Markov chains approximation of jump-diffusion stochastic master equations

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www.imstat.org/aihp 2010, Vol. 46, No. 4, 924–948

DOI:10.1214/09-AIHP330

Markov chains approximation of jump–diffusion stochastic master equations

Clément Pellegrini

Institut C. Jordan, Université C. Bernard, Lyon 1, 21, av Claude Bernard, 69622 Villeurbanne Cedex, France. E-mail:pelleg@math.univ-lyon1.fr Received 28 May 2009; revised 1 July 2009; accepted 16 July 2009

Abstract. Quantum trajectoriesare solutions of stochastic differential equations obtained when describing the random phenomena associated to quantum continuous measurement of open quantum system. These equations, also calledBelavkin equationsor Stochastic Master equations, are usually of two different types: diffusive and of Poisson-type. In this article, we consider more advanced models in which jump–diffusion equations appear. These equations are obtained as a continuous time limit of martingale problems associated to classical Markov chains which describe quantum trajectories in a discrete time model. The results of this article goes much beyond those of [Ann. Probab.36(2008) 2332–2353] and [Existence, uniqueness and approximation for stochastic Schrödinger equation: The Poisson case (2007)]. The probabilistic techniques, used here, are completely different in order to merge these two radically different situations: diffusion and Poisson-type quantum trajectories.

Résumé. Les trajectoires quantiquessont des solutions d’équations différentielles stochastiques décrivant des phénomènes aléa- toires associés aux prinicpes de mesure (quantique) des systèmes quantiques ouverts. Ces équations, également appeléeséquations de Belavkinouéquations maîtresses stochastiquessont habituellement de deux types: soit diffusif soit de type saut. Dans cet article, nous considérons des modèles plus avancés où des équations de type saut-diffusion apparaissent. Ces équations sont obtenues comme solutions de problèmes de martingales. Ces problèmes de martingales sont obtenus comme limites continus (en temps) à partir de chaînes de Markov classiques décrivant des trajectoires quantiques pour des modèles à temps discret. Les résultats de cet article généralisent ceux de [Ann. Probab.36(2008) 2332–2353] et [Existence, uniqueness and approximation for stochastic Schrödinger equation: The Poisson case (2007)]. Ici, les techniques probabilistes utilisés sont complétement différentes afin de pouvoir mixer les deux types d’évolutions: diffusives et poissoniennes.

MSC:60F99; 60G99; 60H10

Keywords:Stochastic master equations; Quantum trajectory; Jump–diffusion stochastic differential equation; Stochastic convergence; Markov generators; Martingale problem

Introduction

In quantum mechanics, an active line of research makes an important use ofQuantum Trajectorytheory with wide applications inQuantum Optics or in Quantum Information theory (cf. [4,6,16,17,22]). A quantum trajectory is a solution of a stochastic differential equation which describes the random evolution of a quantum system undergoing continuous measurement. These equations are calledStochastic Master equationsorBelavkin equations[2–4,10,12, 14].

More precisely, these equations describe situations where the measurement isindirect. The physical setup is the one of an interaction between a small system (atom) and a continuous field (environment). By performing a continuous time quantum measurement on the field, after the interaction, we get a partial information of the evolution of the small system without destroying it. This partial information is governed by stochastic differential equations (Belavkin equations). In the literature, two characteristic equations are usually described as follows:

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1. One is described by a diffusive equation dρt=L(ρ_t)dt+

ρ_tC+Cρ_t−Tr ρ_t

C+C ρ_t

dWt, (1)

whereW_t describes a one-dimensional Brownian motion.

2. The other is given by a stochastic differential equation driven by a counting process dρt=L(ρ_t)dt+

J(ρ_t)

Tr[J(ρ_t)]−ρ_t

dN˜_t−Tr J(ρ_t)

dt

, (2)

whereN˜_tis a counting process with stochastic intensity_t

0Tr[J(ρ_s)]ds.

The solutions of these equations are calledcontinuous quantum trajectories, they are valued in the set of states of the small system (astateordensity matrix is a positive trace class operator with trace one). Such models describe essentially the interaction between a two-level atom and a spin chain [29,30]. More complicated models, with higher degree of liberty, are in general mixing of these two types, that is, they are driven by both a diffusive and a jump process (jump–diffusion model) . . . .

Even in the cases(1)and(2), Belavkin equations pose tedious problems in terms of physical and mathematical justifications. First rigorous results are due to Davies [19] which has described the evolution of a two-level atom undergoing a continuous measurement. Heuristic rules can be used to obtain classical Belavkin equations (1) and (2). There exists different ways to justify these models. In [3,9,11,14],Quantum Filteringtheory is used to obtain stochastic master equations. Such an approach needs high analytic technologies (Von Neumann algebra, conditional expectation in operator algebra) and is based on the quantum stochastic calculus formalism. In [4,5,7,27], jump–diffusion or multi- diffusions models are considered. In these papers, an approach based on classical stochastic differential equations and the notion ofa posteriori stateis used.

Recently, an intuitive approach based on a discrete time model has been used in [29,30] to justify Eqs(1)and(2).

The discrete model is calledQuantum Repeated Measurementsand is based on the setup ofQuantum Repeated Inter- actions. In this context, instead of considering an interaction with a continuous field, the environment is represented as an infinite chain of identical and independent quantum system. Each piece of the chain interacts with the small system during a time interval of lengthτ. After each interaction, a quantum measurement of an observable of the field is performed. From the point of view of the small system, according to the laws of quantum mechanics, each result of an observation gives rise to a random modification of its reference state. The evolution of the state of the small system can be described by classical Markov chains called discrete quantum trajectories. In [29,30], the discrete quantum trajectories are then expressed as solutions of discrete stochastic differential equations which are approximations of Eqs (1)and(2). Next, by using techniques related to convergence of stochastic differential equations, it has been shown that the solutions of(1)and(2)can be obtained as continuous limit from discrete quantum trajectories (when τ goes to zero). However, the techniques used in [29] and [30] are very different and incompatible (for(1)an abstract result of Kurtz and Protter [28] is used whereas for(2)a random coupling method and a comparison with an Euler scheme is employed). Therefore, such methods can not be adapted in situations of mixing of Brownian evolution and jump evolution.

In this article, in order to prove a similar convergence result for jump–diffusion stochastic master equations, we adopt an approach based onMarkov Chain Approximationtheory. We proceed as follows. As the discrete quantum trajectories are Markov chains, we can naturally define their discrete Markov generators which depend on the time parameterτ. The limit,τ goes to zero, of these generators gives rise to infinitesimal generators. These one are then naturally linked with generalmartingale problemsin probability theory [23,25]. Next, we show that such martingale problems are solved by solution of particular jump–diffusion stochastic differential equations, which model continuous time measurement theory. Finally, this approach and these models are physically justified by proving that the solutions of these SDEs can be obtained as continuous time limits (in distribution) of discrete quantum trajectories.

This way, for finite-dimensional systems, we get a general description of jump–diffusion stochastic master equations.

This article is organized as follows.

Section1is devoted to the description of the discrete model of quantum repeated measurements. We remind the presentation of discrete quantum trajectories developed in [29]. Next, we shall focus on the dependence onτ for these Markov chains and we introduce asymptotic assumptions in order to come into the question of convergence.

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In Section2, we present the Markov chain approximation technique. We compute the Markov generators of discrete quantum trajectories and we investigate the limits of these one. Therefore, we obtain infinitesimal generators linked to general martingale problems. We show how to solve these problems in terms of jump–diffusion stochastic differential equations.

Finally in Section3, we show that discrete quantum trajectories converge in distribution to the solutions of stochastic differentials equations described in Section2. This way, the stochastic models of jump–diffusion equations in continuous quantum measurement theory are then justified as limit models of the concrete physical procedure of quantum repeated measurements.

1. Discrete quantum trajectories

1.1. Quantum repeated measurements

In this section, we remind the model of quantum repeated measurements and the description of discrete quantum trajectories in terms of Markov chains [29]. As it is mentioned in theIntroduction, the physical setup is the one a small system in contact with an infinite chain of quantum systems [1] (each piece of the chain is supposed to be independent and identical).

Mathematically, the small system is represented by a finite-dimensional Hilbert spaceH0equipped with an initial stateρ0. The set of states ofH0is denoted byS. The chain is represented by the countable tensor productT Φ =

k≥1Hk, whereHk=H, for allk(His also a finite-dimensional Hilbert space and each copy is equipped with the same reference stateβ). The model of quantum repeated measurements is described as follows. Each copyHk ofH interacts, one after the other, withH0during a time τ. After each interaction, a measurement is performed on the piece of the chain which has just interacted. The sequence of measurements involves random modifications of the state ofH0. These modifications are described by a random sequence of states ofH0denoted by(ρ_k)which is called adiscrete quantum trajectory. Here, we remind the definition and the Markov property of the sequence(ρ_k)(see [29]

for a complete details).

In order to define the sequence(ρ_k), we present the description of the first interaction and the first measurement (this allows namely to define the transitions of(ρ_k)in terms of Markov chain). A single interaction betweenH0and a copy ofHis described by a total HamiltonianH_totacting on the coupling systemH0⊗H. Its general form is given by

H_tot=H₀⊗I+I⊗H+H_int,

where the operatorsH₀andHare the free Hamiltonian of each system. The operatorH_intrepresents the Hamiltonian of interaction. This defines the unitary-operator

U=e^{iτ H}^tot.

The evolution of states ofH0⊗H, in the Schrödinger picture, is given by ρ→UρU.

After this first interaction, a second copy ofHinteracts withH0in the same fashion and so on. Usually the whole procedure is described by the state spaceΓ =H0⊗T Φ and a sequence of unitary operators(U_k). More precisely, the operatorU_k describes thekth interaction betweenH0andHk, it acts asUonH0⊗Hkand it acts as the identity operator on the other copies ofH. Hence, the result of thekfirst interactions is described by the operators(Vk)where V_k=U_kU_k₋₁· · ·U₁, for allk >0 (this is the usual setup of quantum repeated interactions [1]).

We do not need all the details of the setup of quantum repeated interactions in order to describe the discrete quantum trajectory(ρ_k). We just need to describe the transition probabilities which state the Markov character. To this end, we describe the procedure of measurement after the first interaction. Before the interaction the initial state onH0⊗His ρ₀⊗β and after the first interaction, the state onH0⊗His given byμ₁=U (ρ₀⊗β)U.

Now, we shall describe the measurement procedure of an observable ofH. LetAbe any observable onH, with spectral decompositionA=p

j=0λjPj. According to the laws of quantum mechanics, the result of the measurement

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of A is random. In an explicit way, the accessible data are its eigenvalues, and the observation of λ_j obeys the probability law

P[to observeλ_j] =Tr[μ₁P_j], j∈ {0, . . . , p}.

Besides, if we have observed the eigenvalue λj, the wave packet reductionprinciple imposes that the state after measurement becomes

˜

ρ₁(j )=I⊗Pjμ1I⊗Pj

Tr[μ₁I⊗P_j] .

As a consequence, depending on the result of the observation, this defines the new reference state of the system H0⊗H.

In our context, we are only interested in the reduced state of the small systemH0. This state is given by taking a partial trace onH0(see the below definition–theorem). IfHis any Hilbert space, we denote by Tr_H[W]the trace of a trace-class operatorW onH.

Definition–Theorem 1. LetHandK be two Hilbert spaces.Ifαis a state on a tensor productH⊗K,then there exists a unique stateηonHwhich is characterized by the property

Tr_H[ηX] =Tr_H⊗K

α(X⊗I )

for allX∈B(H).This unique stateηis called the partial trace ofαonHwith respect toK.

Letαbe a state onΓ, we denote byE0(α)the partial trace ofαonH0with respect toH. Hence, we define ρ₁(j )=E0

ρ˜₁(j )

(3) for allj =0, . . . , p.The stateρ1(·)is a random state which is the new reference state ofH0after the first interaction and the first measurement. More precisely, each stateρ1(j )appears with probability Tr[μ1I⊗Pj]. This describes the transitions betweenρ₀and the possible statesρ₁(j ),j=0, . . . , p. Now, we can consider a second copy of(H, β) which interacts with(H0, ρ₁)and after the same procedure, we get a new random stateρ₂onH0. Recursively, we describe a random sequence(ρ_k)whose the Markov property is expressed in the following theorem.

Theorem 1. There exists a probability space(Ω,F, P )such that the quantum trajectory(ρk)is a Markov chain with values in the set of states onH0.Ifρ_k=χ_k,thenρ_k₊₁takes one of the values

E0

(I⊗Pi)U (χk⊗β)U(I⊗Pi) Tr[U (χ_k⊗β)U(I⊗P_i)]

, i=0, . . . , p, with probabilityTr[U (χk⊗β)U(I⊗Pi)].

As the operator U depends explicitly on the time parameter τ, it is worth noticing that the Markov chain(ρ_k) depends onτ. In the following, we putτ=1/n. In this way, we write the unitary operatorU (n)(with dependence inn) and we define

ρn(t)=ρ_[nt] (4)

for allt >0. This defines a sequence of processes(ρ_n(t))and we aim to show next that this sequence of processes converges in distribution, whenngoes to infinity. As announced in theIntroduction, such a convergence is obtained from the convergence of generators of Markov chains. The following section is then devoted to present these generators for quantum trajectories.

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1.2. Discrete Markov generators

LetAbe an observable and let(ρn(t))be the process defined from the quantum trajectory describing the successive measurements ofA. In this section, we investigate the explicit computation of the Markov generatorAnof the process (ρ_n(t))(we will make no distinctions between the infinitesimal generators of the Markov chains(ρ_k)and the process (ρ_n(t))generated by this Markov chain). For instance, let us introduce some notations.

We work withH0=C^K⁺¹. The set of operators onH0is identified withR^P forP =2(K+1)²(we do not need to give any particular identification). We setE=R^P and the set of statesS becomes then a compact subset ofE, since a state is a positive operator with trace 1. For all statesρ∈S, we define

L⁽ⁿ⁾_i (ρ)=E0

(I⊗P_i)U (n)(ρ⊗β)U(n)(I⊗P_i) Tr[U (n)(ρ⊗β)U(n)(I⊗P_i)]

,

(5) pⁱ(ρ)=Tr

U (n)(ρ⊗β)U(n)I⊗P_i

, i=0, . . . , p.

Here, we suppose implicitly thatL⁽ⁿ⁾_i (ρ)=0 ifpⁱ(ρ)=0. The operatorsL⁽ⁿ⁾_i (ρ)represent the transition states of the Markov chains described in Theorem1; the termspⁱ(ρ)are the associated probabilities. The Markov generators of (ρ_n(t))are then expressed as follows.

Definition 1. Let(ρ_n(t)) be the process obtained from the repeated measurements of an observableAof the form A=p

i=0λ_iP_i.Let us defineP⁽ⁿ⁾the probability measure which satisfies P⁽ⁿ⁾

ρ_n(0)=ρ₀

=1, (6)

P⁽ⁿ⁾

ρ_n(s)=ρ_k, k/n≤s < (k+1)/n

=1, (7)

P⁽ⁿ⁾

ρ_k₊₁∈Γ/M⁽ⁿ⁾_k

=Π_n(ρ_k,Γ), (8)

whereΠn(ρ,·)is the transition function of the Markov chain(ρk)given by Π_n(ρ,Γ)=

p i=0

pⁱ(ρ)δ_L(n)

i (ρ)(Γ) (9)

for all Borel subsetsΓ ∈B(R^P)andM⁽ⁿ⁾_k =σ{ρ_i, i≤k}.

For all statesρ∈Sand all functionsf ∈C_c²(E)(i.e.C²with compact support),we define Anf (ρ)=n f (μ)−f (ρ)

Πn(ρ,dμ)

=n p

i=0

f

L⁽ⁿ⁾_i (ρ)

−f (ρ)

pⁱ(ρ). (10)

The operatorAnis called theMarkov generatorof the Markov chain(ρk)(or of the process(ρn(t))).

The complete description of generatorsAnneeds the explicit expression ofL⁽ⁿ⁾_i (ρ), for allρand alli∈ {0, . . . , p}.

To this end, we need to compute the partial trace operationE0on the tensor productH0⊗H. A judicious choice of basis for the tensor product allows to simplify the computations. Let(Ω0, . . . , ΩK)be any orthonormal basis ofH0

and(X₀, . . . , X_N)a one ofH. For the tensor product, we choose the basis

B=(Ω₀⊗X₀, . . . , Ω_K⊗X₀, Ω₀⊗X₁, . . . , Ω_K⊗X₁, . . . , Ω₀⊗X_N, . . . , Ω_K⊗X_N).

In this basis, any(N+1)(K+1)×(N+1)(K+1)matrixMonH0⊗Hcan be written by blocks as a(N+1)× (N+1)matrixM=(M_ij)0≤i,j≤N, whereM_ij are operators onH0. The following easy result justifies the choice of this basis.

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Proposition 1. LetW be a state acting onH0⊗H.IfW =(W_ij)₀_≤_i,j_≤_N,is the expression ofW in the basisB, where the coefficientsWij are operators onH0,then the partial trace with respect toHis given by the formula

E0[W] = N i=0

W_ii.

Now, we are in the position to compute L⁽ⁿ⁾_i (ρ), for all statesρ and alli∈ {0, . . . , p}. We choose the reference stateβofHto be the orthogonal projector onCX₀, that is, with physical notationsβ= |X₀ X₀|.This state is called the ground state (or vacuum state) in quantum physics. From general result of G.N.S representation inCalgebra, it is worth noticing that it is not a restriction. Indeed, such a representation allows to identify any quantum system(H, β) with another system of the form(K,|X₀ X₀|), whereX₀is the first vector of an orthonormal basis of a particular Hilbert spaceK(see [26] for details).

The unitary operatorU (n)is described by blocks asU (n)=(U_ij(n))₀_≤_i,j_≤_N, where the coefficientsU_ij are(K+ 1)×(K +1)matrices acting onH0. Fori∈ {0, . . . , p}, we denoteP_i =(p_klⁱ )₀_≤_k,l_≤_N, the eigen-projectors of the observable A. Hence, the non-normalized states E0[I ⊗PiU (n)(ρ⊗β)U (n)I ⊗Pi]and the probabilities pⁱ(ρ) satisfy

E0

I⊗P_iU (n)(ρ⊗β)U (n)I⊗P_i

=

0≤k,l≤N

p_klⁱ U_k0(n)ρU_l0(n),

(11) pⁱ(ρ)=

0≤k,l≤N

pⁱ_klTr

U_k0(n)ρU_l0(n) .

By observing thatL⁽ⁿ⁾_i (ρ)=E0[I ⊗P_iU (n)(ρ⊗β)U (n)I⊗P_i]/pⁱ(ρ), for alli∈ {0, . . . , p}, we get the explicit expression of the generatorAn. The next step is to consider the limit ofAn, whenngoes to infinity. Such limits need appropriate asymptotic assumptions for the coefficientsUij(n). This is the topic of the following section.

1.3. Asymptotic assumptions

The choice of the asymptotic expression ofU (n)=(Uij(n))are based on the works of Attal–Pautrat in [1]. They have namely shown that the operator process defined for allt >0 by

V_[_nt_]=U_[_nt_](n)· · ·U₁(n),

which describes the quantum repeated interactions, strongly converges (in operator sense) to a process(V˜_t)satisfying a Quantum Langevin equation. Moreover, this convergence is non-trivial, only if the coefficientsUij(n)are scaled appropriately. When translated in our context, this expresses that there exist operatorsL_ij such that we have for all (i, j )∈ {0, . . . , N}²

nlim→∞n^ε^ij

U_ij(n)−δ_ijI

=L_ij, (12)

whereε_ij=¹₂(δ0i+δ0j). As the expression(11)ofL⁽ⁿ⁾_i (ρ)only involves the first column ofU (n), we only keep the following asymptotic expressions

U₀₀(n)=I+1 nL₀₀+o

1 n

and U_i0(n)= 1

√nL_i0+o 1

√n

fori >0.

Another fact, which will be important in the computation of limit generators, is the following result (cf. [1]).

Proposition 2. In order thatU=(U_ij)is an unitary-operator,there exists a self-adjoint operatorH₀such that L00= −

iH0+1

2 N i=1

L_i0Li0

.

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Furthermore we have for allρ∈S Tr

L₀₀ρ+ρL₀₀+

1≤k≤N

L_k0ρL_k0

=0,

becauseTr[U (n)ρU(n)] =1,for alln.

We can now apply these considerations to give the asymptotic expression of non-normalized states and probabilities given by the expression(11). For the non-normalized states, we have

E0

I⊗PiU (n)(ρ⊗β)U (n)I⊗Pi

=pⁱ₀₀ρ+ 1

√n

1≤k≤N

p_k0ⁱ L_k0ρ+p_0kⁱ ρL_k0

+1 n

p₀₀ⁱ

L₀₀ρ+ρL₀₀

+

1≤k,l≤N

p_klⁱ L_k0ρL_l0

+o 1

n

. (13)

Moreover, aspⁱ(ρ)=Tr[E0[I⊗P_iU (n)(ρ⊗β)U (n)I⊗P_i]], we get pⁱ(ρ)=pⁱ₀₀+ 1

√nTr

1≤k≤N

pⁱ_k0Lk0ρ+pⁱ_0kρL_k0

+1 nTr

pⁱ₀₀

L00ρ+ρL₀₀

+

1≤k,l≤N

pⁱ_klLk0ρL_l0

+o 1

n

. (14)

The asymptotic expression ofL⁽ⁿ⁾_i (ρ)then follows from(13)and(14). Depending on the fact thatpⁱ₀₀is equal to zero or not, we consider three cases:

1. Ifpⁱ₀₀=0, thenp_0kⁱ =0, for allk >0 and we have L⁽ⁿ⁾_i (ρ)=

1≤k,l≤Np_klⁱ L_k0ρL_l0+o(1) Tr[

1≤k,l≤Np_klⁱ L_k0ρL_l0+o(1)]1_Tr_[

1≤k,l≤Np_klⁱL_k0ρL_l0]=0+o(1). (15) 1. Ifpⁱ₀₀=1, then we have

L⁽ⁿ⁾_i (ρ)=ρ+1 n

L00ρ+ρL₀₀

+

1≤k,l≤N

pⁱ_klLk0ρL_l0

−1 nTr

L00ρ+ρL₀₀

+

1≤k,l≤N

p_klⁱ Lk0ρL_l0

ρ+o 1

n

. (16)

1. Ifpⁱ₀₀∈ {/ 0,1}, then we have

L⁽ⁿ⁾_i (ρ)=ρ+ 1

√n 1

p₀₀ⁱ

1≤k≤N

pⁱ_k0L_k0ρ+p_0kⁱ ρL_k0

− 1 pⁱ₀₀Tr

1≤k≤N

×ρ

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+1 n

1 p₀₀ⁱ

pⁱ₀₀

L00ρ+ρL₀₀

+

1≤k,l≤N

p_klⁱ Lk0ρL_l0

+ 1 (pⁱ₀₀)²Tr

1≤k≤N

p_k0ⁱ L_k0ρ+p_0kⁱ ρL_k0²

×ρ

− 1 p₀₀ⁱ Tr

p₀₀ⁱ

L₀₀ρ+ρL₀₀

+

1≤k,l≤N

p_klⁱ L_k0ρL_l0

×ρ

− 1 (pⁱ₀₀)²

1≤k≤N

p_k0ⁱ L_k0ρ+pⁱ_0kρL_k0

×Tr

1≤k≤N

+o 1

n

. (17)

It is worth noticing that all the o are uniform inρsinceSis a compact set.

The next section is dedicated to the convergence ofAnand the presentation of the continuous models.

2. Jump–diffusion models of quantum measurement

In this section, we show that the limit (n→ ∞) of generatorsAngives rise to explicit infinitesimal generators. We interpret these generators as Markov generators of continuous time processes associated with martingale problems.

Besides, we show that these processes are solution of jump–diffusion stochastic differential equations which are a generalization of the Belavkin equations(1)and(2)presented in theIntroduction.

In our framework, the notion of martingale problem is expressed as follows (see [18,20,23,25] for complete refer- ences). We still consider the identification of the set of states as a compact subset ofE=R^P. LetΠ be a transition kernel onE, leta(·)=(aij(·))be a measurable mapping onEwith values in the set of positive semi-definite symmet- ricP ×P matrices and letb(·)=(b_i(·))be a measurable function fromEtoE. Letf be anyC²_c(E)and letρ∈E.

In this article, we consider infinitesimal generatorsAof the form Af (ρ)=

P i=1

bi(ρ)∂f (ρ)

∂ρ_i +1 2

P i,j=1

aij(ρ)∂f (ρ)

∂ρ_i∂ρ_j

+

E

f (ρ+μ)−f (ρ)− P i=1

μi

∂f (ρ)

∂ρ_i

Π (ρ,dμ). (18)

The notion of martingale problem associated with such generators is expressed as follows.

Definition 2. Letρ₀∈E.We say that a measurable stochastic process(ρ_t)on some probability space(Ω,F, P )is a solution of the martingale problem for(A, ρ₀),if for allf ∈C_c²(E),

M^ft =f (ρ_t)−f (ρ₀)− t

0

Af (ρ_s)ds, t≥0, (19)

is a martingale with respect toFt^ρ=σ (ρ_s, s≤t ).

It is worth noticing that we must also define a probability space(Ω,F, P )to make explicit a solution of a problem of martingale.

In the following section, we show that the Markov generators of discrete quantum trajectories converges to infinitesimal generators of the form(18).

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2.1. Limit infinitesimal generators

Before expressing the theorem which gives the limit infinitesimal generators ofAn, we define particular functions onSwhich appear in the limit. For alliand all statesρ∈S, set

g_i(ρ)= ¹^≤^k,l^≤^Np_klⁱ L_k0ρL_l0 Tr[

1≤k,l≤Np_klⁱ L_k0ρL_l0]−ρ

,

v_i(ρ)=Tr

1≤k,l≤N

pⁱ_klL_k0ρL_l0

,

(20) h_i(ρ)= 1

p₀₀ⁱ 1≤k≤N

−Tr

1≤k≤N

p_k0ⁱ L_k0ρ+pⁱ_0kρL_k0 ρ

,

L(ρ)=L00ρ+ρL₀₀+

1≤k≤N

Lk0ρL_k0.

The next theorem concerning the limit generators follows from Eqs (15)–(17) described in Section1. In this theorem, the termD_ρf denotes the first differential inρof a functionf∈C_c²(E)andD_ρ²f the second differential.

Theorem 2. LetAbe an observable with spectral decompositionA=p

i=0λ_iP_i,whereP_i=(p_klⁱ )0≤k,l≤N are its eigen-projectors.Up to permutation of eigen-projectors,we can suppose thatp⁰₀₀=0.We define the sets

I=

i∈ {1, . . . , p}/pⁱ₀₀=0

and J= {1, . . . , p} \I.

Let(ρ_n^J(t))be the corresponding quantum trajectory obtained from the measurement ofAand letA^Jn be its infinites- imal generator(cf.Definition1).The limit generatorsA^J ofA^J_n exist and are described as follows:

1. IfI= {1, . . . , p},that is,p⁰₀₀=1andJ=∅,we have for allf∈C_c²(E)

nlim→∞sup

ρ∈S

A^Jnf (ρ)−A^Jf (ρ)=0, (21)

whereA^J satisfies A^Jf (ρ)=D_ρf

L(ρ) +

E

f ρ+μ

−f (ρ)−D_ρf (μ)

Π (ρ,dμ), (22)

the transition kernelΠ being defined asΠ (ρ,dμ)=p

i=1vi(ρ)δg_i(ρ)(dμ).

2. IfI= {1, . . . , p},that is,p⁰₀₀=1andJ=∅,we have for allf∈C_c²(E)

nlim→∞sup

ρ∈S

A^Jnf (ρ)−A^Jf (ρ)=0, (23)

whereA^J satisfies A^Jf (ρ)=Dρf

L(ρ) +1

2

i∈J∪{0}

D_ρ²f

hi(ρ), hi(ρ)

(24) +

E

f (ρ+μ)−f (ρ)−D_ρf (μ)

Π (ρ,dμ), the transition kernelΠ being defined asΠ (ρ,dμ)=

i∈Ivi(ρ)δg_i(ρ)(dμ).

(10)

Proof. Recall thatS is the set of states and it is a compact subset ofE. For anyi∈ {0, . . . , p}and for anyρ∈S, let us compute limn→∞A^Jnf (ρ). To this aim, we use the asymptotic results of Section1(Eqs (15)–(17)). There are three different types of limit to consider:

1. Supposepⁱ₀₀=0, we have,

nlim→∞n f

L⁽ⁿ⁾_i (ρ)

−f (ρ) pⁱ(ρ)

=

f ¹^≤^k,l^≤^Np_klⁱ L_k0ρL_l0 Tr[

1≤k,l≤Np_klⁱ L_k0ρL_l0]

−f (ρ)

Tr

1≤k,l≤N

p_klⁱ L_k0ρL_l0

for allρ∈S. This defines a uniformly continuous function onSsincef ∈C_c²andSis compact. As a consequence, the asymptotic concerning this case (and the fact that all the o are uniform onScf. Section1) implies the uniform convergence.

2. Supposepⁱ₀₀=1, by using the Taylor formula of order one, we have

nlim→∞n f

L⁽ⁿ⁾_i (ρ)

−f (ρ) pⁱ(ρ)

=Dρf

L00ρ+ρL₀₀

+

1≤k,l≤N

pⁱ_klLk0ρL_l0

−Tr

L₀₀ρ+ρL₀₀

+

1≤k,l≤N

pⁱ_klL_k0ρL_l0

ρ

.

To obtain the uniform result, we use the asymptotic expressions of Section1and the uniform continuity ofDf onS.

3. Supposepⁱ₀₀∈ {0,/ 1}. By applying the Taylor formula of order two, we get the convergence

i/p₀₀ⁱ ∈{/0,1} nlim→∞n

f

L⁽ⁿ⁾_i (ρ)

−f (ρ) pⁱ(ρ)

=

i/p₀₀ⁱ ∈{/0,1}

D_ρf

p₀₀ⁱ

L₀₀ρ+ρL₀₀

+

1≤k,l≤N

p_klⁱ L_k0ρL_l0

−Tr

p₀₀ⁱ

L₀₀ρ+ρL₀₀

+

1≤k,l≤N

p_klⁱ L_k0ρL_l0

ρ

+ 1 2pⁱ₀₀D²_ρf

1≤k≤N

−Tr

1≤k≤N

pⁱ_k0L_k0ρ+p_0kⁱ ρL_k0 ρ

+

1≤k≤N

pⁱ_k0L_k0ρ+pⁱ_0kρL_k0

−Tr

1≤k≤N

p_k0ⁱ L_k0ρ+pⁱ_0kρL_k0 ρ

. (25)

This last equality needs further explanation. In the Taylor formula, terms of the form G_i(ρ)= 1

√nD_ρf

1≤k≤N

−Tr

1≤k≤N

pⁱ_k0L_k0ρ+p_0kⁱ ρL_k0 ρ

appear for each i such that pⁱ₀₀ ∈ {0,/ 1}. We have

i/p₀₀ⁱ ∈{/0,1}G_i(ρ) = 0, since

i/pⁱ₀₀∈{/0,1}p_k0ⁱ =

i/p₀₀ⁱ ∈{/0,1}pⁱ_0k =p

i=0p_0k=p

i=0p_k0=0, for all k >0 (indeed we have p

i=0P_i =I). Furthermore, this convergence is uniform for the same reasons as previously.

(11)

The different cases of the theorem follows from these three limits. The first case of Proposition2follows from the first two limits described above, the second case follows from the first and the third limits above. Before to describe this in details, we have to notice that

p i=0

1≤k,l≤N

pⁱ_klL_k0ρL_l0=

1≤k≤N

L_k0ρL_k0

sincep

i=0Pi=I. Moreover, we have Tr[L(ρ)] =Tr[L00ρ+ρL₀₀+

1≤k≤NLk0ρL_k0] =0 (see Proposition2).

By using these facts, in the casep₀₀ⁱ =0, the limit can be written as

E

f (ρ+μ)−f (ρ)−Dρf (μ)

vi(ρ)δg_i(ρ)(dμ)+Dρf gi(ρ)

vi(ρ).

Besides, we have Dρf

gi(ρ)

vi(ρ)=Dρf

1≤k,l≤N

p_klⁱ Lk0ρL_l0−Tr

1≤k,l≤N

pⁱ_klLk0ρL_l0

ρ

.

Hence, it implies the first case of Theorem2, that is forI= {1, . . . , p}, we get indeed A^J_f(ρ)=D_ρf

L(ρ) +

E

f (ρ+μ)−f (ρ)−D_ρf (μ)

Π (ρ,dμ).

A similar reasoning gives the expression of the infinitesimal generator in the second case, whereI = {1, . . . , p}

and the proposition is proved.

It is worth noticing that the generatorsA^J are generators of type(18), it suffices to expand the differential terms D_ρf andD_ρ²f in terms of partial derivatives _∂ρ^∂f

j and _∂ρ^∂²^f

i∂ρj. In the next section, we present the continuous time stochastic models.

2.2. Solutions of martingale problems

In all this section, we consider an observableAwith spectral decomposition

A=

i∈I

λ_iP_i+

j∈J∪0

λ_jP_j, (26)

whereI andJ are the subsets of{1, . . . , p}defined in Theorem2. LetA^J be the associated limit generator and let ρ0be a state. In order to solve the martingale problem for(A^J, ρ0), by Definition2, we have to define a probability space(Ω,F, P )and a stochastic process(ρ_t^J)such that the process

M^ft =f ρ_t^J

−f (ρ₀)− _t

0

A^Jf ρ_s^J

ds (27)

is a martingale for the natural filtration of(ρ_t^J). In a usual way, such martingale problems are solved in terms of stochastic differential equations [18,21].

Concerning the suitable probability space, let us consider(Ω,F, P )a probability space which supports a(p+1)- dimensional Brownian motion W =(W₀, . . . , W_p) andp independent Poisson point processes(N_i)₁_≤_i_≤_p on R², independent of the Brownian motion.

As there are two types of limit generators in Theorem2, we define two types of stochastic differential equations in the following way. Letρ0be an initial deterministic state: