this article we shall concentrate on the latter. Roughly speaking CTOQW represents a continuous-time evolution on a graph where a “walker” jumps from node to node at random times. The intensity of jumps depends on the internal degrees of freedom; the latter is modified by the jump, but also evolves continuously between jumps. In both cases the form of the intensity, as well as the evolution of the internal degrees of freedom at jump times and between them, can be justified from a quantum mechanical model. As is well-known, in order to study a continuous-time Markov chain, it is sufficient to study the value of the process at the jump times. Indeed, the time before a jump depends exclusively on the location of the walker, and the destination of the jump is independent of that time. As a consequence, the process restricted to the sequence of jump times is a discrete time Markov chain, and all the properties of that discrete time Markov chain such as irreducibility, period, transience, recurrence, are transferred to the continuous-time process. This is not the case for OQW. In particular, a CTOQW restricted to its jump times is not a (discrete-time) open quantum walk. Therefore, the present study of recurrence and transience cannot be directly derived from the results in [5]. Nevertheless, we can still adopt a similar approach and, for instance, we study irreducibility in the sense of irreducibility for quantum dynamical systems (as defined in [12], see also [18]). As in the discrete case, we obtain a trichotomy, in the sense that CTOQW can be classified in three different possible statuses, depending on the properties of the associated return time and number of visits. The paper is structured as follows. In Section 1, we recall the definition of continuous-time open quantum walks and in particular we introduce useful classical processes attached to CTOQW. Section 2 is devoted to the notion of irreducibility for CTOQW. In Section 3, we address the question of recurrence and transience and give the classification of CTOQW .
2 1 Continuous time open quantum walks and their associated classical processes
This section is devoted to the introduction of continuous-time open quantum walks (CTOQW). In Subsection 1.1, we introduce CTOQW as a special instance of quantum Markov semigroups (QMSs) with generators preserving a certain block structure. Subsection 1.2 is devoted to the exposition of the Dyson expansion associated to a QMS, which will serve as a tool in all remaining sections. It also allows us to introduce the relevant probability space. Finally, in Subsection 1.3 we associate to this stochastic process a Markov process called quantum trajectory which has an additional physical interpretation, and that will be useful in its analysis.
1.1 Definition of continuous-time open quantum walks Let V denotes a set of vertices, which may be finite or countably infinite. CTOQWs are quantum analogues of continuous-time Markov semigroups acting on the set L∞(V ) of bounded functions on V . They are associated to stochastic processes evolving in the composite system = h , (1) H i iM∈V where the hi are separable Hilbert spaces. This decomposition has the following interpretation: the label i in V represents the position of a particle and, when the particle is located at the vertex i V , its internal state is encoded in the space h (see below). Thus, ∈ i in some sense, the space hi describes the internal degrees of freedom of the particle when it is sitting at site i V . When hi does not depend on i, that is if hi h, for all i V , one ∈ 2 ≃ ∈ has the identification h ℓ (V ) and then it is natural to write hi = h i (we use here Dirac’s notation whereH the ≈ ket⊗ i represents the i-th vector in the canonical⊗| basisi of ℓ∞(V ), | i the bra i for the associated linear form, and i j for the linear map f j f i ). We h | | ih | 7→ h | i| i will adopt the notation hi i to denote hi in the general case (i.e. when hi depends on i) to emphasize the position of⊗| thei particle, using the identification h C h . We thus write: i ⊗ ≃ i = h i . (2) H i ⊗| i iM∈V Last, we denote by ( ) the two-sided ideal of trace-class operators on a given Hilbert space I1 K and by the space of density matrices on , defined by: K SK K = ρ ( ) ρ∗ = ρ, ρ 0, Tr(ρ)=1 . SK { ∈ I1 K | ≥ }
A faithful density matrix is an invertible element of K, which is therefore a trace-class and positive-definite operator. Following quantum mechaniS cal fashion, we will use the word “state” interchangeably with “density matrix”. We recall that a quantum Markov semigroup (QMS) on ( ) is a semigroup := I1 K T ( t)t≥0 of completely positive maps on 1( ) that preserve the trace. The QMS is said to beT uniformly continuous if lim IId K= 0 for the operator norm on ( ). It is then t→0kTt − k B K known (see [21]) that the semigroup ( t)t≥0 has a generator = limt→∞( t Id)/t which is a bounded operator on ( ), called theT Lindbladian, and Lindblad’sL TheoremT − characterizes I1 K
3 tL the structure of such generators. One consequently has t = e for all t 0, where the exponential is understood as the limit of the norm-convergeT nt series. ≥ Continuous-time open quantum walks are particular instances of uniformly continuous QMS on ( ), for which the Lindbladian has a specific form. To make this more precise, I1 H we define the following set of block-diagonal density matrices of : H = µ ( ) ; µ = ρ(i) i i . D ∈ S H ⊗| ih | n iX∈V o In particular, for µ with the above definition, one has ρ(i) (h ), ρ(i) 0 and ∈ D ∈ I1 i ≥ Tr ρ(i) = 1. In the sequel, we use the usual notations [X,Y ] = XY YX and i∈V − PX,Y = XY + YX, which stand respectively for the commutator and anticommutator of {two operators} X,Y ( ). ∈ B H Definition 1.1. Let be a Hilbert space that admits a decomposition (1). A continuous-time open quantum walk His a uniformly continuous quantum Markov semigroup on ( ) such I1 H that its Lindbladian can be written: L : 1( ) 1( ) L I H → I H 1 µ i[H,µ]+ 1 SjµSj∗ Sj∗Sj,µ , (3) 7→ − i6=j i i − 2{ i i } i,jX∈V where H and (Sj) are bounded operators on that take the following form: i i,j H H = H i i , with H bounded self-adjoint operators on h , i in V ; • i∈V i ⊗| ih | i i P j for every i = j V , Si is a bounded operator on with support included in hi and with • 6 ∈ H j∗ j range included in hj, and such that the sum i,j∈V Si Si converges in the strong sense. j j Consistently with our notation, we can writeP Si = Ri j i for bounded operators Rj (h , h ). ⊗| ih | i ∈ B i j We will say that the open quantum walk is semifinite if dim h < for all i V . i ∞ ∈ i i From now on we will use the convention that Si = 0, Ri = 0 for any i V . As one can immediately check, the Lindbladian of a CTOQW preserves the set . More∈ precisely, for L D µ = i∈V ρ(i) i i , denoting t(µ) =: i∈V ρt(i) i i for all t 0, we have for all i V ⊗| ih |∈D T ⊗| ih | ≥ ∈P d P 1 ρ (i)= i[H , ρ (i)] + Ri ρ (j)Ri∗ Rj∗Rj, ρ (i) , dt t − i t j t j − 2{ i i t } jX∈V 1.2 Dyson expansion and associated probability space
In this article, our main focus is a stochastic process (Xt)t≥0 that informally represents the position of a particle or a walker constrained to move on V . In order to rigorously define this process and its associated probability space, we need to introduce the Dyson expansion associated to a CTOQW. In particular, this allows to define a probability space on the possible trajectories of the walker. We will recall the result for general QMS as we will use it in the next section. The application to CTOQW is described shortly afterwards. Let ( ) be a uniformly continuous QMS with Lindbladian on ( ) for some Tt t≥0 L I1 K separable Hilbert space . By virtue of Lindblad’s Theorem [21], there exists a bounded K 4 self-adjoint operator H ( ) and bounded operators L on (i I) such that for all ∈ B K i K ∈ µ 1( ), ∈ I K 1 (µ) = i[H,µ]+ L µL∗ L L∗,µ , L − i i − 2{ i i } Xi∈I where I is a finite or countable set and where the series is strongly convergent. The first step is to give an alternative form for the Lindbladian. First introduce 1 G := iH L∗L , − − 2 i i Xi∈I so that for any µ , ∈D (µ)= Gµ + µG∗ + L µ L∗ . (4) L i i Xi∈I ∗ ∗ Remark that G + G + i∈I Li Li = 0, so that (17) is the general form of the generator of ∗ a QMS, as given by LindbladP [21]. The operator (G + G ) is positive semidefinite and t etG defines a one-parameter semigroup of contractions− on by a trivial application of the7→ Lumer-Phillips theorem (see e.g. Corollary 3.17 in [15]). WeK are now ready to give the Dyson expansion of the QMS.
Proposition 1.2. Let ( t)t≥0 be a QMS with Lindbladian as given above. For any initial density matrix µ , oneT has L ∈ SK ∞ ∗ t(µ)= ζt(ξ) µζt(ξ) dt1 dtn , (5) T n 0 Corollary 1.3. Let ( t)t≥0 be a CTOQW with Lindbladian given by Equation (7). For any initial density matrixT µ , one has L ∈D ∞ ∗ t(µ)= Tt(ξ) ρ(i0)Tt(ξ) dt1 dtn in in , (8) T n 0 (t−tn)Gin in (tn−tn−1)Gi − (t2−t1)Gi i1 t1Gi Tt(ξ) := e R e n 1 e 1 R e 0 . (9) in−1 ··· i0 5 Note the small discrepancy between ξ = (i1,...,in; t1,...,tn) in (5) and ξ = (i0,...,in; t1,...,tn) in (8), where the additional index i0 is due to the decomposition of µ. Remark 1.4. Equation (5) is also called an unravelling of the QMS. It was first introduced in [13, 30], with a heuristic interpretation as the average result of trajectory ξ = (i0,...,in; t1,...,tn) on the state µ, averaged over all possible such trajectories. We discuss ∗ later connections with an operational interpretation of Tt(ξ)ρ(i0)Tt(ξ) in Section 1.4. The decomposition described in Equation (9) will allow us to give a rigorous definition of the probability space associated to the evolution of the particle on V . The goal is to introduce the probability measure Pµ that models the law of the position of the particle, when the initial density matrix is µ . The following is inspired by [4, 7, 19]. ∈D (n) First define the set of all possible trajectories up to time t [0, ] as Ξt := Ξt , ∈ ∞ n⊔∈N (n) where Ξt is the set of trajectories on V up to time t comprising n jumps: Ξ(n) := ξ =(i ,...,i ; t ,...,t ) V n+1 Rn, 0 < t < < t < t . t { 0 n 1 n ∈ × 1 ··· n } (n) (t) (n) For t R+, the set Ξt is equipped with the σ-algebra Σt and with the measure νt , which is induced∈ by the map I : V n+1 [0, t)n, (V n+1) ([0, t)n), δn+1 1 λ Ξ(n), Σ(n), ν(n) , n × P × B × n! n → t t t (i ,...,i ; s ,...,s ) (i ,...,i ; s ,...,s ) 0 n 1 n 7→ 0 n min max where δ is the counting measure on V , ([0, t)n) is the Borel σ-algebra on [0, t)n and λ is the B n Lebesgue measure on ([0, t)n) for all n 0. These measures are σ-finite and this allows us to B ≥ (t) apply Carathéodory’s extension Theorem. We first define the σ-algebra Σt := σ(Σt , n N) (n) (n) ∈ and the measure νt on Ξt such that νt = νt on Ξt . For a given µ = i∈V ρ(i) i i in , one can then define the probability measure Pt on (Ξ , Σ ) such that, for all E ⊗|Σih, | D µ t t P ∈ t Pt ∗ t µ(E) := Tr Tt(ξ) µ Tt(ξ) dν (ξ) E Z∞ ∗ = 1ξ∈E Tr Tt(ξ)ρ(i0)Tt(ξ) dt1 dtn , (10) n 0 6 1.3 Quantum trajectories associated to CTOQW Quantum trajectories are another convenient way to describe the distribution of the position process (Xt)t≥0 associated to the CTOQW. Actually, the combination of quantum trajectories and of the Dyson expansion will be essential tools for the main result of this article. Formally speaking, quantum trajectories model the evolution of the state when a continuous measurement of the position of the particle is performed. The state at time t can be described by a pair (X , ρ ) with X V the position of the particle at time t (as recorded by t t t ∈ the measuring device) and ρt H the density matrix describing the internal degrees of freedom, given by the wave function∈ S collapse postulate and thus constrained to have support on hi alone. The stochastic process (Xt, ρt)t≥0 is then a Markov process, and this will allow us to use the standard machinery for such processes. However, their rigorous description is less straightforward than the one for discrete time OQWs. It makes use of stochastic differential equations driven by jump processes. We refer to [25] for the justification of the below description and for the link between discrete and continuous-time models. Remark that we denote by the same symbol the stochastic process (Xt)t≥0 appearing in this and the previous section. This will be justified in Section 1.4 below. In order to present the stochastic differential equation satisfied by the pair (Xt, ρt) we need a usual filtered probability space Ω, , ( ) , P , where we consider independent F Ft t Poisson point processes N i,j, i, j V, i = j on R2. These Poisson point processes will govern the jump from site i to site j on∈ the graph6 V . Definition 1.5. Let ( ) be a CTOQW with Lindbladian and let µ = ρ(i) i i be Tt t≥0 L i∈V ⊗| ih | an initial density matrix in . The quantum trajectory describing the indirect measurement D P of the position of the CTOQW is the Markov process (µt)t≥0 taking values in the set such that D µ = ρ X X , 0 0 ⊗| 0ih 0| where X0 and ρ0 are random with distribution ρ(i) P (X0, ρ0)= i, = Tr(ρ(i)) for all i V Tr(ρ(i))!! ∈ and such that µt =: ρt Xt Xt satisfies for all t 0 the following stochastic differential equation: ⊗| ih | ≥ t t j j∗ Si µs− Si i,j µt = µ0 + (µs−) ds + µs− 1 j j∗ N (dy, ds) R j j∗ 0 The connection between the process (Xt, ρt) defined in this section and the Dyson expansion has been deeply studied in the literature. We do not give all the details of this construction and instead refer to [4, 7] for a complete and rigorous justification. The main point is that the ∞ ∞ process (Xt, ρt) defined in Section 1.3 can be constructed explicitly on the space (Ξ , Σ , P), as we now detail. Recall the interpretation of ξ = (i0,...,in; t1,...,tn) as the trajectory of a particle, ∞ ∞ initially at i0 and jumping to ik at time tk. First, on (Ξ , Σ , P) define the random variable ˜ i,j Nt by N˜ i,j(ξ) = card k =0,...,n 1 (i , i )=(i, j) t − | k k+1 n o for ξ =(i0,...,in; t1,...,tn) as above. Now, let ik if tk t < tk+1 X˜t(ξ)= ≤ ( in if tn t. ≤ ∗ (15) Tt(ξ)ρ0 Tt(ξ) ρ˜t(ξ)= ∗ Tr(Tt(ξ)ρ0Tt(ξ) ) (recall that Tt(ξ) is defined in Equation (9)) and µ˜ =ρ ˜ X˜ X˜ . t t ⊗| tih t| Differentiating Equation (15), one can show that the process (˜µt)t satisfies j j∗ Si µ˜s−Si ˜ i,j d˜µt = (˜µt−) dt + j j∗ µ˜t− dN (t). (16) M Tr(S µ˜t−S ) − Xi6=j i i It is proved in [4] that the processes t i,j i,j ˜ 1 j j∗ (Nt )t and 0 9 rather, the map ρ T (ξ)ρT (ξ)∗) as describing the effect of the trajectory where jumps 7→ t t (up to time t) occur at times t1,. . . ,tn and i0,. . . ,in is the sequence of updated positions: as long as the particle sits at ik V , the evolution of its internal degrees of freedom ∈ tGi is given by the semigroup of contraction (e k )t≥0 and, as the particle jumps to ik+1, it ik+1 undergoes an instantaneous transformation governed by Rik (this Tt(ξ) as the analogue for continuous-time OQW of the operator Lπ of [5, 10]). Therefore, the expression for ρt(ξ) in Equation (15) encodes the effect of the reduction postulate, or postulate of the collapse of the wave function, on the state of a particle initially at i0 and with internal state ρ0. This rigorous connection of the unravelling (9) to (indirect) measurement was first described in [4] (see also [23, 24], as well as [14] for a connection to two-time measurement statistics). To summarize this section and the preceding one, we have defined a Markov process (µt)t as µt = ρt Xt Xt , where Xt V and ρt h , of which the law can be computed ⊗| ih | ∈ ∈ S Xt in two ways: either by the Dyson expansion of the CTOQW as in Equation (10) or by use of the stochastic differential equation (11). 2 Irreducibility of quantum Markov semigroups In this section, we state the equivalence between different notions of irreducibility for general quantum Markov semigroup. Our main motivation is the fact that we could not find a complete proof in the case of an infinite-dimensional Hilbert space, as is required e.g. for CTOQW with infinite V . We then discuss irreducibility for CTOQW. Theorem 2.1. Let := ( ) be a quantum Markov semigroup with Lindbladian T Tt t≥0 (µ)= Gµ + µG∗ + L µ L∗ . (17) L i i Xi∈I The following assertions are equivalent: 1. is positivity improving: for all A ( ) with A 0 and A =0, there exists t> 0 T ∈ I1 K ≥ 6 such that etL(A) > 0. 2. For any ϕ 0 , the set C[ ] ϕ is dense in where C[ ] is the set of polynomials in etG for t>∈ K\{0 and} in L for iL I. K L i ∈ 3. For any ϕ 0 , the set C[G, L] ϕ is dense in where C[G, L] is the set of ∈ K\{ } K polynomials in G and in the L for i I. i ∈ 4. is irreducible, i.e. there exists t > 0 such that t admits no non-trivial projection PT ( ) with P ( )P P ( )P . T ∈ B K Tt I1 K ⊂ I1 K From now on, any quantum Markov semigroup which satisfies any one of the equivalent statements of Theorem 2.1 is simply called irreducible. Remark 2.2. Positivity improving maps are also called primitive. We therefore call primitivity the property of being positivity improving. Remark also that one can replace “there exists t> 0” by “for all t> 0” in assertions 1. and 4. above to get another equivalent formulation of irreducibility and primitivity. This follows from the observation that assertion 3. does not depend of t. 10 Proof. We first prove the equivalence 1. 2. Note that 1. holds if and only if for every ϕ = 0, there exists t > 0 such that ϕ, et⇔L( ϕ ϕ )ϕ > 0 for all ϕ = 0. Now remark that 0 6 0 h | 0ih 0| i 6 from Equation (8), ∞ tL 2 ϕ, e ( ϕ0 ϕ0 )ϕ = ϕ,ζt(ξ)ϕ0 dt1 dtn (18) h | ih | i 0 The projection P being subharmonic for t which is irreducible by assumption, P is trivial. As it is non-zero, P = Id. Since P is theT orthogonal projection on C[ ]ϕ, this shows that C[ ]ϕ is dense in . L L K Remark 2.3. An immediate corollary of Theorem 2.1 is that a quantum Markov semigroup =( ) is irreducible if and only if its adjoint ∗ =( ∗) is irreducible. T Tt t T Tt t We now introduce the notion of irreducibility of a CTOQW, focusing on the trajectorial formulation. Let := ( t)t≥0 be a CTOQW on a set V . For i, j in V and n N, we denote by n(i, j) the setT of continuous-timeT trajectories going from i to j in n jumps:∈ P n(i, j)= ξ =(i ,...,i ; t ,...,t ) Ξ(n) i = i, i = j P { 0 n 1 n ∈ ∞ | 0 n } 11 n and we set (i, j) = N (i, j). For any ξ = (i,...,j; t ,...,t ) in (i, j), we denote by P ∪n∈ P 1 n P R(ξ) the operator from hi to hj defined by j (t −t − )G in−1 (t −t )G i t G R(ξ)= R e n n 1 in−1 R e 2 1 i1 R 1 e 1 i . (20) in−1 in−2 ··· i This is almost the same as the operator Tt(ξ) defined in Equation (9) but here we do not include the evolution between the time tn and t, that is, (t−tn) Gj Tt(ξ)= e R(ξ) . The following proposition is a direct application of Theorem 2.1, and will constitute our definition of irreducibility for continuous-time open quantum walks. The criterion here is equivalent to any other formulation proposed in Theorem 2.1. Proposition 2.4. The CTOQW defined by the quantum Markov semigroup := ( ) is T Tt t≥0 irreducible if and only if, for every i and j in V and for any ϕ in hi 0 , the set R(ξ) ϕ, ξ (i, j) is total in h . \{ } { ∈ P } j The proposition below gives a sufficient condition for irreducibility of a quantum Markov ∗ semigroup. We recall (see [27]) that a map on 1( ) of the form µ i LiµLi is called C I K 7→ irreducible if for any ϕ0 = 0, the set [L]ϕ0 is dense in (this is simplyP a discrete-time analogue of the present notion6 of irreducibility). K Proposition 2.5. Let := ( ) be a quantum Markov semigroup with Lindbladian T Tt t≥0 (µ)= Gµ + µG∗ + L µ L∗ . L i i Xi If Φ(µ)= L µ L∗ is irreducible, then := ( ) is irreducible as well. i i i T Tt t≥0 Proof. ThisP is obvious from Lemma 3.3 of [27] and the third characterization of irreducibility in Theorem 2.1. One can compare the present notion of irreducibility with the usual one for classical Markov chain. Actually if Q =(qi,j)i,j∈V denotes the generator of a continuous-time Markov chain, then the irreducibility of the chain depends on the transitions (q , i = j). In particular i,j 6 the Markov chain is irreducible if and only if for all i, j there exists a path i0 = i, i1,...,in = j such that q . . . q − > 0. Unfortunately in the case of CTOQW only one implication i0,i1 × × in 1,in is valid since one can find := ( t)t≥0 irreducible where the corresponding Φ is not irreducible (see the example below).T T Example 2.6. We focus on an example on = C2 1 + C2 2 , where the Lindbladian is H ⊗| i ⊗| i defined by (7) with: 1 1 2 2 1 0 1 G1 = G2 = − , R = R = . 2 2 1 1 2 1 0 − − ! ! One can easily check that R(ξ) ϕ, ξ (i, j) = hj for all i, j 1, 2 and ϕ hi 0 but { 2 ∈ P1 } ∈{ } ∈ \{ } the discrete OQW defined by R1 and R2 is not irreducible. This is an example of CTOQW where etL is irreducible even though Φ is not. 12 3 Transience and recurrence of irreducible CTOQW In the classical theory of Markov chains on a finite or countable graph, an irreducible Markov chain can be either transient or recurrent. Transience and recurrence issues are central to the study of Markov chains and help describe the Markov chain’s overall structure. In the case of CTOQW, transience and recurrence notions are made more complicated by the fact that the process (Xt)t alone is not a Markov chain. In the present section, we define the notion of recurrence and transience of a vertex in our setup and prove a dichotomy similar to the classical case, based on the average occupation time at a vertex. However, compared to the classical case, the relationship between the occupation time and the first passage time at the vertex is less straightforward. Recall that the first passage time at a given vertex i V is defined as ∈ τ = inf t T X = i i { ≥ 1| t } where T1 is defined in Section 1.3. Similarly the occupation time is given by ∞ 1 ni = Xt=i dt . Z0 In the discrete-time and irreducible case (Theorem 3.1. of [5]), the authors prove that there exists a trichotomy rather than the classical dichotomy. We state a similar result for continuous-time semifinite open quantum walks (we recall that an OQW is semifinite if dim h < for all i V ). i ∞ ∈ Theorem 3.1. Consider a semifinite irreducible continuous-time open quantum walk. Then we are in one (and only one) of the following situations: 1. For any i, j in V and ρ in , one has E (n )= and P (τ < )=1. Shi i,ρ j ∞ i,ρ j ∞ 2. For any i, j in V and ρ in , one has E (n ) < and P (τ < ) < 1. Shi i,ρ j ∞ i,ρ i ∞ 3. For any i, j in V and ρ in , one has E (n ) < , but there exist i in V and ρ, ρ′ Shi i,ρ j ∞ in (ρ necessarily non-faithful) such that P (τ < )=1 and P ′ (τ < ) < 1. Shi i,ρ i ∞ i,ρ i ∞ Note that in the sequel we only focus on semifinite case. Recall that when hi is one-dimensional for all i V , we recover classical continuous-time Markov chains. In this case, the Markov chain falls∈ in one of the first two categories of this theorem; that is, the third category is a specifically quantum situation. The rest of this section is dedicated to the proof of Theorem 3.1. More precisely, in Subsection 3.1 we prove the dichotomy between infinite and finite average occupation time. This allows us to define transience and recurrence of CTOQW. We also give examples of CTOQW that fall in each of the three classes of Theorem 3.1. In Section 3.2 we state technical results that give closed expressions to the occupation time and the first passage time. Finally, the proof of Theorem 3.1 is given in Subsection 3.3. 3.1 Definition of recurrence and transience We begin by proving that for an irreducible CTOQW, the average occupation time Ei,ρ(nj) of site j starting from site i is either finite for all i, j or infinite for all i, j. 13 Proposition 3.2. Consider a semifinite irreducible continuous-time open quantum walk. Suppose furthermore that there exist i0, j0 V and ρ0 h such that Ei ,ρ (nj ) = . ∈ ∈ S i0 0 0 0 ∞ Then, for all i, j V and ρ one has E (n )= . ∈ ∈ Shi i,ρ j ∞ Proof. Fix i, j V and ρ . Then one has ∈ ∈ Shi ∞ ∞ tL Ei,ρ(nj)= Pi,ρ(Xt = j) dt = Tr e (ρ i i )(I j j ) dt . 0 0 ⊗| ih | ⊗| ih | Z Z By hypothesis, ( t)t≥0 is irreducible and thus positivity improving by Theorem 2.1; by T ∗ Remark 2.3 the same is true of ( t )t≥0. Therefore, since for any i V , hi is finite-dimensional, for any s> 0 there exist scalarsT α,β > 0 such that ∈ ∗ esL(ρ i i ) α ρ i i and esL (I j j ) β I j j . ⊗| ih | ≥ 0 ⊗| 0ih 0| ⊗| ih | ≥ ⊗| 0ih 0| We then have, fixing s> 0, ∞ (t−2s)L sL sL∗ Ei,ρ(nj) Tr e e (ρ i i ) e (I j j ) dt ≥ 2s ⊗| ih | ⊗| ih | Z ∞ uL αβ Tr e (ρ0 i0 i0 )(I j0 j0 ) du ≥ 0 ⊗| ih | ⊗| ih | Z αβ E (n ) . ≥ i0,ρ0 j0 This concludes the proof. The above proposition leads to a natural definition of recurrent and transient vertices of V . Definition 3.3. For any continuous-time open quantum walk, we say that a vertex i in V is: recurrent if for any ρ , E (n )= ; • ∈ Shi i,ρ i ∞ transient if there exists ρ such that E (n ) < . • ∈ Shi i,ρ i ∞ Thus, by Proposition 3.2, for an irreducible CTOQW, either all vertices are recurrent or all vertices are transient. Furthermore, in the transient case, E (n ) < for all ρ in . i,ρ i ∞ Shi We conclude this section by illustrating Theorem 3.1 by simple examples. The n-th example corresponds to the n-th situation in Theorem 3.1. Example 3.4. 1. For V = 0, 1 and h = h = C, consider the CTOQW characterized by the following { } 0 1 operators: 1 G = G = , R1 = R0 =1 . 0 1 −2 0 1 Then the process (Xt)t≥0 is a classical continuous Markov chain on 0, 1 , where the walker jumps from one site to the other after an exponential time of{ parameter} 1. 2. For V = Z and hi = C for all i Z, consider the CTOQW described by the transition operators: ∈ 1 √3 1 G = , Ri+1 = , Ri−1 = , for all i Z . i −2 i 2 i 2 ∈ The process (Xt)t≥0 is a classical continuous Markov chain on Z where after an 3 exponential time of parameter 1, the walker jumps to the right with probability 4 1 or to the left with probability 4 . 14 3. Consider the CTOQW defined by V = N with h = C2 and h = h = C for i 2, and 1 0 i ≥ 1 1 1 2 1 0 2 G0 = , R0 = , G1 = I2 , R1 = 0 1 , R1 = 1 0 , −2 √5 1! −2 1 1 1 1 i+1 √3 i−1 1 R2 = , Gi = , Ri = for i 2 and Ri = for i 3 . 2√2 1! −2 2 ≥ 2 ≥ 0 0 This is an example of positivity improving CTOQW where, for ρ = , one has 0 1! P (τ < )=1 1,ρ 1 ∞ but P ′ (τ < ) < 1 i,ρ i ∞ 0 0 for any ρ′ = . This example therefore exhibits “specifically quantum” behavior. 6 0 1! This example is inspired from [5]. 3.2 Technical results Proposition 3.5 below is essential, as it expresses the probability of reaching a site in finite time as the trace of the initial state, evolved by a certain operator. Proposition 3.5. For any continuous-time open quantum walk, there exists a completely positive linear operator P from (h ) to (h ) such that for every i, j V and ρ , i,j I i I j ∈ ∈ Shi P (τ < ) = Tr P (ρ) . i,ρ j ∞ i,j Furthermore, the map Pi,j can be expressed by: ∞ ∗ Pi,j(ρ)= R(ξ) ρ R(ξ) dt1 . . . dtn , n 0 in (tn−t − )G in−1 i n 1 in−1 1 t1Gi0 where we recall that R(ξ) = Rin−1 e Rin−2 ... Ri0 e for ξ = (i0,...,in; t1,...,tn). Note that we do not require the hi to be finite-dimensional here. Proof. We have the trivial identity: ∞ (t−tn)L ∗ Pi,ρ(τj < t)= Tr e R(ξ) ρ R(ξ) j j dt1 . . . dtn . 0 Then, since e(t−tn)L is trace preserving, ∞ ∗ Pi,ρ(τj < t)= Tr R(ξ) ρ R(ξ) dt1 . . . dtn , n 0 15 and since both sides of the identity are nondecreasing in t, taking the limit t + yields → ∞ ∞ ∗ Pi,ρ(τj < )= Tr R(ξ) ρ R(ξ) dt1 . . . dtn . ∞ n 0 It remains to show that Pi,j is well defined. Let us denote by (Vn)n∈N an increasing sequence of subsets of V such that Vn = min(n, V ) and n∈N Vn = V . For any X (hi) 0 write the canonical decomposition| |X = X |X | + iX iX of X as a linear combination∈ I \{ } of four 1 − 2 3S− 4 nonnegative operators. We get N ∗ Tr R(ξ) X R(ξ) dt1 . . . dtn 0 Consequently, by the Banach-Steinhaus Theorem, the operator on (h ) to (h ) defined by I i I j ∞ ∗ Pi,j(X)= R(ξ) X R(ξ) dt1 dtn 0 Corollary 3.6. For every i, j V and ρ , we have ∈ ∈ Shi ∞ E (n )= Tr Pk P (ρ) . (22) i,ρ j j,j ◦ i,j kX=0 16 Proof. Let i, j V and ρ . Then ∈ ∈ Shi ∞ ∞ tL Ei,ρ(nj)= Pi,ρ(Xt = j) dt = Tr e (ρ i i )(Id j j ) dt 0 0 ⊗| ih | ⊗| ih | ∞ Zn Z ∗ = Tr ΥρΥ dt1 dtn dt , 0 in (tn−tn− )G in−1 im +1 (tm −tm )Gi im i 1 in−1 1 1+1 1 m1 1 1 t1Gi0 where Υ = R − e R − R e R − R e . The above in 1 in 2 im1 im1 1 i0 expression corresponds to a decomposition··· of any path from i to j as a··· concatenation of a path from i to j and k paths from j to j which do not go through j. This yields Equation (22). The next corollary allows us to link the quantity P (τ < ) to the adjoint of the i,ρ j ∞ operator Pi,j. In particular, as we shall see, it is a first step towards linking the properties of P (τ < ) and E (n ). i,ρ j ∞ i,ρ j Corollary 3.7. Let i and j be in V . One has Id 0 P (τ < )=1 P∗ (Id) = in the decomposition h = Ran ρ (Ran ρ)⊥ . i,ρ j ∞ ⇐⇒ i,j 0 i ⊕ ∗! P P ′ In particular, if there exists a faithful ρ in hi such that i,ρ(τj < )=1, then i,ρ (τj < )=1 for any ρ′ in . S ∞ ∞ Shi P ∗ P Proof. By Proposition 3.5, one has i,ρ(τj < ) = Tr(ρ Pi,j(Id)). Therefore, if i,ρ(τj < ) = 1, then P∗ (Id) has the following form in∞ the decomposition h = Ran ρ (Ran ρ)⊥: ∞ i,j i ⊕ ∗ Id A Pi,j(Id) = . A B! Besides, the fact that Id P∗ (Id) forces A to be null. In particular, if ρ is faithful, then ≥ i,j P∗ (Id) = Id and therefore P ′ (τ < )=1 for any ρ′ in . i,j i,ρ j ∞ Shi 3.3 Proof of Theorem 3.1 Let i and j be in V . As we can see in Corollary 3.7, if we suppose that Pi,ρ(τj < )=1 ∗ ∞ for a faithful density matrix ρ, we necessary have Pi,j(Id) = Id. This will be used in the following proposition, which in turn explains the statement regarding non-faithfulness in the third category of Theorem 3.1. Proposition 3.8. Let i be in V . If there exists a faithful ρ in such that P (τ < )=1, Shi i,ρ i ∞ then one has E ′ (n )= for any ρ′ in . i,ρ i ∞ Shi i (n) Proof. We set τ1 = τi and, for all n> 1, we define τi as the time at which (Xt)t≥0 reaches i for the n-th time: τ (n) = inf t > τ i X = i and X = i . i { n−1| t t− 6 } P ′ ′ From Corollary 3.7, one has i,ρ (τi < ) = 1 for all ρ in hi . This implies that for all (n) ∞ S P ′ ′ i n> 0, τi is i,ρ -almost finite for any ρ hi . For n 0, let Tn be the occupation time in i i ∈ S ≥ i between τn and τn+1: i Tn = inf u Xτ (n)+u = i { | i 6 } 17 (0) with the convention that τi = 0. Since we have i i E ′ E ′ E i,ρ (ni) i,ρ Tn inf i,ρˆ(Tn), ≥ ≥ ρˆ∈Shi nX≥1 nX≥1 E i it will be enough to obtain a lower bound for i,ρˆ(Tn) which is uniform in n and inρ ˆ. To P i this end, we use the quantum trajectories defined in (11). We first compute i,ρˆ(Tn > t) for all t 0. To treat the case of n = 1 we consider the solution of ≥ t ρˆ ρˆ ρˆ ∗ ρˆ ρˆ ρˆ ∗ ηt =ρ ˆ + Gi ηs + ηs Gi ηs Tr(Gi ηs + ηs Gi ) ds . (23) 0 − Z Using the independence of the Poisson processes N i,j involved in (11) we get P i P i,ρˆ(T1 > t)= i,ρˆ(no jump has occurred before time t) ∗ = P N i,j u,y 0 u t, 0 y Tr(RjηρˆRj ) =0 j = i i,ρˆ { | ≤ ≤ ≤ ≤ i u i } ∀ 6 ∗ = P N i,j u,y 0 u t, 0 y Tr(RjηρˆRj ) =0 i,ρˆ { | ≤ ≤ ≤ ≤ i u i } jY6=i t j ρˆ j∗ = exp Tr(Ri ηs Ri ) ds − 0 jY6=i n Z o t ∗ ρˆ = exp Tr (Gi + Gi ) ηs ds (24) 0 Z where we used relation (6). Similarly, using the strong Markov property, i P E 1 i i,ρˆ(Tn > t)= i,ρˆ( Tn>t) E E i = i,ρˆ i,ρ i (1T >t) τn 1 t ρ i E ∗ τn = i,ρˆ exp Tr (Gi + Gi ) ηs ds 0 ∗ Z t G G ∞ e− k i+ i k . ≥ E i ∞ P i Now, using the fact that i,ρˆ(Tn) = 0 i,ρˆ(Tn > t) dt, this gives us the expected lower bound: R 1 E (T i ) . i,ρˆ n ≥ G + G∗ k i i k∞ This concludes the proof. The next proposition is connected to the first point of Theorem 3.1. Proposition 3.9. Consider a semifinite irreducible continuous-time open quantum walk. If E P ′ there exist i, j in V and ρ hi such that i,ρ(nj)= , then one has j,ρ (τj < )=1 for any ρ′ in . ∈ S ∞ ∞ Shj Proof. By Proposition 2.4, there is no nontrivial invariant subspace of hj left invariant by R(ξ) for all ξ (j, j). Since any such ξ is a concatenation of paths from j to j that ∈ P remain in V j except for their start- and endpoints, there is also no nontrivial invariant \{ } subspace of hj left invariant by the operator Pj,j of Proposition 3.5. The latter is therefore a completely positive irreducible map acting on the set of trace-class operators on hj. By the Russo-Dye Theorem (see [26]), one has P = P∗ (Id) 1, so that the spectral radius k j,jk k j,j k ≤ 18 λ of Pj,j satisfies λ 1. By the Perron-Frobenius Theorem of Evans and Hoegh-Krøhn ≤ ′ (see [16] or alternatively Theorem 3.1 in [27]), there exists a faithful density matrix ρ on hj ′ ′ such that Pj,j(ρ ) = λρ . If λ < 1, then by Corollary 3.6 one has Ej,ρ′ (nj) < , but then Proposition 3.2 contradicts our running assumption that E (n )= . Therefore∞ λ = 1 and i,ρ j ∞ ρ′ is a faithful density matrix such that P ′ (τ < ) = Tr P (ρ′) = Tr(ρ′) = 1. We then j,ρ j ∞ j,j conclude by Corollary 3.7. Proposition 3.10. Consider a semifinite irreducible continuous-time open quantum walk; if there exists i V such that for all ρ′ one has P ′ (τ < )=1, then P (τ < )=1 ∈ ∈ Shi i,ρ i ∞ i,ρ j ∞ for any j V and ρ . ∈ ∈ Shi Proof. Fix i and j in V . Observe first that, by irreducibility, for any ρ in , there exists Shi ξ =(i = i0, i1,...,in−1, in = j; t1,...,tn) ∗ such that Tr R(ξ)ρR(ξ) > 0. We denote by t(ξ) the element tn of ξ. Using the continuity of Tr R(ξ)ρR(ξ)∗ in ρ and the compactness of , we obtain a finite family ξ ,...,ξ , of Shi 1 p paths, again going from i to j, such that ∗ inf max Tr R(ξk)ρR(ξk) > 0 . ρ∈Sh k=1,...,p i ∗ Let δ > maxk=1,...,p t(ξk). By continuity of each Tr R(ξi)ρR(ξi) in the underlying jump times t1,...,tn and using expression (21), we have α := inf Pi,ρ(τj δ) > 0 . ρ∈Shi ≤ Now, if P (τ < ) = 1 for all ρ in , then the discussion in Section 1.3 imply that i,ρ i ∞ Shi almost-surely one can find an increasing sequence (tn)n of times with tn and xtn = i. Choose a subsequence (t ) such that t t > δ for all n. Since→ ∞ never reaching ϕ(n) n ϕ(n) − ϕ(n−1) j means in particular not reaching j between tϕ(n) and tϕ(n+1) for n = 1,...,k, the Markov property of (X , ρ ) and the lower bound t t > δ imply that for all ρ , t t t≥0 ϕ(n) − ϕ(n−1) ∈ Shi P (τ = ) P 0 n k , t [t , t ] , X = j i,ρ j ∞ ≤ i,ρ ∀ ≤ ≤ ∀ ∈ ϕ(n) ϕ(n+1) t 6 k sup Pi,ρ(τj > δ) ≤ ρ∈h i (1 α)k . ≤ − Since the above is true for all k, we have P (τ < )=1. i,ρ j ∞ Now we combine all the results of Section 3.2 to prove Theorem 3.1. Proof of Theorem 3.1. Proposition 3.2 shows that either E (n ) = for all i, j and ρ, or i,ρ j ∞ Ei,ρ(nj) < for all i, j and ρ. Proposition 3.9 combined with Proposition 3.10 shows that in the former∞ case, P (τ < )=1 for all i, j and ρ as well. Proposition 3.8 shows that, in i,ρ j ∞ the latter case, Pi,ρ(τj < ) = 1 may only occur for non-faithful ρ, and this concludes the proof. ∞ Acknowledgments The authors are supported by ANR project StoQ ANR-14-CE25-0003-01. 19 References [1] S. Attal, N. 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