Central limit theorem for reducible and irreducible open quantum walks

In this work we aim at proving central limit theorems for open quantum walks on $\mathbb{Z}^d$. We study the case when there are various classes of vertices in the network. Furthermore, we investigate two ways of distributing the vertex classes in the network. First we assign the classes in a regular pattern. Secondly, we assign each vertex a random class with a uniform distribution. For each way of distributing vertex classes, we obtain an appropriate central limit theorem, illustrated by numerical examples. These theorems may have application in the study of complex systems in quantum biology and dissipative quantum computation.

In this work we analyze the asymptotic behavior of open quantum walks. Especially we consider the possibility to determine the time limit properties of walks with non-homogeneous structure. The theorems for the homogeneous case are proven [43]. In this work we consider two different approaches: the possibility to reduce the walk to the homogeneous one and provide walk's asymptotic properties as it is. In the first case, we construct a set of rules and methods that allows to determine when it is possible to reduce a walk. In the second case, we state a new central limit theorem that allows us to derive asymptotic distribution under certain conditions. We illustrate this approaches with appropriate numerical examples.

Quantum states and channels
Definition 1 We call an operator ρ ∈ L(X ) for some Hilbert space X a density operator iff ρ ≥ 0 and Trρ = 1. We denote the set of all density operators on X by Ω(X ).
Definition 2 A superoperator Φ is a linear mapping acting on linear operators L(X ) on a finite dimensional Hilbert space X and transforming them into operators on another finite dimensional Hilbert space Y i. e. Φ : L(X ) → L(Y). (1)

Definition 3 Given superoperators
we define the product superoperator to be the unique linear mapping that satisfies: for all operators A 1 ∈ L(X 1 ), A 2 ∈ L(X 2 ). The extension for operators not in the tensor product form follows from linearity.
Definition 4 A quantum channel is a superoperator Φ : L(X ) → L(Y) that satisfies the following restrictions: 1. Φ is trace-preserving, i.e. ∀A ∈ L(X ) Tr(Φ(A)) = Tr(A), 2. Φ is completely positive, that is for every finite-dimensional Hilbert space Z the product of Φ and an identity mapping on L(Z) is a nonnegativity preserving operation, i.e.
Note that quantum channels map density operators to density operators.

Definition 5
The Kraus representation of a quantum channel Φ : L(X ) → L(Y) is given by a set of operators K i ∈ L(X , Y). The action of the superoperator Φ on A ∈ L(X ) is given by: with the restriction that Definition 6 Given a superoperator Φ : Note, that the conjugate to a completely positive superoperator is completely positive, but is not necessarily trace-preserving.

Open quantum walks
The model of the open quantum walk was introduced by Attal et al. [1] (see also [44]).
To introduce the open quantum walk (OQW) model, we consider a random walk on a graph with the set of vertices V and directed edges {(i, j) : i, j ∈ V }. The dynamics on the graph is described in the space of states V = C V with an orthonormal basis {|i } i∈V . We model an internal degree of freedom of the walker by attaching a Hilbert space X to each vertex of the graph. Thus, the state of the quantum walker is described by an element of the space Ω(X ⊗ V).
To describe the dynamics of the quantum walk, for each directed edge (i, j) we introduce a set of operators {K ijk ∈ L(X )}. These operators describe the change in the internal degree of freedom of the walker due to the transition from vertex j to vertex i. Choosing the operators K ijk such that we get a Kraus representation of a quantum channel for each vertex j ∈ V of the graph. As the operators K ijk act only on X , we introduce the operators where|i , |j ∈ V which perform the transition from vertex j to vertex i and internal state evolution. It is straightforward to check that ijk M † ijk M ijk = 1l X ⊗V .

Asymptotic behavior of open quantum walks
Recently Attal et al. [43] provided a description of asymptotic behavior of open quantum walks in the case when the behavior of every vertex is the same i. e. all vertices belong to one class. We call such networks homogeneous.
In order to describe asymptotic properties of an open quantum walk we will use the notion of quantum trajectory process associated with this open quantum walk.

Definition 8
We define the quantum trajectory process as a classical Markov chain assigned to an open quantum walk constructed as a simulation of the walk with measurement at each step. The initial state is (ρ 0 , X 0 ) ∈ Ω(X )×Z d with probability 1. The state (ρ n , X n ) at step n evolves into one of the 2d states corresponding to possible directions ∆ j , j = ±1, . . . , ±d: with probability p(j) = Tr(K j ρK † j ). We also define a transition operator for a Markov chain (ρ, ∆X) associated with this trajectory process We define an auxiliary channel Φ : L(X ) → L(X ) that mimics the behavior of the walk when all the internal states are the same as We assume that the channel has a unique invariant state ρ ∞ ∈ Ω(X ). Additionally we define a vector that approximates the estimated asymptotic transition for the channel Φ: where |j ∈ R d and for j > d we put |j = −|j − d .
Let us recall the theorem by Attal et al. [43].

Theorem 1
Consider a quantum open walk on Z d associated with transition operators {K 1 , . . . , K 2d }. We assume that a channel Φ admits a unique invariant state. Let (ρ n , X n ) n≥0 be the quantum trajectory process associated with this open quantum walk, then and probability distribution of normalized random variable X n converges in law to the Gaussian distribution in R d .

Results
We are mainly interested in open quantum walks that are defined on networks with many classes of vertices. In this paper we assume that there is a finite number of vertex classes Γ = {C 1 , . . . , C n }. The transitions in each vertex is given by Kraus operators defined for each class separately {K We define a transition operator of the Markov chain as in Eq. (13): Next, we define a channel Φ C for each class C as in Eq. (14) Again, we assume that Φ C has a unique invariant state ρ C ∞ ∈ Ω(X ). Additionally for each class C we define a vector as in Eq. (15) where |j ∈ R d and for j > d we put |j = −|j − d .
In order to provide a description of distribution evolution of open quantum walks on non-homogeneous networks we analyze two cases. First in Section 3.1, we model a walk with vertices defined in such a way that it is possible to reduce the network to the homogeneous case. Secondly in Section 3.2, we study a network which is irreducible in the above sense but satisfies some basic properties that allow us to develop other techniques.

Reducible open quantum walks
Let us consider an open quantum walk with several classes of vertices. We aim to analyze the possibility to construct a new walk that that behaves the same way in the asymptotic limit.

Definition 9
We call an open quantum walk reducible if there is a class A that for some integer l each l-step path from a vertex of type A always leads to a vertex of type A.
When considering a reducible OQW we can consider these paths as edges and reduce the network to the homogeneous case.
Definition 10 For a reducible quantum walk with N possible paths we construct a new set of Kraus operators {K R 1 , .., K R N } ⊂ L(X ) such that each operator is a composition of all the operators corresponding to the consecutive steps composing one of the paths from vertex A to another vertex A, i. e. for a path q consisting of vertices X 1 , . . . , X l and direction changes ∆ 1 , . . . , ∆ l the corresponding operator is We call the OQW based on these operators a reduced open quantum walk.
The simplest example of a reducible open quantum walk is a walk on Z 2 presented in Fig. 1. Starting in a vertex of class A, after two steps we always end up in a vertex of class A. We use that property to construct a new walk with only one vertex type and exactly the same asymptotic behavior. In Fig. 2 we present a more complex example of a network with these properties.

Central limit theorem and its proof
Theorem 2 Consider a reducible quantum open walk on Z d . By P we denote the abstract class of vertices constructed as described in Definition 10. We assume that a channel constructed with these paths Φ P has a unique invariant state ρ ∞ ∈ Ω(X ) with average transition vector |m P . Let (ρ n , X n ) n≥0 be the quantum trajectory process associated to this open quantum walk, then and probability distribution of normalized random variable X n converges in law to the Gaussian distribution in R d . Proof. We apply the Theorem 1 to the reduced OQW as in Def. 10. As all the path's lengths are equal and describe all possible paths starting from a vertex of type A we have that N q=1 K R † q K R q = N q=1 K q 1 . . . K q l † K q 1 . . . K q l = 1l X . Thus the new walk satisfies assumptions of the Theorem 1. One step of this walk corresponds exactly to l steps of the original walk. The one-to-one correspondence assures that the asymptotic behavior is the same.

Example
We show the application of Theorem 2 by considering a walk on a network presented in the Fig. 1. The Kraus operators for vertices of type A are defined as follows: Figure 2: An example of a 2D reducible OQW. The arrows show possible transitions. Each path from one vertex of type A leads to another vertex of this type with exactly 4 steps.
The operators for vertices of type B are: In our example we set α = 0.81. The behavior of this particular walk is presented in Fig. 3. As expected, after a sufficiently large number of steps, the distribution is Gaussian and moves towards the left and down.

Irreducible OQWs
The assumptions introduced in Theorem 2 allow us to analyze some nonhomogeneous OQW, but the class of such walks is still very limited. In this section we aim to provide a way to determine asymptotic behavior of less restricted family of OQWs.

Theorem and proof
Lets consider an OQW on a network composed with several types of vertices on an infinite lattice. The main assumption of the following theorem is that the distribution of vertices is regular over the lattice. In other words, the probability of finding a vertex of particular type is transition invariant.

Definition 11
A regular network is a network where density of every vertex class C ∈ Γ is transition invariant. In other words for any ∈ R there is fixed neighborhood size (ball radius) δ ∈ R that for any two neighborhoods the distance of vertex class probability distribution is bounded by . Thus, when restricted to any finite neighborhood of the considered hypercube, there is fixed probability p C of finding a vertex of class C in any random direction from any randomly chosen vertex. 2d } X∈Z d ⊂ L(X ) we construct for each class of vertices C ∈ Γ a quantum channel Φ C as in Eq. (19) with a unique invariant state ρ C ∞ ∈ Ω(X ) and an average position vector |m = C∈Γ p C |m C , where |m C is obtained from Eq. (20) and p C from Def. 11. Let (ρ n , X n ) n≥0 be the quantum trajectory process associated with this open quantum walk, then and probability distribution of normalized random variable X n converges in law to the Gaussian distribution.
Before we prove Theorem 3, let us introduce three technical lemmas. Lemma 2 Given a channel Φ C corresponding to vertex class C with associated Kraus {K C 1 . . . K C 2d } ⊂ L(X ) which has a unique invariant state ρ ∞ , for every l ∈ R d there exists L C l ∈ L(X ) such that

Lemma 1 For every superoperator
Proof. First we compute m C |l . We get Next, we move all the terms to one side of the equation and write all terms under the trace where we multiplied m C |l by ρ C ∞ 1l X . Finally, we use the fact that trace is cyclic and linear and get: Thus we obtain that the term under the bracket in Eq. (31) is orthogonal to ρ C ∞ and as it is the only invariant state of Φ C we get that ker(1l L(X ) − Φ C ) = ρ C ∞ . Then, from Lemma 1, the states orthogonal to the kernel are in the image of the conjugated superoperator, hence we get: (32) Hence, we have shown that L C l exists.

Lemma 3
For each class C and a vector l ∈ R d a function given by the explicit formula where P C is given by Eq. (18).
Proof. We apply the P C operator as defined in Eq. 18. Let us note that . Applying the definition of P C to (34) we get: where Id is an identity operation on the space of functions R d → R. Now, using Lemma 2 we get which completes the proof. Proof of Theorem 3. For a random variable X n we expand the formula F l = X n |l − n m|l : Recall that C∈Γ p C = 1, m = C∈Γ p C m C and we denote X k |− X k−1 | = ∆X k |, we get: From Lemma 3 we get (|X − |m C )|l = (Id − P C )f C (ρ, |X ) for some ρ ∈ Ω(X ), hence: After rearranging the sum in the formula for F l we get: Now we consider M n and R n separately. First we discuss M n : We notice that M n is a centered martingale i.e.
where ∆M n = M n − M n−1 and F denotes filtering for stochastic process M n [45,46]. This follows from the definition of P C . Additionally |∆M n | is bounded from above i.e. |∆M n | < M max as ∆M n includes terms corresponding to one step of the walk.
In the case of R n we have: From the definition of f C we notice that R n is bounded as the first two terms are constant and the last one P C f C (ρ, |∆X n ) = T r(ρL C l ) + ∆X n |l is clearly bounded, hence|R n | < R max and R n does not influence the asymptotic behavior.
Now it suffices to show that the following two equalities hold (for proof see Theorem 3.2 and Corollary 3.1 in [46]): and to obtain that M n / √ n converges in distribution to N (0, σ 2 ), where introduces restricted expectation values. We prove Eq. (45) using the fact that |∆M k | is bounded, hence the sum in Eq. (45) terminates for some N ∈ N.
In order to prove the equality in Eq. (46) we expand (∆M k ) 2 : where ∆M C k = (Tr(ρ k L C l ) − Tr(ρ k−1 L C l ) + ( ∆X k | − m C |)|l ). Next, we expand the product We divide this expression into three terms ∆M C k ∆M C k = T (1,k) C,C . Henceforth, we will drop indexes C, C , k when unambiguous. The term T (1) is equal to: We compute E(T (1) |F k−1 ) by adding the term ±Tr(ρ k L C l )Tr(ρ k L C l ) we obtain a sum of two terms that can be interpreted as an increment part of a martingale and an increment part of a sum respectively. Thus after a summation over k both terms are bounded and we get the equality The term T (2) is given by: We note that E(∆M k |F k−1 ) = 0. Thus after summation over C and C we get the the expectation value of the whole term T (2),k C,C : We will calculate the term T (3) using the definition of the expectation value. We write the probability of |∆X being equal to |j and ρ k being . This can be expressed in a nice trace form: where c(k − 1) is the class of X k−1 . Thus we can define Ξ After summation over C and C the value is equal to: where Ξ c(k−1) = C,C ∈Γ p C p C ΞC, C c(k−1) . By the ergodic theorem (Th. 4.2 in [43]) this converges to: with Ξ = c p c Ξ c . Finally, after summing of all of the terms we get: which completes the proof.

Example
As an example of a walk consistent with description in Section 3.2.1 we consider a walk with the same vertex types as in the reducible case, that is: and Although, in this case we assign the type to a vertex randomly with a uniform distribution. The channels formed from Kraus operators A x and B x where x ∈ U, R, L, D both have a unique invariant state. The behavior of the network is presented in the Fig. 4. We obtain a similar behavior as in the reducible case, although the convergence to a Gaussian distribution is slower.

Conclusions
The aim of this paper was to provide formulas describing the behavior of the open quantum walk in the asymptotic limit. We described two cases: networks that are reducible to the 1-type case and networks with random, uniformely distributed vertex types. This result allows one to analyze behavior of walks with a more complex structure compared to the known results. We have illustrated our claims with numerical examples that show possible applications and correctness of our theorems. The networks are still restricted to vertices that exhibits invariant states.
We provided examples showing that the theorems are valid in the case of a 2D regular lattice with two vertex types. In Section 3.1.2 we shown application to the reducible case, when the assignment of vertex types is regular and translation invariant. Next, in Section 3.2.2 we turned to a random, uniformly distributed assignment of vertex types.
These theorems can also be applied to the non-lattice graphs. Different types of vertices allow also to apply this in the case of graphs with nonconstant degrees. This may be very useful in modeling complex structures, especially of regular definition as in the case of Apollonian networks. These