Central Limit Theorem and Large Deviation Principle for Continuous Time Open Quantum Walks
Open Quantum Walks (OQWs), originally introduced in , are quantum generalizations of classical Markov chains. Recently, natural continuous time models of OQW have been developed in . These models, called Continuous Time Open Quantum Walks (CTOQWs), appear as natural continuous time limits of discrete time OQWs. In particular they are quantum extensions of continuous time Markov chains. This article is devoted to the study of homogeneous CTOQW on . We focus namely on their associated quantum trajectories which allow us to prove a Central Limit Theorem for the "position" of the walker as well as a Large Deviation Principle.
Open Quantum Walks concern evolution on lattices driven by quantum operations. They describe Markovian dynamics influenced by internal degrees of freedom. They have been introduced originally by  (see also ). These OQWs are promising tools to model physical problems, especially in computer science (see ). They can also model a variety of phenomena, as energy transfer in biological systems ().
Continuous time models have been developed as a natural continuous time limit of discrete time models [24, 5]. In particular in , a natural extension of Brownian motion called Open Quantum Brownian Motion has been constructed. In this article, we focus on the continuous time open quantum walks (CTOQWs) model presented in . More precisely, we focus on CTOQWs on . Briefly speaking, CTOQWs on concern the evolution of density operators of the form
where the "-component" represents the "position" of the walker and is a Hilbert space describing the internal degrees of freedom. In particular, if denotes the set of density operators of the form (1), CTOQWs are described by a semigroup such that, preserves for all .
In the context of quantum walks, one is mainly interested in the position of the walker. At time , starting with density matrix in as (1), the quantum measurement of the "position" gives rise to a probability distribution on , such that, for all ,
As well, after evolution, if
In , it has been shown that usual classical continuous time Markov chains are particular cases of CTOQWs. In particular one can easily construct models where the distribution corresponds to the one of a classical continuous time Markov chain. Contrary to continuous time Markov chains, the distribution of CTOQWs cannot be in general recovered by the knowledge of the initial distribution . One needs to have access to the full knowledge of the initial state . In this sense, this justifies the name quantum walks.
Our models of continuous time quantum walks are rather different from the usual models of unitary quantum walks. An essential difference concerns the large time behaviour of the corresponding distribution . Let be a random variable of law , in the unitary quantum walk theory it has been shown that satisfies a Central Limit Theorem of the type
where has distribution
Note that such behaviour is not usual in classical probability where usually one expects speed in and Gaussian law as limit in the Central Limit Theorem (CLT). In our context, the distributions express a rather classical behaviour in large time in the sense that a more usual CLT holds. In particular this paper is devoted to show that for CTOQWs one has the following weak convergence
where denotes usual Gaussian law. Such phenomena have also been observed in the discrete setting of OQWs . A key point to show this result is the use of the quantum trajectories associated to the CTOQWs. In general, quantum trajectories describe evolutions of quantum system undergoing indirect measurements (see  for an introduction). In the context of CTOQWs, quantum trajectories describe the evolution of the states undergoing indirect measurements of the position of the walker. In particular these quantum trajectories appear as solution of jump-type stochastic differential equations called stochastic master equations (see  for link between discrete and continuous time models in the context of OQW, one can also consult  for such an approach in the context of Open Quantum Brownian Motion). In the physic literature, note that such models appear also naturally in order to describe non-Markovian evolutions. They are called non-Markov generalization of Lindblad equations (see [6, 25, 4]).
After establishing the CLT, our next goal is to investigate a Large Deviation Principle (LDP) for the position of the walker. In particular under additional assumptions, one can apply the Gärtner-Ellis Theorem in order to obtain the final result (one can consult  for a similar result for discrete time OQWs).
The article is structured as follows. In Section 2, we present the model of CTOQWs on . Next we develop the theory of quantum trajectories which describe the continuous measurement of the position. In Section 3, we present the Central Limit Theorem. Section 4 is devoted to the Large Deviation Principle (LDP). Finally in Section 5, we present some examples which illustrate the CLT and the LDP.
2 Continuous Time Open Quantum Walks
2.1 Main setup
The models of Continuous Time Open Quantum Walks have been formalized in . They arise as continuous limits of discrete time OQWs (we do not recall the discrete time models and we refer to ). These limits processes are described by particular types of Lindblad master equations. Originally, these equations appear in the "non-Markovian generalization of Lindblad theory" from Breuer . In this article, we focus on nearest neighbors, homogeneous CTOQWs on .
In the sequel, denotes a finite dimensional Hilbert space and denotes the space of density matrix on :
We put where stands for the position of a particle while corresponds to the internal degree of freedom of this particle. We consider the canonical basis of , we set and for all . The canonical basis of is denoted by .
As announced we focus on particular diagonal density matrices of :
In the sequel we shall consider evolutions on which preserve . To this end we consider a family of operators on and we define the operators on such that
Now as announced the CTOQWs are generated by particular Lindblad master equations. Let the following Lindblad operator on ,
is a self-adjoint operator on which is called the Hamiltonian.
Let us introduce the operator
The next computation shows that preserves the set .
for all .
The following proposition describes precisely our model of CTOQWs.
 Let , the equation
with initial condition admits a unique solution with values in .
More precisely, is of the form such that:
for all .
The evolution (2) is called a Continuous Time Open Quantum Walk on .
This definition is justified by the following. The operator transcribes the idea that the particle localized in can only jump to one of its nearest neighbors , and in this case, the transformation on is governed by . In the case the particle stands still, the evolution on is governed by . It is the exact analogue of the usual OQWs for continuous time evolutions. An interesting fact has been pointed out in , usual continuous time classical Markov chains can be realized within this setup.
Now let us describe the probability distributions associated to CTOQWs.
Let . Let be the solution of the equation
and we denote the random variable on of law , that is
for all .
As we can see in Section 3 and as it was announced in the introduction, the shape of seems to converge to Gaussian shape. This is exactly the result pointed out by the CLT in Section 3. In order to prove this, we shall need the theory of quantum trajectories for CTOQWs.
2.2 Quantum trajectories
As in the discrete case, quantum trajectories are essential tools for showing the CLT and the LDP. The description of quantum trajectories is less straightforward than the one in OQWs. It makes use of stochastic differential equations driven by jump processes. We refer to  for the justification of the below description and the link between discrete and continuous time models. One can also consult  where general indirect measurements for non-markovian generalization of Lindblad equations have been developped.
Let be an initial state on . The quantum trajectory describing the indirect measurement of the position of the CTOQWs led by is modeled by a Markov process . This Markov process is valued in the set
and such that the following differential equation is satisfied:
where are independent Poisson point processes on .
In particular the Markov process is valued in and satisfies
and with probability .
Remark: The second expression of the description of quantum trajectories is the exact continuous time analogue of the one described in  for OQWs. Let us briefly explain how the quantum trajectories evolve in time. To this end we introduce:
The processes are Poisson processes with intensity . In particular the processes
are martingales with respect to the filtration induced by . The evolution described by (2.2.1) is deterministic and interrupted by jumps occurring at random time, it is typically a Piecewise Deterministic Markov Process. The jumps are generated by the Poisson processes (7). As we can check from Eq. (2.2.1), if for some and (that is ), the deterministic evolution let the position unchanged until a jump occurs. Since the Poisson processes are indepedent, only one Poisson process is involved. If denotes the time of the first jump and assume the process is involved, the internal degree of freedom is updated by and the position is changed and becomes . This means that the particle has jumped from the position to the position . In other words we have , for all and . Next, the deterministic evolution starts again with the new initial condition until a new jump occur and so on.
The following result allows us to make the connection between CTOQWs and their associated quantum trajectories.
Moreover, for all , the random variables and have the same distributions .
The first part is proved in . For the second part, let a bounded continuous map on , we get:
and the result holds. ∎
In the next section, we state the CLT.
3 Central Limit Theorem
This section is devoted to prove the Central Limit Theorem for CTOQWs. The result holds under some assumption concerning the Lindblad operator on . This operator is defined below.
Our main assumption for the CLT is the following.
There exists a unique density matrix such that
In particular .
Under the condition , we have the following ergodic theorem which is a particular case of the Ergodic Theorem of . In particular this theorem shall be useful in the proof of the CLT.
Now, our strategy to show the CLT consists in reducing the problem to a CLT for martingales with the help of the solution of the Poisson equation. To this end let us introduce the generator of the process .
We denote the Markov generator of the process and its domain. For all , and , we get
where for all and where denotes the partial differential of with respect to .
Remark: Note that in the sequel we do not need to make precise the exact domain of . Actually we shall apply the Markov generator on functions.
We shall also need the following quantity,
The following lemma shall be used in the proof.
For all , the equation
admits a solution and the difference between any couple of solutions of (9) is a multiple of the identity.
First, let us remark that
which implies that . But by hypothesis, we have . Moreover, since , we finally get that
which proves the existence of the lemma. Now we prove the second part. To this end consider and two solutions of (9) and set . It is then clear that
Therefore . Since , we get and since , the operator is necessarily a multiple of the identity. ∎
From now on, for , we denote the unique solution of (9) such that . Moreover, if , then we simply write . Using the linearity of , one can notice that:
The next lemma concerns the Poisson equation in our context (see  for more details on the Poisson equation).
For all and , let set
Then is solution of the Poisson equation:
For all and , we complete the following computation:
so is solution of the Poisson equation (11). ∎
Now we have found the solution of the Poisson equation, we express the CLT for martingales that we shall use.
Theorem 3.0.4 ().
Let be a real, càdlàg, and square integrable martingale. Suppose the following conditions:
for some , then
We shall also use the following lemma which is a straightforward consequence of the law of large numbers for martingales (see ).
Let a real, càdlàg, and square integrable martingale which satisfies for a constant , then
The last lemma below shall be useful in this part as well as in the next one. From now on, we denote the Euclidean norm of .
For all and all , we have
Let and ,
Since are independent Poisson point processes on , we get
In the same way, one can prove that
Now, we are in the position to state the main result of this section.
Assume holds. Let the Markov process defined in Proposition 2.2.1 then
where such that for all ,
Remark: Proposition 2.2.2 implies then the CLT for the process as it holds for .
As announced, the proof is a combination of Lemma 3.0.3 and Theorem 3.0.4. Let and the function defined in Lemma 3.0.3. Since is the generator of , following the theory of problem of martingale, the process defined by
is a local martingale with respect to the filtration associated to (see [26, 13] for more details on problem of martingale). In order to apply Theorem 3.0.4, we shall show that is a true martingale. To this end it is sufficient to show that (see  for more details). This way, since for all , one can check with the help of Lemma 3.0.6 that
Now we shall see that fulfills the conditions of Theorem 3.0.4. The first one is the easiest one. Indeed,
Now we shall make the martingales appear in the first and the last term of the above expression. Concerning the second term, we recognize and to get