Abstract
We study continuous-time open quantum walks in one dimension through a matrix representation, focusing on nearest-neighbor transitions for which an associated weight matrix exists. Statistics such as site recurrence are studied in terms of matrix-valued orthogonal polynomials and explicit calculations are obtained for classes of Lindblad generators that model quantum versions of birth-death processes. Emphasis is given to the technical distinction between the cases of a finite or infinite number of vertices. Recent results for open quantum walks are adapted in order to apply the folding trick to continuous-time birth-death chains on the integers. Finally, we investigate the matrix-valued Stieltjes transform associated to the weights.
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1 Introduction
Random walks have been a fundamental concept in the study of stochastic processes and probability theory for many decades [1, 21, 26]. In the field of quantum mechanics, the concept of quantum walks has emerged as a robust tool for exploring quantum systems’ behavior and dynamics [4, 25, 27]. Quantum walks can be categorized into various types, and one particularly intriguing category is open quantum walks (OQWs) [4]. This process introduces the influence of the environment, which leads to a richer set of dynamics and behaviors when compared to the classical random walks induced by Markov chains, making them an exciting area of research in quantum information and quantum computation. Continuous-time open quantum walks (CTOQWs) [24] represent a specific class of OQWs where the evolution of a quantum walker in a graph is continuous and influenced by an initial quantum state.
Inspired by the classical birth–death processes (BDPs), this article develops a generalization from the perspective of CTOQWs, exploring promising valuable insights into the behavior of quantum walkers in systems that exhibit birth and death processes. Introducing a matrix representation for the generator of a CTOQW, we apply the theory of matrix orthogonal polynomials to tridiagonal block matrices. The matrix orthogonal polynomial approach provides a powerful framework for analyzing the representation of those generators, enabling us to gain a deeper understanding of CTOQWs and their connection to BDPs. This technique has been applied in the case of unitary quantum walks, where the relevant orthogonal polynomials are described in terms of the theory of CMV matrices [9, 10]. Regarding the setting of open quantum dynamics [6, 8, 11], the problem of obtaining orthogonal polynomials and associated weights is an interesting one as well, although we would have to consider operators which are no longer unitary. A first step in this direction has been discussed in [19], where a procedure for obtaining weight matrices associated with open quantum walks (OQWs) [4] on the half-line was described. In [13], it was studied the case of discrete-time quantum Markov chains on the line, as defined by S. Gudder [17] and gave a collection of some nontrivial examples where the spectral representation can be explicitly achieved.
Analogously to the references above, we can employ the matrix orthogonal polynomial framework to explore various statistical aspects of CTOQWs, including recurrence patterns and transition probabilities. We utilize the Stieltjes transform as a key tool to analyze these statistics, offering an effective method to understand the intricate dynamics of quantum walkers in quantum systems influenced by tridiagonal block matrices. We are particularly interested in the recurrence of CTOQWs in this work. A first step in this direction can be seen in [23].
Let us recall the classical BDPs. Birth–death processes on \(\mathbb {Z}_{\ge 0}\) are continuous-time Markov chains characterized by a set of birth–death rates \(\{(\lambda _n,\mu _n), n\ge 0\}\) such that \(\lambda _n>0,n\ge 0\), \(\mu _n>0,n\ge 1\) and \(\mu _0\ge 0\) (see [2]). The transition function \(P(t)=(P_{ij}(t))\) satisfies the following conditions as \(t\rightarrow 0^+\):
The matrix corresponding to the infinitesimal operator associated with the process is given by
Following the classical work of S. Karlin and J. McGregor [20, 21], we can apply Favard’s Theorem to the Jacobi matrix (1) and assure the existence of a probability spectral measure \(\psi \) supported on \([0,\infty )\) associated with \(\mathcal {A}\). Moreover, if we define the sequence of polynomials \(\{Q_n(x)\}_{n\ge 0}\) by the three-term recurrence relation
that is, \(-xQ(x)=\mathcal {A}Q(x),\) where \(Q(x)=(Q_0(x),Q_1(x),\ldots )^T,\) then we have that the polynomials \(\{Q_n(x)\}_{n\ge 0}\) are orthogonal with respect to \(\psi \). This provides the so-called Karlin–McGregor formula which gives an integral representation of the probability of reaching vertex j at time t given that the process started at vertex i, i.e., \(P_{ij}(t)\). This formula is given by
The main purpose of this paper is to analyze the spectral representation of some continuous-time open quantum walks (CTOQWs) by using the basic theory of matrix-valued orthogonal polynomials. The theory of orthogonal polynomials can be applied to the open quantum walks through an appropriate matrix representation that rises from the “vec” application, whose role is to stack a density matrix in a unique bigger vector, and then a reversion of this application is made after an application of the matrix representation (see [13, 19]).
This dynamic is described by a quantum Markov semigroup with a specific Lindblad generator and performs an evolution of the initial density operator. Roughly speaking, the state at instant t can be described by a pair \((X_t,\rho _t)\) with \(X_t\) being the position of the particle at time t, and \(\rho _t\) is the density operator describing the internal degrees of freedom of the corresponding vertex. We concentrate our results on CTOQWs whose vertices have all the same internal degrees of freedom, thereby the operators that describe the Lindblad generator will be acting on the same Hilbert space, and the matrices that describe the probability transitions will be squares.
The main result of this work is Eq. (20), which expresses a formula for the Stieltjes transform of a CTOQW on the integer line in terms of Stieltjes transforms on the integer half-line. This transform associates a weight with a real function, enables us to evaluate the recurrence of CTOQWs, and offers a method for the construction of matrix weights that influence the orthogonality of the polynomials. We remark that this result is valid for any semigroup having a matrix representation of the form (9), thus the folding trick is not retained to CTOQWs. For instance, we can also apply those formulas to quasi-birth-and-death processes.
In Sect. 2, we review the “vec” representation for completely positive maps and the matrix representation for maps of the form \(\Psi (\rho )=G\rho +\rho G^*,\) where G is the part of the Lindblad generator which is not completely positive. In Sect. 3, we discuss the model of CTOQWs and present its matrix representation. In Sect. 4, we recall the concept of matrix-valued orthogonal polynomials and show how the recurrence of CTOQWs can be associated to the Stieltjes transform. Section 5 develops the matrix representation for CTOQWs in the integer line, leading to the main result of this work, Eq. (20), which associates the weight matrix of the walk in the half-line with the walk on the integer line. Section 6 illustrates the results with examples, giving explicit probabilities for different classes of Lindblad generators. In Sect. 1, an appendix is dedicated to recalling properties related to the existence of a matrix weight associated with the Lindblad generator.
2 General settings
Let \(\mathcal {H}\) be a separable Hilbert space with inner product \(\langle \,\cdot \,|\,\cdot \,\rangle \), whose closed subspaces will be referred to as subspaces for short. The superscript \({}^*\) will denote the adjoint operator. The Banach algebra \(\mathcal {B}(\mathcal {H})\) of bounded linear operators on \(\mathcal {H}\) is the topological dual of its ideal \(\mathcal {I}(\mathcal {H})\) of trace-class operators with trace norm
through the duality [3, Lec. 6]
If \(\dim \mathcal {H}=k<\infty \), then \(\mathcal {B}(\mathcal {H})=\mathcal {I}(\mathcal {H})\) is identified with the set of square matrices of order k, denoted \(M_k(\mathbb {C})\). The duality (2) yields a useful characterization of the positivity of an operator \(\rho \in \mathcal {I}(\mathcal {H})\),
and similarly for the positivity of \(X\in \mathcal {B}(\mathcal {H})\). In this work, we assume that we have a quantum particle walking either on the integer line, the integer half-line, or on a finite segment, that is, we have that the set of vertices V is labeled by \(\mathbb {Z}\), \(\mathbb {Z}_{\ge 0}\) or a finite set \(\{0,1,\dots ,N\}\), respectively. We will also call vertices sites. The state of the system is described by a column vector
The vector representation vec(A) of \(A\in M_k(\mathbb {C})\), given by stacking together its rows, will be a useful tool. For instance,
Let \(\overline{B}=[\overline{b}_{ij}]\) for \(B=[{b}_{ij}],\) that is, the entries of \(\overline{B}\) are the complex conjugate entries of B. The vec mapping satisfies \(vec(AXB^T)=(A\otimes B)\,vec(X)\) for any square matrices A, B, X, with \(\otimes \) denoting the Kronecker product. In particular, \(vec(BXB^*)=vec(BX\overline{B}^T)=(B\otimes \overline{B})\,vec(X)\), from which we can obtain the matrix representation \({\widehat{\Phi }}\) for a completely positive (CP) map \(\sum _i B_i\cdot B_i^*\) when the underlying Hilbert space \(\mathcal {H}\) is finite-dimensional:
Here, the operators \(B_i\) are identified with some matrix representation. We have that 
, where \(B^*\) denotes the Hermitian transpose (also known as conjugate transpose) of a matrix B. The same idea can be applied to maps of the form \(\Psi (\rho )=G\rho +\rho G^*.\) On this case, the map \(\Psi \) has matrix representation
For more details, we refer the reader to the reference [18].
3 Continuous-time open quantum walks
An operator semigroup \(\mathcal {T}\) on a Hilbert space \(\mathcal {H}\) is a family of bounded linear operators \((T_t)\) acting on \(\mathcal {H}\), \(t\ge 0,\) such that
If \(t\mapsto T_t\) is continuous for the operator norm of \(\mathcal {H}\), then \(\mathcal {T}\) is said to be uniformly continuous. This class of semigroups is characterized by the following result:
Theorem 1
([7], page 161) The following assertions are equivalent for a semigroup \(\mathcal {T}\) on \(\mathcal {H}:\)
-
(1)
\(\mathcal {T}\) is uniformly continuous;
-
(2)
There exists a bounded operator L on \(\mathcal {H}\) such that
$$\begin{aligned} T_t=e^{tL},\quad t\in \mathbb {R}^+. \end{aligned}$$Further, if the conditions are satisfied, then
$$\begin{aligned} L=\lim _{t\rightarrow 0^+}\frac{T_t-I_\mathcal {H}}{t}. \end{aligned}$$
The operator L is called the generator of \(\mathcal {T}.\)
A trace-preserving semigroup \(\mathcal {T}:=(\mathcal {T}_t)_{t\ge 0}\) of CP maps acting on \(\mathcal {I}_1(\mathcal {H}),\) set of trace-class operators on \(\mathcal {H},\) is called a Quantum Markov Semigroup (QMS) on \(\mathcal {I}_1(\mathcal {H}).\) When \(\lim _{t\rightarrow 0}||\mathcal {T}_t- Id||=0,\) \(\mathcal {T}\) has a generator \(\mathcal {L}=\lim _{t\rightarrow 0^+}(\mathcal {T}_t-\text {Id})/t\) (see [22]), which is a bounded operator on \(\mathcal {I}_1(\mathcal {H}),\) also known as Lindblad operator.
We consider a finite or countable set of vertices V and then take the composite system
where each \(\mathfrak {h}_i\) denotes a separable Hilbert space. The label \(i\in V\) is interpreted as being the position of the walker and, when the walker is located at the vertex \(i\in V\), its internal state is encoded in the space \(\mathfrak {h}_i,\) describing the internal degrees of freedom of the particle when it is sitting at site \(i\in V.\) Since we will be considering only examples with \(\mathfrak {h}_i=\mathfrak {h}_j\) for all \(i,j\in V,\) we let \(\mathfrak {h}_i=\mathfrak {h}\) for every \(i\in V.\)
The set of diagonal density operators acting on \(\mathcal {H}\) will be denoted by
Definition 1
([24]) A Continuous-time Open Quantum Walk (CTOQW) is a uniformly continuous QMS on \(\mathcal {I}_1(\mathcal {H})\) with Lindblad operator of the form
where, consistently with the notation, we write \(S_i^j=R_i^j\otimes \vert j\rangle \langle i\vert \) for bounded operators \(R_i^j\in \mathcal {B}(\mathfrak {h}_i,\mathfrak {h}_j).\) Moreover, H and \(S_i^j\) are bounded operators on \(\mathcal {H}\) of the form \(H=\sum _{i\in V}H_i\otimes \vert i\rangle \langle i\vert ,\) \(H_i\) is self-adjoint on \(\mathfrak {h}_i,\) \(S_i^j\) is a bounded operator on \(\mathcal {H}\) with \(\sum _{i,j\in V}S_i^{j*}S_i^j\) converging in the strong sense. Also, \([A,B]\equiv AB-BA\) is the commutator between A and B and \(\{A,B\}\equiv AB+BA\) is the anti-commutator between A and B.
Then, we have \(\rho =\sum _{i\in V}\rho (i)\otimes \vert i\rangle \langle i\vert \in \mathcal {D},\) \(e^{t\mathcal {L}}(\rho )=\mathcal {T}_t(\rho )=\sum _{i\in V}\rho _t(i)\otimes \vert i\rangle \langle i\vert ,\forall t\ge 0,\) with
An alternative way to rewrite (3) is given by equation (18.7) in [5]:
where
Further, we will present the matrix representation for CTOQWs, and this will be done by taking the representation given in Eq. (4).
The label \(i\in V\) represents the position of the walker and, when the walker is located at \(i\in V,\) its internal state is encoded in \(\mathfrak {h}_i,\) that is, \(\mathfrak {h}_i\) describes the internal degrees of freedom of the walker when it is at site \(i\in V.\)
Starting the walk on site \(\vert i\rangle \) with initial density operator \(\rho \in \mathcal {S}(\mathfrak {h}_i)=\sum _{i\in V}\rho (i)\vert i\rangle \langle i\vert ,\) the quantum measurement of the position gives rise to a probability distribution \(p_0\) on V, such that
and for evolution on time \(t\ge 0,\)
where
The vector and matrix representation of states and CP maps may be easily adapted to CTOQWs. In fact, since any element of \(\mathcal {I}_1(\mathcal {H})\) is block diagonal, when \(\dim \mathcal {H}<\infty \), it may be represented by combining the vector representations of the finite diagonal blocks,
The CTOQW (4) admits the block matrix representation
where
We will often identify the Lindblad generator \(\mathcal {L}\) with its block matrix representation and omit the hat, as the usage of such object will be clear from the context. Also, we will sometimes write X instead of 
in contexts where no confusion arises.
It is worth noting that although the above definitions concern CTOQWs on general graphs, in this paper, we will deal exclusively with the one-dimensional situation which we may also call the quantum birth–death process, and represent the generator by
for certain operators \(A_i, B_i, C_i\), and the remaining operators being equal to zero. The above representation is for a quantum particle walking on the integer half-line \(\mathbb {Z}_{\ge 0}\), but we will also study examples acting on a finite set \(\{0,1,\ldots ,N\}\) or the integer line \(\mathbb {Z}\).
4 Matrix-valued orthogonal polynomials
In this section, we introduce the Karlin–McGregor Formula for CTOQW with set of vertices of the forms \(V=\{0,1,2,\ldots ,N\}\) and \(V=\mathbb {Z}_+=\{0,1,2\ldots \}.\) Then, we will be able to give a recurrence criterion for vertex \(\vert 0\rangle \) based on the Stieltjes transform of the associated weights.
Following [14], we pick \(d\in \{1,2,3,\ldots \},\) \((A_n)_{n\ge 0}, \;(B_n)_{n\ge 0},\) and \((C_n)_{n\ge 1}\), such that the block tridiagonal matrix
represents a Lindblad generator of a CTOQW \(\Lambda .\) Then, define recursively the associated matrix-valued polynomials from the matrix \(\hat{\mathcal {L}}\) on (5) by
that is, \(Q(x)=(Q_0(x),Q_1(x),\ldots )\) are solutions of the equation \(-xQ(x)=Q(x)\hat{\mathcal {L}}.\) Here, we denote \(I_d\) and \(0_d\) the identity and the null matrix of dimension \(d\times d\).
We recall that \(\Lambda _t'=\hat{\mathcal {L}}\Lambda _t,\) where \(\Lambda _t=e^{t\hat{\mathcal {L}}}\) and define the two-variable function
One has
whose solution is \(f(x,t)=e^{-xt}Q(x).\) Hence, \(e^{-xt}Q(x)=Q(x)\Lambda _t.\) Componentwise,
where \(\Lambda _{ki}(t)\) is the (k, i)-th block of \(\Lambda (t)\).
If there exists a weight matrix \(\Sigma \) such that the matrix-valued polynomials \(\{Q_n(x)\}_{n\ge 0}\) are orthogonal with respect to \(\Sigma ,\) in the following sense
then multiplying on the left side of (7) by \(Q_j^*(x)\) and integrating with respect to \(\Sigma \), we obtain
therefore for any \(i,j\in V,\) we have the Karlin–McGregor Formula for CTOQWs:
\(\Lambda (t)=(\Lambda _{ji}(t))_{j,i=0,1,\ldots }.\) For more details about how to construct this formula, see [14].
Sometimes, we will write (8) as
Let \(p_{ji;\rho }(t)\) represent the probability of reaching site \(\vert j\rangle \) at instant t, given that we started at site \(\vert i\rangle \) with initial density \(\rho \) concentrated at i. Then,
For simplicity, we write the transition probabilities by
in contexts where no confusion arises.
Consider a CTOQW with set of vertices V. Given \(\vert i\rangle \in V\) and \(\rho \in \mathcal {S}(\mathfrak {h}_i),\) we say that \(\vert i\rangle \) is \(\rho \)-recurrent if
When \(\vert i\rangle \) is recurrent for all densities, then we say that \(\vert i\rangle \) is recurrent. This concept is associated with the weight matrices by the following theorem.
Theorem 2
Consider a tridiagonal CTOQW on \(\mathbb {Z}_{\ge 0}=\{0,1,2,\ldots \}\) and let \(\Sigma \) be its associated weight matrix. Vertex \(\vert j\rangle \) is \(\rho \)-recurrent if and only if
Proof
For each pair \(i,j\in V\), we have
\(\square \)
We recall the Stieltjes transform associated with \(\Sigma :\)
thus we obtain the straightforward consequence of Theorem 2:
Corollary 1
Consider a tridiagonal CTOQW on \(\mathbb {Z}_{\ge 0}=\{0,1,2,\ldots \}\) and let \(\Sigma \) be its associated weight matrix. Vertex \(\vert 0\rangle \) is \(\rho \)-recurrent if and only if
It is crucial to note that not all polynomials induced by block tridiagonal matrices are orthogonalizable under any matrix weight. The nontrivial nature of establishing orthogonality in this context necessitates a discerning criterion for the existence of such weights. Within the framework of Sect. 1, the appendix recalls a criterion for the orthogonality of polynomials induced by block tridiagonal matrices, and a precise expression for a specific type of weight.
5 Walks on \(\mathbb {Z}\): the folding trick
Consider the generator of a tridiagonal CTOQW on \(\mathbb {Z}\), given by
where all blocks are matrices of order \(d^2,\) thus \(\dim (\mathfrak {h})=d.\)
We assume that there exists a sequence of \(d^2\times d^2\) Hermitian matrices \((E_n)_{n\in \mathbb {Z}}\) and nonsingular matrices \((R_n)_{n\in \mathbb {Z}}\) such that
Consider the two independent families of matrix-valued polynomials defined recursively from (9) as
where we have the block vector \(Q^\alpha (x)=\bigg (\ldots ,Q_{-2}^\alpha (x),Q_{-1}^\alpha (x),Q_{0}^\alpha (x),Q_{1}^\alpha (x),Q_{2}^\alpha (x),\ldots \bigg ),\) \(\alpha =1,2,\) satisfying \(-xQ^\alpha (x)=Q^\alpha (x)\hat{\mathcal {L}}\).
As in the classical case, we introduce the block tridiagonal matrix
where each block entry is a \(2d^2\times 2d^2\) matrix, given by
The term folding trick comes from the transformation of the original generator \(\hat{\mathcal {L}}\), whose graph is represented in Fig. 1,
Generator \(\hat{\mathcal {L}}\) of a CTOQW on \(\mathbb {Z}\)
to the generator described by \(\breve{\mathcal {L}},\) which is represented by the folded walk in Fig. 2.
Folded walk of \(\hat{\mathcal {L}}\) on \(\mathbb {Z}_{\ge 0}\times \{1,2\}\) given by \(\breve{\mathcal {L}}\)
Note that \(\breve{\mathcal {L}}\) is a block tridiagonal matrix on \(\mathbb {Z}_{\ge 0},\) thereby we can apply all the properties we have seen in previous sections. The following \(2d^2\times 2d^2\) matrix polynomials are defined in terms of (11),
and these satisfy
The leading coefficient of \(\mathcal {Q}_{n}(x)\) is always a nonsingular matrix. Moreover, for
we see that the block matrices of \(\breve{\mathcal {L}}\) satisfy the conditions (10) for \(n\ge 0:\)
where matrices \(\breve{R}_{n}\) are nonsingular and \(\breve{E}_n\) are Hermitian for all \(n\ge 0\). Defining
and using Eq. (28) to note that
we obtain the correspondence between \(\breve{\Pi }_j\) and \(\Pi _j:\)
By [14], there exists a weight matrix W leading to the Karlin–McGregor formula for \(\breve{\Lambda }=e^{t\breve{\mathcal {L}}}:\)
Once we have found the weight matrix appearing on (13), we can also obtain the blocks \(\Lambda _{ji}(t)\) of the original walk generated by \(\hat{\mathcal {L}}.\) The key for this operation is the following proposition:
Proposition 1
Assume that \(\hat{\mathcal {L}}\) is the generator of a CTOQW of the form (9). The relation between \(\breve{\Lambda }_{ji}(t)\) and \(\Lambda _{ji}(t)\) is
Proof
First, we use [13, Proposition 7.1] (replace \(\breve{\Phi }_{ji}^{(n)}\) and \({\hat{\Phi }}_{ji}^{(n)}\) by \(\breve{\mathcal {L}}^n_{ji}\) and \(\hat{\mathcal {L}}^n_{ji}\), respectively) to obtain that
hence we obtain for every \(i,j\in \mathbb {Z}_{\ge 0}\) the expression
\(\square \)
Note that we can evaluate \(\breve{\Lambda }_{ji}(t)\) by (13) and then extract the block \(\Lambda _{ji}(t)\) as in (14). Further, for a density operator \(\rho \), we have
However, we would like to obtain the probability above avoiding the evaluation of \(\breve{\Lambda }_{ji}(t).\) This can be done via a generalization of the Karlin–McGregor formula on \(\mathbb {Z}_{\ge 0}\). We proceed as follows: First, write the decomposition
where \(\text {d}W_{21}(x)=\text {d}W_{12}^*(x)\), since \(\text {d}W(x)\) is positive definite. Then, one has for \(i,j\in \mathbb {Z}_{\ge 0},\)
Joining equation above and Proposition 1, we obtain the Karlin–McGregor formula for a CTOQW on \(\mathbb {Z}\), given by
Conversely, if there exist weight matrices \(\text {d}W_{11}(x),\text {d}W_{12}(x),\text {d}W_{22}(x)\) such that \(\Lambda _{ji}(t)\) is of the form (15), then \(\breve{\Lambda }_{ji}(t)\) is of the form
The weight matrix
is called the spectral block matrix of \(\mathcal {L}.\)
Remark 1
Extending Theorem 1 to the CTOQW on \(\mathbb {Z},\) we observe that since \(Q_0^1=Q_{-1}^2=I_d\) and \(Q_0^2=Q_{-1}^1=0_d\), the following limits hold
where B(z; W) is the Stieltjes transform of the weight matrix W. Analogously,
Let us write the matrix \(\hat{\mathcal {L}}\) in the form
Our goal now is to write the Stieltjes transforms associated with the weight matrices \(W_{\alpha \beta },\alpha ,\beta =1,2,\) in terms of the Stieltjes transforms associated with \(W_{\pm }\), the weight matrices associated with \(\breve{\mathcal {L}}^{\pm }\).
We introduce the generating function of \(\hat{\mathcal {L}}\)
to obtain an explicit form for the Laplace transform of \(\Lambda (t)\) on the following way:
Using equations (48), (49), (50) and (51) of [13], applied to \(\Phi _{ji}(s^{-1})=s{\hat{\Lambda }}_{ji}(s)\), we obtain
We notice that the block matrices of both \(\breve{\mathcal {L}}^+\) and \(\breve{\mathcal {L}}^-\) satisfy the conditions of Eq. (10), thus there are positive weight matrices \(W_\pm \) associated with \(\breve{\mathcal {L}}^\pm \) for which the associated polynomials are orthogonal. Then, we can write
The Laplace transform of \(\Lambda _{ji}(t)\) can be associated with the Stieltjes transform using that
\(s>0,\) that is,
thereby we recall that \(Q_0^1=Q_{-1}^2=I_{d^2},\) \(Q_0^2=Q_{-1}^1=0_{d^2}\) in order to obtain the relations
Joining with the identities (16), (17), (18), (19), the new Stieltjes transform identities are obtained:
Sometimes the operators \(\Pi _i^+\) and \(\Pi _i^-\) are equal to the identity operator. In this case, (20) are reduced to
Equations (20) and (21) allow us to obtain the Stieltjes transform of the CTOQW with \(V=\mathbb {Z}\) when we know the Stieltjes transform associated to the walks on \(\mathbb {Z}_\ge 0\) and \(\mathbb {Z}_\le 0.\) Since we are interested in the recurrence and transience of the CTOQWs, those equations are enough to obtain this information as it will be seen on the next section.
Remark 2
A sufficient condition for \(\Pi _i^+=\Pi _i^-=I\) is to have \(A_n=C_{n+1}^*\) and \(B_n=B_n^*\) for every \(n\in \mathbb {Z},\) since we will always have \(G_n=G_n^*\) for all \(n\in \mathbb {Z}\) in this case, and therefore we can take \(R_i=I\) for all \(i\in \mathbb {Z}\) (see Eq. (10)). On the other hand, those conditions are not necessary, since we can find examples with \(R_n\) being any unitary matrices for each n.
Most of our examples consider \(R_i^i=0\) for all \(i\in V.\) In this case the Hamiltonian part does not contribute to the probabilities, as it will be seen as a consequence of the following Proposition. Moreover, this Proposition gives equivalence to a condition that the diagonal of the matrix representation of the generator has negative-semidefinite matrices.
Proposition 2
Let us consider a tridiagonal CTOQW in \(Z_{\ge 0}\) (or a finite V) satisfying the conditions of Eq. (27). Then, \({\textbf {G}}_n+\lceil B_n\rceil \le 0\) if and only if
where
and
Proof
Let us suppose that \(T_n:={\textbf {G}}_n+\lceil B_n\rceil \) for every \(n\ge 0\) satisfies the conditions of Eq. (27).
Firstly, we suppose that \(T_n\le 0,\) thus there exists an orthonormal basis \(\{v_1,\ldots ,v_{d^2}\}\) of \(\mathbb {C}^{d^2}\) constituted by eigenvectors of \(T_n\) with \(T_nv_k=t_kv_k,\;k=1,\ldots ,d^2.\)
Denote \(S_n=\frac{1}{2}\left( A_n^*A_n+B_n^*B_n+C_n^*C_n\right) \) to obtain
We have \(F_{1,k,k},F_{2,k,k}\in \mathbb {R},\) thus \(\langle v_k,\left( -H_n\otimes I+I\otimes \overline{H_n}\right) v_k\rangle =-Im(\langle v_k,\lceil B_n\rceil v_k\rangle ),\) thereby the entries of the diagonal of \(-H_n\otimes I+I\otimes \overline{H_n}\) coincide with the entries of the imaginary part of the diagonal of \(-\lceil B_n\rceil .\)
For \(j\ne k,\) we have
thus, denoting by \([X]_{jk}\) the (j, k)-th entry of a matrix X on the basis \((v_k)_k,\) we obtain the identity
completing the first part of the proof.
On the other hand, we suppose that there exists an orthonormal basis such that Eq. (22) is valid. In this case, we have
and
This shows that \([T_n]\) is diagonal with respect to this orthonormal basis and its entries are all real, thus it is Hermitian. \(\square \)
Corollary 2
Let us consider a tridiagonal CTOQW in \(Z_{\ge 0}\) (or a finite V) with a positive matrix weight associated with this CTOQW. Then, \(H_n=h_nI\) for some \(h_n\in R\) if and only if \(B_n\) is Hermitian. In this case, \(H_n\) does not contribute to the probability of the walk.
Proof
We suppose that there exists a positive matrix weight associated with the CTOQW, thus Eq. (22) is valid. We have that \(B_n\) is Hermitian if and only if
The matrix \(-H_n\otimes I+I\otimes \overline{H_n}\) is a multiple of the identity if and only if \(h_{jk}^{(n)}=0\;\forall j\ne k\) and \(h_{kk}^{(n)}=h\;\forall k\) for some \(h\in \mathbb {R},\) where the second statement is valid by Proposition 2. Moreover,
which is equivalent to have \(\lceil B_n\rceil \) to be Hermitian, since S is Hermitian. \(\square \)
6 Examples
6.1 Diagonal and simultaneously diagonalizable transitions
First, we will consider a homogeneous CTOQW on the \(N+1\) nodes indexed as \(V=\{0,1,\dots , N\}\), where we add two absorbing barriers (\(\vert -1\rangle ,\;\vert N+1\rangle \)) on the extreme nodes, \(R_i^i=0\) for each site, and the generator \(\mathcal {L}\) is given by
The classical symmetrization
where \(K=\lceil \sqrt{AC}\rceil =\textrm{diag}\left( a_1c_1,\;\sqrt{a_1c_1a_2c_2},\;\sqrt{a_1c_1a_2c_2},\;a_2c_2\right) ,\) gives
The matrix-valued polynomials \(\{Q_n\}_{n\ge 0}\) are recursively defined by
which can be identified with the Chebyshev polynomials of the second kind \(\{U_n\}_{n\ge 0}\). Indeed, we have
Now, if we define
we have that the zeros of \(\det (R_{N+1}(x))\) coincide with the eigenvalues of \(-J\). A simple calculation shows that
We would like to solve the equation det\((R_{N+1}(x))=0\). Recalling the representation
we obtain, for the matrix-valued case at hand,
thus
where we have put \(G=-\textrm{diag}(g_1,g_2,g_3,g_4)\) and \(K=-\textrm{diag}(k_1,k_2,k_3,k_4).\) Since \(g_2=g_3\) and \(k_2=k_3,\) \(\text{ det }(R_{N+1}(x))\) is a polynomial of degree \(4(N+1)\) having \(3(N+1)\) distinct roots, which are of the form
each \(y_j\) being of multiplicity 2. There can be cases of eigenvalues with a greater multiplicity, which happens when the collection of zeros \(x_N,y_N\) and \(z_N\) overlap, so the multiplicity changes accordingly.
Let us compute the weight matrixes on the zeros above. We have
an expression which can be deduced from (see [16])
and noting that this corresponds to the Laurent sum of the operator on the left-hand side except for the sign change \(\lambda _k-\lambda =-(\lambda -\lambda _k)\). With formula (23), a calculation shows that for every N, we have a corresponding set of multiples of the matrices given by
More precisely, we have a collection of \(3(N+1)\) roots with weights
For a specific instance of the above take \(N=2\) (3 sites), so we have nine roots, with weights
associated with zeros \(\;a_1^2+c_1^2-2a_1c_1,\;\sqrt{a_1c_1a_2c_2}-(a_1^2+c_1^2+a_2^2+c_2^2)\) and \(\;a_2^2+c_2^2-2a_2c_2\;\) respectively; weights
associated with zeros \(\;a_1^2+c_1^2-\sqrt{2}a_1c_1,\;\sqrt{a_1c_1a_2c_2}-\sqrt{2}(a_1^2+c_1^2+a_2^2+c_2^2)/2\) and \(\;a_2^2+c_2^2-\sqrt{2}a_2c_2\;\) respectively; and weights
associated with zeros \(\;a_1^2+c_1^2,\;\sqrt{a_1c_1a_2c_2}\) and \(\;a_2^2+c_2^2\;\) respectively.
Now, let us consider the walk on the half-line.
We will consider a CTOQW whose set of vertices is \(V=\{0,1,2,\ldots \}\) and the walker can jump to its nearest neighbor; however, there is an absorbing barrier (\(\vert -1\rangle \)). Therefore, this walk can be interpreted as a BDP in which the population may become extinct. The matrix
is a valid generator of a CTOQW. Also,
The CTOQWs of these first examples are entirely described by diagonal matrices; therefore, the parameter b in the density matrix \(\rho \) has no influence on these random walks with the specific transitions A and C.
If we take \(K:=\lceil (AC)\rceil ^{1/2}\), then we obtain the symmetrization
where K is positive definite,
Let us obtain the weight matrix associated with \({\tilde{J}},\)
using the results of A.J. Durán ( [15]).
Since \({\textbf {G}}\) and K commute, it is easy to see that the matrix \(H_{A,B}(x)\) given by [15] is
The associated weight matrix to \({\tilde{J}}\) is
where \(h_j\) represents the j-th diagonal entry of the diagonal appearing on the representation of H(x) and
Here, we are using the notation \([f(x)]_+ = f(x)\) if \(f(x)\ge 0\) and 0 otherwise.
We are interested on the transitions of the CTOQW, thus only \(d_1(x)\) and \(d_4(x)\) contribute for the calculus of the trace when we evaluate
thereby we will avoid the massive calculations using terms as \(d_2(x)\) and \(d_3(x)\) appearing on the sequel of this section.
The Stieltjes transform is
where \(w_2(z)=w_3(z)\) is a function that does not vanish and
Since the weight is obtained on the terms of [15], we must have \(\Pi _0=I_4,\) then we use equation (2.20) of [14] to obtain the Stieltjes transform of the weight matrix associated with J:
where
It is a simple calculation to verify that \(\lim _{z\uparrow 0}\sigma _j(z)=\infty \Leftrightarrow a_j\le c_j,\) thus, given a density operator \(\rho =\begin{bmatrix}a &{} b \\ b^* &{} 1-a\end{bmatrix},\) we have
where \(\pi _1,\pi _2>0.\) Therefore, if \(\{\vert e_0\rangle ,\vert e_1\rangle \}\) is the canonical basis of \(\mathbb {C}^2\), then an application of Corollary 1 shows that
-
\(a_1\le c_1\) and \(a_2\le c_2\;\Rightarrow \) vertex \(\vert 0\rangle \) is recurrent;
-
\(a_1\le c_1\) and \(a_2> c_2\;\Rightarrow \) vertex \(\vert 0\rangle \) is \(\vert e_1\rangle \langle e_1\vert \)-transient and \(\rho \)-recurrent for \(\rho \ne \vert e_1\rangle \langle e_1\vert ;\)
-
\(a_1> c_1\) and \(a_2\le c_2\;\Rightarrow \) vertex \(\vert 0\rangle \) is \(\vert e_0\rangle \langle e_0\vert \)-transient and \(\rho \)-recurrent for \(\rho \ne \vert e_0\rangle \langle e_0\vert ;\)
-
\(a_1> c_1\) and \(a_2>c_2\;\Rightarrow \) vertex \(\vert 0\rangle \) is transient.
The Perron–Stieltjes inversion formula (Proposition 1.1 of [12]) gives
thus an application of the Karlin–McGregor formula for CTOQWs gives for \(\rho =\begin{bmatrix}a &{} b \\ b^* &{} 1-a\end{bmatrix},\)
The particular case of \(r:=a_1=c_1\) and \(s:=a_2=c_2\) gives the weight matrix
where
Finally, we describe the associated walk on the integer line.
Let us consider the homogeneous CTOQW on \(\mathbb {Z}.\) In this case, the quantum walker’s dynamics are uniform across different positions on the integer lattice and could be explored in various physical systems, such as trapped ions, superconducting circuits, or photonic systems.
We take
In this case, we have
Using the first equation on (21) with \(A_{-1}=A\) and \(C_0=C,\) we obtain
where we used \(\text {d}W_+=\text {d}W_-=d{\tilde{\Sigma }}(x),\) \(\;d{\tilde{\Sigma }}(x)\) being the weight matrix given by (24).
It is easily seen that
therefore, for \(\rho =\begin{bmatrix}a &{} b \\ b^* &{} 1-a\end{bmatrix},\) we obtain that
-
\(a_1=c_1\) and \(a_2=c_2\) implies that the walk is recurrent;
-
\(a_1\ne c_1\) and \(a_2\ne c_2\) implies that the walk is transient;
-
\(a_1=c_1\) and \(a_2\ne c_2\) implies that the walk is \(\rho \)-transient for \(a=0\) and \(\rho \)-recurrent for \(a>0\);
-
\(a_1\ne c_1\) and \(a_2= c_2\) implies that the walk is \(\rho \)-transient for \(a=1\) and \(\rho \)-recurrent for \(a<1\).
We observe that the walker returns infinitely often, in mean, to vertex \(\vert 0\rangle \) for any initial density operator when \(a_1=c_1\) and \(a_2=c_2\). In the contrasting scenario, where \(a_j\ne c_j,\;j=1,2\), the walker returnees is finite in mean. Lastly, when only one of the values \(a_j\) equals \(c_j\), then the walk returns a finite number of times to \(\vert 0\rangle \), in mean, for one only density, and infinitely often for all others.
Moreover, the weight \(\text {d}W_{11}\) is obtained by applications of the Perron–Stieltjes inversion formula:
6.2 The case of simultaneous unitarily diagonalizable transitions
The above analysis can be applied to the simultaneous unitary diagonalizable coins, that is, we can take a unitary matrix U and coins given by
to obtain analogous conclusions about the recurrence of vertex \(\vert 0\rangle .\) In this case, we have
-
\(a_1\le c_1\) and \(a_2\le c_2\;\Rightarrow \) vertex \(\vert 0\rangle \) is recurrent;
-
\(a_1\le c_1\) and \(a_2> c_2\;\Rightarrow \) vertex \(\vert 0\rangle \) is \(U\vert e_1\rangle \langle e_1\vert U^*\)-transient and \(\rho \)-recurrent for \(\rho \ne U\vert e_1\rangle \langle e_1\vert U^*;\)
-
\(a_1> c_1\) and \(a_2\le c_2\;\Rightarrow \) vertex \(\vert 0\rangle \) is \(U\vert e_0\rangle \langle e_0\vert U^*\)-transient and \(\rho \)-recurrent for \(\rho \ne U\vert e_0\rangle \langle e_0\vert U^*;\)
-
\(a_1> c_1\) and \(a_2>c_2\;\Rightarrow \) vertex \(\vert 0\rangle \) is transient.
Let us describe an example of this and, in addition, let us consider a perturbation on the first vertex. In this case, the walk can be represented by Fig. 3, where \(B_0\) represents the rate of jumping from vertex \(\vert 0\rangle \) to itself.
A slight modification on the first vertex
Let \(U\in \mathbb {M}_2(\mathbb {C})\) be an unitary matrix and consider the CTOQW with generator
and the Hamiltonian operator
The diagonal block matrices of \(\hat{\mathcal {L}}\) are \({\textbf {G}}=-\mathcal {U}diag (8,5,5,2)\mathcal {U}^*\) and
where \(\mathcal {U}=U\otimes U,\) thus \({\textbf {G}}_{0}\) is Hermitian.
The Stieltjes transform of the matrix weight associated with \(\tilde{\mathcal {L}}\) (\(\hat{\mathcal {L}}\) with \({\textbf {G}}_0+[B_0]\) switched by \({\textbf {G}}\)) is then, by Eq. (25),
The Stietjes transform of \(\hat{\mathcal {L}}\) is obtained by
After some calculus using the limit given on Corollary 1, we obtain that this walk is recurrent for any choices of \(h_1\in \mathbb {C},h_2\in \mathbb {R}.\)
A perturbation on the vertex \(\vert 0\rangle \) for the CTOQW on \(\mathbb {Z}:\) We consider a CTOQW on \(\mathbb {Z}\) with the same transitions as above but with a perturbation on vertex \(\vert 0\rangle .\) That is, we are taking the walk given by Fig. 4, where
Each position on the lattice behaves similarly, except \(\vert 0\rangle \). The perturbation, characterized by a different matrix rate for a self-loop at \(\vert 0\rangle \), introduces a localized influence on the walker’s behavior. This can be seen as a quantum interference effect, where the perturbation disrupts the otherwise uniform evolution of the quantum state. Specifically, a self-loop alters the probability amplitudes associated with staying at vertex \(\vert 0\rangle \) versus transitioning to neighboring vertices, leading to a non-trivial modulation in the walker’s probability distribution along the lattice. Moreover, we can understand this as a quantum-level interference phenomenon, where the perturbation introduces phase relationships among probability amplitudes, influencing the walker’s behavior and creating distinctive patterns in the quantum state evolution.
Physically, this setup could be implemented in a quantum system where the different vertices of the integer lattice correspond to distinct quantum states, and the perturbation arises from a modification in the local dynamics at one specific position. This might be achieved through controlled interactions or external fields acting on the quantum system. Such perturbations can be leveraged in quantum algorithms and simulations, providing a way to encode specific information or perform quantum operations selectively at certain positions in the lattice.
Generator \(\hat{\mathcal {L}}\) of a CTOQW on \(\mathbb {Z}\) with a perturbation on vertex \(\vert 0\rangle \)
We want to apply Eq. (20) to verify if vertex \(\vert 0\rangle \) is recurrent. To do this, we notice that
where \({\textbf {G}}=-\mathcal {U}diag (8,5,5,2)\mathcal {U}^*\) and now
which is a Hermitian matrix.
The Stieltjes transform of the matrix weight associated with \(\breve{\mathcal {L}}^-\) is given in Eq. (26) (since \(A=C\)), while the Stietjes transform of the matrix weight associated with \(\breve{\mathcal {L}}^+\) is
where
Some calculus shows that
for any choice of \(h_1\in \mathbb {C},\;h_2\in \mathbb {R}\) and \(\rho \in \mathbb {M}_2(\mathbb {C}),\) therefore vertex \(\vert 0\rangle \) is always recurrent for this CTOQW.
The same can be done with vertex \(\vert -1\rangle ;\) however, on this case, we have to evaluate \(-\lim _{z\uparrow 0}Tr \left( B(z,W_{22})\rho \right) =\infty ,\) which is always infinite for any choice of \(h_1\in \mathbb {C},\;h_2\in \mathbb {R}\) and \(\rho \in \mathbb {M}_2(\mathbb {C}),\) therefore vertex \(\vert -1\rangle \) is also always recurrent for this CTOQW.
6.3 Noncommuting transitions
Let
where
Consider the CTOQW with \(V=\{0,1,2,3\}\) induced by the generator
Note that this generator satisfies the conditions (27) with \(R_n=I_4,\;n=0,1,2,3,\) thus there exists a positive weight matrix associated to \(\hat{\mathcal {L}},\) which will be evaluated now.
The eigenvalues of \(-\hat{\mathcal {L}}\) are
(\(\lambda _1,\lambda _4,\lambda _5,\lambda _6,\lambda _7,\lambda _8\) and \(\lambda _9\) have multiplicity 2) with weights
where
For instance, we have for \(\rho =\begin{bmatrix}a &{} b \\ b^* &{} 1-a\end{bmatrix},\)
where \(v_1=\dfrac{\sqrt{5}}{40}(1-2a+4Re (b))\) and \(v_2=\dfrac{\sqrt{7}}{28}(2-a+2Re (b)).\)
Non-diagonal matrices often lead to more intricate transition probabilities in CTOQWs. Therefore, these distinct values introduce more complex interference effects and allow the walker’s behavior dependence on b. Furthermore, the variation of the initial density distribution can be understood as a conceptual analogy for decoherence. Initially, the uniform distribution between quantum states exemplifies a coherent starting point. However, the introduction of decoherence involves changing this initial distribution, with the subsequent evolution of CTOQW reflecting the influence of this perturbation, similar to the loss of coherence observed in decoherence phenomena. This change in initial density alters the probabilistic evolution, illustrating the sensitivity of the quantum system to its initial conditions. See Fig. 5 to compare the transition probabilities for different initial densities.
Different transition probabilities for Example 6.3
6.4 Antidiagonal transitions: another approach
Let us discuss an example with antidiagonal transitions. We do this in terms of preliminary reasoning with a generator that has alternating matrices. More precisely, we consider a block matrix of the form
where
We notice that J may not be a valid generator of a CTOQW; however, this block matrix will be auxiliary to obtain a weight matrix associated with a specific kind of generator later. Then, we use [14, Theorem 2.4] to obtain the following equality associated to the Stieltjes transform of the weight matrix \(\text {d}\Sigma (x)\) associated with \({\tilde{J}},\) which is the equivalent of J with \({\textbf {G}}_0\) switched by \({\textbf {G}}:\)
where \(R_i=I_4\) for every \(R_i\) appearing on [14, Theorem 2.4] is a consequence of \(\lceil P_0\rceil =\lceil P_0\rceil ^T\) and \(\lceil P_1\rceil =\lceil P_1\rceil ^T.\)
The known matrices of the equality are all diagonal, thus we assume that
and then each \({\tilde{f}}_k(z)\) is a solution of
where \({\textbf {G}}=diag ({\tilde{g}}_1,{\tilde{g}}_2,{\tilde{g}}_3,{\tilde{g}}_4),\;\;\lceil P_j\rceil =diag (m_{j,1},m_{j,2},m_{j,3},m_{j,4}),\;j=0,1.\) Some algebra gives
Therefore,
As usual, the next step is to obtain the Stieltjes transform of \(\text {d}\Sigma ,\) the weight matrix associated with J. By equation (2.20) of [14], we have
where
and we have put \({\textbf {G}}_0=\textrm{diag}(g_1,g_2,g_3,g_4,)\) \(\psi _k(z)=-(z+g_k)^2+m_{1,k}^2-m_{0,k}^2,\;\;\gamma _k(z)=(z+g_k)m_{1,k}.\)
Now, we are able to consider an antidiagonal transition in the following terms: consider a CTOQW on \(\mathbb {Z}_+\) whose generator is of the form
We have the symmetrization
where
and \(P_0\) and \(P_1\) are the ones given above. Thus, J and \(\hat{\mathcal {L}}\) have the same associated weight matrix and we obtain, for \(\text {d}\Sigma (x)\) given above that
where \(\rho =\begin{bmatrix}a &{} b \\ b^* &{} 1-a\end{bmatrix}.\) After some calculus, we obtain that
giving the following conclusion (see Corollary 1):
-
\(a_1=\sqrt{\frac{2c_2^4-a_2^2c_1^2+a_2^4+3a_2^2c_2^2}{a_2^2+2c_2^2}}\) and \(a_2=\sqrt{\frac{2c_1^4-a_1^2c_2^2+a_1^4+3a_1^2c_1^2}{a_1^2+2c_1^2}}\Rightarrow \) vertex \(\vert 0\rangle \) is recurrent;
-
\(a_1=\sqrt{\frac{2c_2^4-a_2^2c_1^2+a_2^4+3a_2^2c_2^2}{a_2^2+2c_2^2}}\) and \(a_2\ne \sqrt{\frac{2c_1^4-a_1^2c_2^2+a_1^4+3a_1^2c_1^2}{a_1^2+2c_1^2}}\Rightarrow \) vertex \(\vert 0\rangle \) is \(\rho \)-transient when \(a=0\) and \(\rho \)-recurrent when \(a>0;\)
-
\(a_1\ne \sqrt{\frac{2c_2^4-a_2^2c_1^2+a_2^4+3a_2^2c_2^2}{a_2^2+2c_2^2}}\) and \(a_2=\sqrt{\frac{2c_1^4-a_1^2c_2^2+a_1^4+3a_1^2c_1^2}{a_1^2+2c_1^2}}\Rightarrow \) vertex \(\vert 0\rangle \) is \(\rho \)-transient when \(a=1\) and \(\rho \)-recurrent when \(a<1;\)
-
\(a_1\ne \sqrt{\frac{2c_2^4-a_2^2c_1^2+a_2^4+3a_2^2c_2^2}{a_2^2+2c_2^2}}\) and \(a_2\ne \sqrt{\frac{2c_1^4-a_1^2c_2^2+a_1^4+3a_1^2c_1^2}{a_1^2+2c_1^2}}\Rightarrow \) vertex \(\vert 0\rangle \) is transient.
Hence, as in Example 6.1, the recurrence only depends on the density once the values of A and C are defined. Although the conclusion is similar to the case in Example 6.1, the more complex values above arise from the representations of \(\lceil P_0\rceil \) and \(\lceil P_1\rceil \), which are not diagonal.
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The datasets generated during the current study are available from the corresponding author on reasonable request.
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Acknowledgements
The author wishes to thank his PhD advisor Carlos F. Lardizabal. The author is also grateful to M. Domínguez de la Iglesia for interesting discussions around the topic. This work is financially supported by CAPES (Coordenação de Aperfeiçoamento de Pessoal de Nível Superior) and Instituto Federal de Mato Grosso do Sul.
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Appendix
Appendix
In this appendix, we will recall some well-known results from the theory of matrix orthogonal polynomials, underlining their relevance and application in the context of this work.
1. Let \(\Sigma \) be a \(d^2\times d^2\) weight matrix and denote by
the corresponding moments. The block Hankel matrices are defined by
Theorem 3
(Slight adaptation of Theorem 2.1 [14]) Consider the block matrix \(\hat{\mathcal {L}}\) given by Eq. (5), assume that \(A_n,C_{n+1},\;n\ge 0,\) are nonsingular matrices and \(B_n\le 0\;\forall n\). Now, let \(\{Q_n(x)\}_{n\ge 0}\) be the sequence of matrix-valued polynomials defined by (6). Then, there exists a weight matrix \(\Sigma \) with positive definite block Hankel matrices \({\underline{H}}_{2m}, m\ge 0\), such that the sequence of polynomials \(\{Q_n(x)\}_{n\ge 0}\) is orthogonal with respect to \(\Sigma \) if and only if there is a sequence of nonsingular matrices \((R_n)_{n\ge 0}\) such that
Moreover, \(S_0=\left( R_0^*R_0\right) ^{-1},\) and
2. Perturbation of the Stieltjes transform:
Theorem 4
(Theorem 2.3 of [14]) Consider the block matrix \(\hat{\mathcal {L}}\) given by Eq. (5) and the matrix \(\tilde{\mathcal {L}}\) which is the same as \(\hat{\mathcal {L}}\) but with a perturbation on the first block, that is,
If \(\Sigma \) is the weight matrix associated with \(\hat{\mathcal {L}}\) with positive definite block Hankel matrices such that \(R_0{\tilde{B}}_0R_0^{-1}\) is symmetric and such that \((R_n)_{n\ge 0}\) is a sequence of matrices which satisfies condition (27), then there exists a weight matrix \(\Sigma \) corresponding to \(\tilde{\mathcal {L}}.\) If the weight matrix \(\Sigma \) and \({\tilde{\Sigma }}\) are determined by their moments, then the Stieltjes transforms of the weights satisfy
3. Explicit weight matrix for a class of walks on the half-line. The following is a restatement of a result due to A.J. Durán [15]: let A be positive definite and define
Such matrix is diagonalizable except for at most finitely many complex numbers z, so that we can write \(-H(z)=U(z)D(z)U^{-1}(z)\), where D(z) is a diagonal matrix with diagonal entries \(\{d_{ii}(z)\}\). For x real, we have that \(-H(z)\) is Hermitian, so it is unitarily diagonalizable, that is, we can have U(x) such that \(U(x)U^*(x)=I\). Also, D(z) has real entries. With such matrices defined, we have:
Theorem 5
[15] If A is positive definite and B Hermitian, the weight matrix for the matrix-valued polynomials defined by
is the matrix of weights given by
where \(D^+(z)\) is a diagonal matrix with diagonal entries \(d_{ii}^+(z)=\max \{d_{ii}(x),0\}\).
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Loebens, N. Continuous-time open quantum walks in one dimension: matrix-valued orthogonal polynomials and Lindblad generators. Quantum Inf Process 23, 96 (2024). https://doi.org/10.1007/s11128-024-04303-2
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DOI: https://doi.org/10.1007/s11128-024-04303-2