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01.03.2021 | Ausgabe 3/2021 Open Access

Quantum Information Processing 3/2021

Convergence theorems on multi-dimensional homogeneous quantum walks

Zeitschrift:
Quantum Information Processing > Ausgabe 3/2021
Autor:
Hiroki Sako
Wichtige Hinweise

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1 Introduction

A quantum walk is a dynamical system given by a unitary operator U on a Hilbert space \(\mathcal {H}\). The space \(\mathcal {H}\) is often associated with some metric spaces like \(\mathbb {Z}^d\) and \(\mathbb {R}^d\). The first purpose of this paper is to propose a general framework for quantum walks on d-dimensional spaces. See Sect. 2.1. This new framework includes the quantum walks studied in [ 17], and in many other papers.
The second purpose is to observe asymptotic behavior of such walks. For an initial unit vector \(\xi \) in \(\mathcal {H}\), we obtain a sequence \(\{U^t \xi \}_{t \in \mathbb {N}}\) of unit vectors. It gives a sequence of probability measures \(\{p_t\}_{t \in \mathbb {N}}\), which is related to the probability interpretation of quantum mechanics. It is proved in Theorem 3.10 that the probability measures are asymptotically concentrated on some small region. This theorem can be applied to walks which is not necessarily space-homogeneous. In this theorem, we need some very weak condition on the walk U called smoothness defined in Sect. 2.2.
The third purpose is to show existence of limit distributions for analytic homogeneous quantum walks with respect to an arbitrary initial unit vector in Theorem 4.3. Analyticity is so mild condition that almost all the known quantum walks satisfy it. For the proof, we study 1-cocycles of such a walk in Sect. 3.1. 1-cocycles are related to Heisenberg representation of time evolution of observables. Let U be a quantum walk acting on a Hilbert space \(\mathcal {H}\), and let D be some observable of position. The sequence of observables \(\{U^{-t} D U^t\}_t\) stands for the time evolution of the observable D. The sequence \(\{c_t = U^{-t} D U^t - D\}_t\) is the most important example of 1-cocycle and is called a logarithmic derivative.
For more than 20 years, the most important subject of quantum walks is the following form of unitary operator:
$$\begin{aligned} U = \left( \begin{array}{cc} S &{}\quad 0\\ 0 &{}\quad S^{-1} \end{array} \right) \left( \begin{array}{cc} \alpha &{}\quad - \overline{\beta }\\ \beta &{}\quad \overline{\alpha } \end{array} \right) \end{aligned}$$
(1)
and their higher-dimensional version. Here, \(\alpha \) and \(\beta \) are complex numbers satisfying \(|\alpha |^2 + |\beta |^2 = 1\), and the operator S is the bilateral shift on the Hilbert space \(\ell _2 (\mathbb {Z})\) consisting of the square summable functions on the integer group \(\mathbb {Z}\). The operator U acts on the Hilbert space \(\ell _2(\mathbb {Z}) \otimes \mathbb {C}^2\). It has been already shown that several kinds of space-homogeneous quantum walks have limit distributions \(\lim _{t \rightarrow \infty } p_t\). Although their argument was not rigorous, Grimmett, Janson, and Scudo presented an excellent idea to show such a convergence theorem in [ 3]. This paper improves their result in the following aspects:
  • Our paper clarifies what kind of property of the operator U really works in the proof of the convergence theorem. It turns out that analyticity of the operator works.
  • We no longer need smooth eigenvalue functions of the inverse Fourier transform of the quantum walk. 1
  • Our theorem can be applied to many kinds of homogeneous quantum walks, because analyticity for the quantum walks is a weak condition. For example, a quantum walk on arbitrary crystal lattice with translation symmetry with finite propagation satisfies this condition.
  • Our theorem does not require locality of its initial unit vector.

2 Definition of quantum walks and their regularity

2.1 Definition of multi-dimensional QWs

Throughout this paper, \(\mathcal {B}(\mathcal {H})\) stands for the set of all the bounded linear operators on a Hilbert space \(\mathcal {H}\).
Definition 2.1
Let d be a natural number. A triple \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \) is said to be a d-dimensional quantum walk, if the following conditions hold:
(1)
\(\mathcal {H}\) is a Hilbert space,
 
(2)
\(\left( U^t \right) _{t \in \mathbb {Z}}\) is a unitary representation of \(\mathbb {Z}\) on \(\mathcal {H}\).
 
(3)
E is a Borel measure on \(\mathbb {R}^d\) whose values are orthogonal projections in \(\mathcal {B}(\mathcal {H})\) such that \(E(\mathbb {R}^d) = \mathrm {id}_{\mathcal {H}}\).
 
The measure E in the last item is called a spectral measure.
In many references, researchers concentrate on quantum walks on the Hilbert space \(\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^d\).
Example 2.2
For the Hilbert space \(\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^d\), the measure E is defined as follows: For every subset \(\Omega \subset \mathbb {R}^d\), the operator \(E(\Omega )\) is the orthogonal projection from \(\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^d\) onto \(\ell _2(\Omega \cap \mathbb {Z}^d) \otimes \mathbb {C}^d\). A unitary operator U on \(\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^d\) defines a triplet \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \), which we call a quantum walk in this paper. \(\square \)
Throughout this paper, we often make use of the following items.
  • For \(i \in \{1, \cdots , d\}\), a self-adjoint operator \(D_i\) on \(\mathcal {H}\) is defined by
    $$\begin{aligned} D_i = \int _{(x_i) \in \mathbb {R}^d} x_i E(d x_1 \cdots d x_i \cdots d x_d). \end{aligned}$$
  • We denote by \(( \cdot , \cdot )\) the standard real-valued inner product on \(\mathbb {R}^d\). The projection-valued measure E gives a unitary representation of \(\mathbb {R}^d\):
    $$\begin{aligned} \sigma :\mathbf {k}\mapsto \int _{\mathbf {x}\in \mathbb {R}^d} \exp (\mathbf {i}(\mathbf {k}, \mathbf {x})) E(d\mathbf {x}). \end{aligned}$$
    This representation is given by \(\sigma ((k_i)_i) = \prod _{i = 1}^d \exp (\mathbf {i}k_i D_i)\). The representation \(\sigma \) is continuous with respect to the strong operator topology. Throughout this paper, \(\mathbf {i}\) stands for the imaginary unit, and i is a natural number.
  • We denote by \(\alpha _\mathbf {k}\) the d-parameter automorphism group on \(\mathcal {B}(\mathcal {H})\) defined by
    $$\begin{aligned} \alpha _\mathbf {k}(X) = \sigma (\mathbf {k}) X \sigma (- \mathbf {k}). \end{aligned}$$
In many cases, \(D_i\) stands for the position observable. However, Definition 2.1 allows us to treat quantum walks in more flexible manner. The self-adjoint operator \(D_i\) can be other quantum mechanical observables like momentum.

2.2 Regularity on bounded operators

Definition 2.3
Let X be a bounded operator on \(\mathcal {H}\). The operator X is said to be uniform with respect to E, if the mapping \(\mathbb {R}^d \ni \mathbf {k}\mapsto \alpha _\mathbf {k}(X) \in \mathcal {B}(\mathcal {H})\) is continuous with respect to the operator norm on \(\mathcal {B}(\mathcal {H})\).
If the operator X is uniform, then the mapping \(\mathbf {k}\mapsto \alpha _\mathbf {k}(X)\) is uniformly continuous. It is easy to see that the set of uniform operators is a closed subset \(\mathcal {B}(\mathcal {H})\). Using the operator norm on \(\mathcal {B}(\mathcal {H})\), we may consider partial derivatives.
Definition 2.4
For \(i \in \{1, \cdots , d\}\), define the partial derivative \(\partial _i (X)\) by the norm limit
$$\begin{aligned} \partial _i (X) = \lim _{k \rightarrow 0} \frac{\exp (\mathbf {i}k D_i) X \exp (- \mathbf {i}k D_i) - X}{k}, \end{aligned}$$
(2)
if it exists.
Example 2.5
Let \(\ell _2(\mathbb {Z}^d)\) be the Hilbert space of all the square summable functions on \(\mathbb {Z}^d\). For \(\mathbf {x}\in \mathbb {Z}^d\), denote by \(\delta _\mathbf {x}\) the definition function of \(\{\mathbf {x}\} \subset \mathbb {Z}^d\). Let E be the standard spectral measure on \(\mathbb {R}^d\) defined in Example 2.2. Let j be an element of \(\{1, \cdots , d\}\). Define the unitary operator \(S_j\) by \(\delta _{\mathbf {x}} \mapsto \delta _{\mathbf {x}+ \mathbf {e}_j}\), where \(\{\mathbf {e}_1, \cdots , \mathbf {e}_j, \cdots , \mathbf {e}_d\}\) stands for the standard basis of \(\mathbb {Z}^d\).
By the straightforward calculation, for every \(\mathbf {k}= (k_1, \cdots , k_j, \cdots , k_d) \in \mathbb {R}^d\), we have
$$\begin{aligned} \alpha _\mathbf {k}(S_j) = \exp (\mathbf {i}k_j) S_j. \end{aligned}$$
Since \(\exp (\mathbf {i}k_j)\) is a continuous function on \(\mathbb {R}^d\), it turns out that \(S_j\) is uniform.
By Eq. ( 2), for \(i \ne j\), we have
$$\begin{aligned} \partial _i (S_j) = \lim _{k \rightarrow 0} \frac{\exp (\mathbf {i}k D_i) S_j \exp (- \mathbf {i}k D_i) - S_j}{k} = 0. \end{aligned}$$
We also have
$$\begin{aligned} \partial _j (S_j)= & {} \lim _{k_j \rightarrow 0} \frac{\exp (\mathbf {i}k_j D_i) S_j \exp (- \mathbf {i}k_j D_i) - S_j}{k_j} \\= & {} \lim _{k_j \rightarrow 0} \frac{\exp (\mathbf {i}k_j) - 1}{k_j} S_j\\= & {} \mathbf {i}S_j. \end{aligned}$$
We can also calculate the higher derivatives as above. It follows that the unitary operator \(S_j\) is smooth.
In particular, the bilateral shift S on \(\ell _2(\mathbb {Z})\) is smooth. \(\square \)
Example 2.6
Consider the case that \(d = 1\). Let U be the most famous 1-dimensional quantum walk presented in Eq. ( 1) acting on \(\ell _2(\mathbb {Z}) \otimes \mathbb {C}^2\). Let E be the standard spectral measure on \(\ell _2(\mathbb {Z}) \otimes \mathbb {C}^2\). Since the bilateral shift S on \(\ell _2(\mathbb {Z})\) is smooth, U is smooth with respect to E. \(\square \)
Lemma 2.7
If \(\partial _i (X)\) exists, then \(X (\mathrm {dom}(D_i)) \subset \mathrm {dom}(D_i)\), and \(\partial _i (X)\) is the unique extension of the commutator \([\mathbf {i}D_i, X] = \mathbf {i}(D_i X - X D_i)\) defined on the domain \(\mathrm {dom}(D_i)\). For every \(\mathbf {k}\in \mathbb {R}^d\), \(\partial _i (\alpha _\mathbf {k}(X))\) exists and is equal to \(\alpha _\mathbf {k}( \partial _i (X))\).
Proof
The first assertion is well known. See [ 8, Lemma 2.4] for example. We calculate \(\partial _i (\alpha _\mathbf {k}(X))\) as follows:
$$\begin{aligned}&\lim _{k \rightarrow 0} \frac{\exp (\mathbf {i}k D_i) \sigma (\mathbf {k}) X \sigma (- \mathbf {k}) \exp (- \mathbf {i}k D_i) - \sigma (\mathbf {k}) X \sigma (- \mathbf {k})}{k}\\&\quad = \sigma (\mathbf {k}) \lim _{k \rightarrow 0} \frac{\exp (\mathbf {i}k D_i) X \exp (- \mathbf {i}k D_i) - X}{k} \sigma (- \mathbf {k}) = \alpha _\mathbf {k}(\partial _i(X)). \end{aligned}$$
\(\square \)
Definition 2.8
Let X be an operator on \(\mathcal {H}\) which is uniform with respect to E. The operator X is said to be smooth with respect to E, if for every sequence \(\{i(m)\}\) of \(\{1, \cdots , d\}\), the higher-order partial derivatives
$$\begin{aligned} \partial _{i(1)}(X), \quad \partial _{i(2)} ( \partial _{i(1)}(X)), \quad \partial _{i(3)} ( \partial _{i(2)} ( \partial _{i(1)}(X))), \quad \cdots \end{aligned}$$
exist and are uniform with respect to E.
Lemma 2.9
Suppose that \(X, Y \in \mathcal {B}(\mathcal {H})\) are smooth with respect to E. Then, \(X^*\) and XY are also smooth and satisfy
$$\begin{aligned} \partial _i(X^*) = \partial _i(X)^*, \quad \partial _i(X Y) = \partial _i(X) Y + X \partial _i(Y) \end{aligned}$$
for every \(i \in \{1, \cdots , d\}\).
Proof
By Eq. ( 2), we have the following equation of limits in norm topology:
$$\begin{aligned} \partial _i (X)^*= & {} \left( \lim _{k \rightarrow 0} \frac{\exp (\mathbf {i}k D_i) X \exp (- \mathbf {i}k D_i) - X}{k} \right) ^*\\= & {} \lim _{k \rightarrow 0} \left( \frac{\exp (\mathbf {i}k D_i) X \exp (- \mathbf {i}k D_i) - X}{k} \right) ^*\\= & {} \lim _{k \rightarrow 0} \frac{\exp (\mathbf {i}k D_i) X^* \exp (- \mathbf {i}k D_i) - X^*}{k}\\= & {} \partial _i (X^*). \end{aligned}$$
Therefore, every partial derivatives of \(X^*\) exist. Since \(\partial _i (X)\) is uniform, its adjoint is also uniform. Repeating this calculation of partial derivatives, every higher derivatives of \(X^*\) also exist and are uniform. This means that \(X^*\) is also smooth.
Since Y is uniform, the equation
$$\begin{aligned} Y = \lim _{k \rightarrow 0} \exp (\mathbf {i}k D_i) Y \exp (- \mathbf {i}k D_i) \end{aligned}$$
holds. Combining with Eq. ( 2) for X and for Y, we have
$$\begin{aligned}&\partial _i(X) Y + X \partial _i(Y) = \partial _i(X) \left( \lim _{k \rightarrow 0} \exp (\mathbf {i}k D_i) Y \exp (- \mathbf {i}k D_i) \right) + X \partial _i(Y) \\&\quad = \lim _{k \rightarrow 0} \frac{\exp (\mathbf {i}k D_i) X Y \exp (- \mathbf {i}k D_i) - X\exp (\mathbf {i}k D_i) Y \exp (- \mathbf {i}k D_i)}{k} \\&\qquad + \lim _{k \rightarrow 0} X \frac{\exp (\mathbf {i}k D_i) Y \exp (- \mathbf {i}k D_i) - Y}{k}\\&\quad = \lim _{k \rightarrow 0} \frac{\exp (\mathbf {i}k D_i) X Y \exp (- \mathbf {i}k D_i) - XY}{k}\\&\quad = \partial _i(X Y). \end{aligned}$$
Therefore, every partial derivatives of XY exist. Since \(\partial _i(X), Y, X, \partial _i(Y)\) are uniform, \(\partial _i(X Y)\) is also uniform. Repeating this calculation of partial derivatives, every higher derivatives of XY also exist and are uniform. This means that XY is also smooth. \(\square \)
Definition 2.10
Let X be a bounded operator on \(\mathcal {H}\). The operator X is said to be analytic with respect to E, if the mapping \(\mathbf {k}\mapsto \alpha _\mathbf {k}(X)\) can be extended to a holomorphic mapping defined on a neighborhood
$$\begin{aligned} \{(\kappa _1, \cdots , \kappa _d) \in \mathbb {C}^d \ |\ - \delta< \mathrm {Im}(\kappa _i) < \delta \mathrm {\ for\ } 1 \le i \le d\} \end{aligned}$$
of \(\mathbb {R}^d\), where \(\delta \) is some positive number.
Note that every analytic operator is smooth.
Definition 2.11
A d-dimensional quantum walk \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \) is said to be uniform (smooth, analytic), if U is uniform (smooth, analytic) with respect to E.
We note that analyticity on quantum walks is so weak that the class contains all the multi-dimensional quantum walks which have been studied.
Example 2.12
Let j be in \(\{1, \cdots , d\}\). Let \(S_j\) be the unitary operator on \(\ell _2(\mathbb {Z}^d)\) given by the shift as in Example 2.5. The action \(\alpha _\mathbf {k}(S_j)\) of \(\mathbf {k}= (k_1, \cdots , k_d) \in \mathbb {R}^d\) is given by Eq. (3), \(\alpha _\mathbf {k}(S_j) = \exp (\mathbf {i}k_j) S_j\). This action can be extended to \(\mathbf {k}= (k_1, \cdots , k_d) \in \mathbb {C}^d\) by \(\mathbf {k}\mapsto \exp (\mathbf {i}k_j) S_j\). This map is holomorphic. It follows that \(S_j\) is analytic. In particular, the bilateral shift S on \(\ell (\mathbb {Z})\) is analytic. Therefore, the most famous 1-dimensional quantum walk presented in Eq. ( 1) is also analytic.

2.3 Similarity between multi-dimensional QWs

Let \(\mathcal {H}_1\) and \(\mathcal {H}_2\) be Hilbert spaces. For \(i = 1, 2\), let \(E_i\) be a Borel measure on \(\mathbb {R}^d\) whose values are orthogonal projections in \(\mathcal {B}(\mathcal {H}_i)\). Define a new projection-valued measure \(E_1 \oplus E_2\) by
$$\begin{aligned} E_1 \oplus E_2(\Omega ) = E_1(\Omega ) \oplus E_2(\Omega ), \quad \mathrm{\ a \ Borel \ subset \ } \Omega \subset \mathbb {R}^d \end{aligned}$$
A bounded operator \(X :\mathcal {H}_1 \rightarrow \mathcal {H}_2\) is said to be smooth with respect to \(E_1\) and \(E_2\), if \( \left( \begin{array}{cc} 0 &{} 0\\ X &{} 0 \end{array} \right) :\mathcal {H}_1 \oplus \mathcal {H}_2 \rightarrow \mathcal {H}_1 \oplus \mathcal {H}_2 \) is smooth with respect to \(E_1 \oplus E_2\).
Definition 2.13
Two d-dimensional quantum walks \((\mathcal {H}_1, (U_1^t)_{t \in \mathbb {Z}}, E_1)\), \((\mathcal {H}_2, (U_2^t)_{t \in \mathbb {Z}}, E_2)\) are said to be similar, if there exists a unitary operator \(V :\mathcal {H}_1 \rightarrow \mathcal {H}_2\) satisfying the following conditions 2 :
  • \(V U_1 = U_2 V\),
  • V is smooth with respect to \(E_1\) and \(E_2\).
Since \(V^{-1}\) is equal to \(V^*\), by Lemma 2.9, similarity between quantum walks is a reflexive relation. To see similarity between quantum walks is transitive, assume that \(V_1\) is a smooth unitary intertwiner from a quantum walk \(U_1\) to a quantum walk \(U_2\), and that \(V_2\) is a smooth unitary intertwiner from a \(U_2\) to a quantum walk \(U_3\). Since we have
$$\begin{aligned} V_2 V_1 U_1 = V_2 U_2 V_1 = U_3 V_2 V_1, \end{aligned}$$
\(V_2 V_1\) is a unitary intertwiner from \(U_1\) to \(U_3\). Again by Lemma 2.9, the unitary \(V_2 V_1\) is also smooth. It follows that \(U_1\) and \(U_3\) are similar. Therefore, similarity is a equivalence relation. If two quantum walks are similar and if one of the walks is smooth, then the other walk is also smooth.
Remark 2.14
As in Proposition 3.12, if two quantum walks are similar, and if one of them has a limit distribution, then the other walk also has a limit distribution. Then, the limit is the same as that of the other. For the study of limit distributions of a quantum walk U, we can replace it with another walk which is similar to U. For a Hilbert space \(\mathcal {H}\) with a d-dimensional coordinate system E, we can modify the information of position in the following sense.
Example 2.15
Let \(X \subset \mathbb {R}^d\) is a discrete subset. Consider the case that for every \(x \in X_1\), a Hilbert space \(\mathcal {H}_x\) is given. Define a Hilbert space \(\mathcal {H}_X\) by \(\oplus _{x \in X} \mathcal {H}_x\). Let \(E_X\) be the spectral measure on \(\mathbb {R}^d\) defined by the following: For \(\Omega \subset \mathbb {R}^d\), \(E_X(\Omega )\) is the orthogonal projection from \(\mathcal {H}_X\) to \(\oplus _{x \in \Omega \cap X} \mathcal {H}_x\).
Let \(f :X \rightarrow \mathbb {R}^n\) be a map satisfying that there exists a constant \(0 < R\) such that for every \(x \in X\), the distance between x and f( x) is at most R. The map f stands for the modification of position. Define Y by the image f( X). For \(y \in Y\), define \(\mathcal {K}_y\) by \(\oplus _{x \in f^{-1}(y)} \mathcal {H}_x\). Define \(\mathcal {K}_Y\) by \(\oplus _{y \in Y} \mathcal {K}_y\). Let \(E_Y\) be the spectral measure on \(\mathbb {R}^d\) defined by the following: For \(\Omega \subset \mathbb {R}^d\), \(E_Y(\Omega )\) is the orthogonal projection from \(\mathcal {K}_Y\) to \(\oplus _{y \in \Omega \cap Y} \mathcal {K}_y\).
Define a unitary V from \(\mathcal {H}_X = \oplus _{x \in X} \mathcal {H}_x\) to \(\mathcal {K}_Y = \oplus _{y \in Y} \mathcal {K}_y = \oplus _{y \in Y} \oplus _{x \in f^{-1}(y)} \mathcal {H}_x\) by the direct sum of \(\mathrm {id}_{\mathcal {H}_x} :\mathcal {H}_x \rightarrow \mathcal {H}_x\). Then, the unitary V is smooth or more strongly analytic with respect to \(E_X\) and \(E_Y\).
Let U be a quantum walk acting on \((\mathcal {H}_X, E_X)\). Then, we also have a walk \(V U V^{-1}\) acting on \((\mathcal {K}_Y, E_Y)\). Since V is analytic (therefore smooth and uniform), the walk \(V U V^{-1}\) is analytic (smooth, or uniform), if and only if U has the same regularity. Proposition 3.12 shows that asymptotic behavior of \(V U V^{-1}\) is the same as U.
We can apply this example to every crystal lattice. Let d be 2 or 3. Assume that \(X \subset \mathbb {R}^{d}\) forms a crystal lattice. More precisely, there exists an additive subgroup \(G \subset \mathbb {R}^d\) which is isomorphic to \(\mathbb {Z}^d\) such that for every \(g \in G\), \(g + X = X\). We can chose some bounded fundamental domain \(\Xi \subset \mathbb {R}^d\) of the additive action of G on \(\mathbb {R}^d\); that is, the family \(\{g + \Xi \}_{g \in G}\) is disjoint and its union is \(\mathbb {R}^d\). The intersection of \(\Xi \) and X stands for the unit of the crystal structure. Choose a point \(y_0\) in \(\Xi \). Define \(Y \subset \mathbb {R}^d\) by \(\{ g + y_0 \ | \ g \in G\}\). Define \(f :X \rightarrow Y\) as follows: For every \(x \in (g + \Xi ) \cap X\), \(f(x) = g + y_0\). This map f stands for the modification of position.
For a quantum walk U on a Hilbert space associated to X, we can consider a quantum walk \(V U V^{-1}\) on a Hilbert space associated with Y which is similar to U. In this way, we reduce the study of quantum walks on arbitrary crystal lattice to those on the integer lattice \((G + y_0) \cong \mathbb {Z}^d\). \(\square \)

3 General theory for asymptotic behavior of quantum walks

3.1 Logarithmic derivatives and their asymptotic behavior

Definition 3.1
Let \(\left( U^t \right) _{t \in \mathbb {Z}}\) be a unitary representation of \(\mathbb {Z}\) on a Hilbert space \(\mathcal {H}\). A two-sided sequence \(\{c_t\}_{t \in \mathbb {Z}}\) of bounded operators on \(\mathcal {H}\) is called a 1-cocycle of \(\left( U^t \right) _{t \in \mathbb {Z}}\), if for every \(s, t \in \mathbb {Z}\), \(c_{s + t} = U^{-t} c_s U^t + c_t\).
Lemma 3.2
If \(\{c_t\}_{t \in \mathbb {Z}}\) is a 1-cocycle of \(\left( U^t \right) _{t \in \mathbb {Z}}\), then for every \(t \ge 1\), \(\Vert c_t \Vert \le t \Vert c_1\Vert \).
Proof
For every \(s, t \in \mathbb {Z}\), we have
$$\begin{aligned} \Vert c_{s + t} \Vert \le \Vert U^{-t} c_s U^t\Vert + \Vert c_t\Vert = \Vert c_s\Vert + \Vert c_t\Vert . \end{aligned}$$
(3)
Repeating this decomposition, we obtain \(\Vert c_t\Vert \le t \Vert c_1\Vert \). \(\square \)
Lemma 3.3
The limit \(\lim _{t \rightarrow \infty } \frac{\Vert c_t\Vert }{t}\) exists and is less than \(\Vert c_1 \Vert \).
Proof
By the inequality ( 3), the sequence \(\{\Vert c_t\Vert \}\) is subadditive. For every subadditive sequence \(\{\gamma _t\}_{t = 1}^\infty \) of real numbers, the sequence \(\left\{ \gamma _t / t \right\} \) converges to its infimum. \(\square \)
For the rest of this paper, the 1-cocycle associated with the observable of position plays a key role. Let \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \) be a d-dimensional smooth quantum walk. Let \(\mathbf {w}= (w_i)\) be a vector in \(\mathbb {R}^d\). Denote by D the self-adjoint operator
$$\begin{aligned} \sum _{i = 1}^d w_i D_i = \sum _{i = 1}^d \int _{(x_i) \in \mathbb {R}^d} w_i x_i E(d x_1 \cdots d x_d). \end{aligned}$$
Since \(U^t\) is smooth with respect to E, by Lemma 2.7, the commutator of \(U^t\) with \(\mathbf {i}D_i\) is bounded. It follows that the operator
$$\begin{aligned} U^{-t} D U^t - D = - \mathbf {i}\sum _{i = 1}^d U^{-t} w_i [\mathbf {i}D_i, U^t] \end{aligned}$$
(4)
uniquely defines a bounded operator \(c_t\) on \(\mathcal {H}\). We note that the sequence \(U^{-t} D U^t\) stands for the time evolution of the observable D in the Heisenberg representation. The two-sided sequence \(\{c_t\}_{t \in \mathbb {Z}}\) is a 1-cocycle of \(\left( U^t \right) _{t \in \mathbb {Z}}\). By Lemma 2.7, and by Eq. ( 4), the 1-cocycle is equal to 3
$$\begin{aligned} c_t = - \mathbf {i}\sum _i w_i U^{-t} \partial _i(U^t). \end{aligned}$$
(5)
Definition 3.4
The 1-cocycle \(\left\{ c_t = U^{-t} D U^t - D \right\} \) is called the logarithmic derivatives of the quantum walk \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \) with respect to the operator \(D = \sum _i w_i D_i\). The operator \(\left\{ c_t / t \right\} \) is called the average of logarithmic derivatives of the quantum walk.
In most cases, the norms of the logarithmic derivatives \(\left\{ \Vert c_t\Vert \right\} \) linearly increase. Asymptotic behavior of the average \(\left\{ c_t / t \right\} \) of the logarithmic derivatives is an important tool in the study of limit distribution of the quantum walk (Theorem 4.3). The following proposition means that if two walks are similar, then averages of logarithmic derivatives are similar.
Proposition 3.5
Let \((\mathcal {H}_1, (U_1^t)_{t \in \mathbb {Z}}, E_1)\) and \((\mathcal {H}_2, (U_2^t)_{t \in \mathbb {Z}}, E_2)\) be d-dimensional smooth quantum walks. Let \(D_i^{(1)}\) be the self-adjoint operator \(\int _{\mathbb {R}^r} x_i E_1(d \mathbf {x})\). Let \(D_i^{(2)}\) be the self-adjoint operator \(\int _{\mathbb {R}^r} x_i E_2(d \mathbf {x})\). Let \(V :\mathcal {H}_1 \rightarrow \mathcal {H}_2\) be a smooth unitary operator which gives similarity between \(U_1\) and \(U_2\). For \(i = 1, \cdots , d\), denote by \(\partial _i^{(1)}\) the i-th partial derivative with respect to \(E_1\) and denote by \(\partial _i^{(2)}\) the i-th partial derivative with respect to \(E_2\). Then for \(i = 1, \cdots , d\), the sequence
$$\begin{aligned} \left\{ \frac{U_2^{-t} \partial _i^{(2)} (U_2^t)}{\mathbf {i}t} - V \frac{U_1^{-t} \partial _i^{(1)} (U_1^t)}{\mathbf {i}t} V^{-1} \right\} _{t} \end{aligned}$$
converges to 0 in the norm topology.
Proof
By Lemma 2.7, the operator \(\frac{U_2^{-t} \partial _i^{(2)} (U_2^t)}{\mathbf {i}t}\) is equal to the closure of
$$\begin{aligned} \frac{V U_1^{-t} V^{-1} D_i^{(2)} V U_1^t V^{-1} - D_i^{(2)}}{t}. \end{aligned}$$
Since V is smooth with respect to \(E_1\) and \(E_2\), by Lemma 2.7 for \(E_1 \oplus E_2\), the operator \(D_i^{(2)} V - V D_i^{(1)}\) is bounded. This means that distance between \(V^{-1} D_i^{(2)} V\) and \(D_i^{(1)}\) is small. If t is large, the above operator is almost equal to
$$\begin{aligned} \frac{V U_1^{-t} D_i^{(1)} U_1^t V^{-1} - V D_i^{(1)} V^{-1}}{t} = \frac{V ( U_1^{-t} D_i^{(1)} U_1^t - D_i^{(1)}) V^{-1}}{t}. \end{aligned}$$
This is equal to the operator \(V \frac{U_1^{-t} \partial _i^{(1)} (U_1^t)}{\mathbf {i}t} V^{-1}\). \(\square \)

3.2 General theory for asymptotic behavior of the distribution \(p_t\)

For the quantum walk \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \), and for a unit vector \(\xi \in \mathcal {H}\), a sequence \(\{p_t\}_{t \in \mathbb {Z}}\) of Borel probability measures on \(\mathbb {R}^d\) is defined by
$$\begin{aligned} p_t(\Omega ) = \langle E(t \Omega ) U^t \xi , U^t \xi \rangle , \quad \ \mathrm{for \ every \ Borel\ subset\ } \Omega \subset \mathbb {R}^d. \end{aligned}$$
(6)
The unit vector \(\xi \in \mathcal {H}\) is called an initial vector. In many concrete examples of homogeneous quantum walks, the existence of the weak limit of \(\{p_t\}\) has already been studied. See [ 4, 5, 7], for example.
Example 3.6
To see what the measure \(p_t\) means, let us look at a quantum walk U acting on \(\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^n\). Take the spectral measure E as in Example 2.2. Express \(U^t \xi \in \ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^n\) by \((\Psi _t(\mathbf {x}))_{\mathbf {x}\in \mathbb {Z}^d}\), where \(\Psi _t(\mathbf {x})\) is a vector in \(\mathbb {C}^n\). By Eq. ( 6), for every subset \(\Omega \) of \(\mathbb {R}^d\), we have
$$\begin{aligned} p_t(\Omega ) = \sum _{\mathbf {x}\in t \Omega \cap \mathbb {Z}^d} \Vert \Psi _t(\mathbf {x})\Vert ^2 = \sum _{\mathbf {v}\in \Omega \cap \mathbb {Z}^d / t} \Vert \Psi _t(t \mathbf {v})\Vert ^2. \end{aligned}$$
Therefore, the measure \(p_t\) is a sum of scalar multiple of point masses \(\{\delta _\mathbf {v}\}_{\mathbf {v}\in \mathbb {Z}^d / t}\), and the coefficient of \(\delta _\mathbf {v}\) is \(\Vert \Psi _t(t \mathbf {v})\Vert ^2\). \(\square \)
Lemma 3.7
Let f be a bounded Borel function. The mean of f with respect to the measure \(p_t\) is given by
$$\begin{aligned} \int _{\mathbf {v}\in \mathbb {R}^d} f(\mathbf {v}) p_t(d \mathbf {v}) = \left\langle \int _{\mathbf {x}\in \mathbb {R}^d} f \left( \frac{\mathbf {x}}{t} \right) E(d \mathbf {x}) U^t \xi , U^t \xi \right\rangle . \end{aligned}$$
(7)
Proof
In the case that f is a definition function of a Borel subset of \(\mathbb {R}^d\), the above equation holds, by the definition of \(p_t\). It follows that for every Borel step function f, the above equation holds. For a general Borel function f, we have only to use a sequence \(f_n\) of Borel step functions which are uniformly close to f. \(\square \)
Definition 3.8
A vector \(\xi \) in \(\mathcal {H}\) is said to be smooth, if it is in the domain of \(D_{i(m)} D_{i(m -1)} \cdots D_{i(1)}\) for every natural number m and for every sequence \(\{i(1)\), \(\cdots \), \(i(m)\}\) of \(\{1, \cdots , d\}\).
If the quantum walk \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \) is smooth and if the initial unit vector \(\xi \) is smooth, then \(U^t \xi \) is also smooth. The proof is given by Lemma 2.7 and by the Leibniz rule: \(D_i(U^t \xi ) = -\mathbf {i}\partial _i(U^t) \xi + U^t (D_i \xi )\).
Lemma 3.9
Suppose that the quantum walk \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \) is smooth and that the vector \(\xi \) is smooth. Let t be an arbitrary natural number. Then, every polynomial function g on \(\mathbb {R}^d\) is integrable with respect to \(p_t\), and the integral is given by
$$\begin{aligned} \int _{\mathbf {v}\in \mathbb {R}^d} g(\mathbf {v}) p_t(d \mathbf {v}) = \left\langle \int _{\mathbf {x}\in \mathbb {R}^d} g \left( \frac{\mathbf {x}}{t} \right) E(d \mathbf {x}) U^t \xi , U^t \xi \right\rangle . \end{aligned}$$
Proof
We may assume that \(g(x_1, x_2 \cdots , x_d)\) is of the form \(x_{i(m)} x_{i(m -1)} \cdots x_{i(1)}\). In this case, we have
$$\begin{aligned} \int _{\mathbf {x}\in \mathbb {R}^d} g \left( \frac{\mathbf {x}}{t} \right) E(d \mathbf {x}) = \frac{D_{i(m)}}{t} \cdots \frac{D_{i(1)}}{t}. \end{aligned}$$
Since U and \(\xi \) are smooth with respect to E, \(U^t \xi \) is in the domain of the above self-adjoint operator. Therefore, the right-hand side is well defined. There exists a countable sum \(f(\mathbf {v})\) of scalar multiples of definition functions which is Borel and uniformly close to \(g(\mathbf {v})\). Then, the difference between
$$\begin{aligned} \int _{\mathbf {x}\in \mathbb {R}^d} g \left( \frac{\mathbf {x}}{t} \right) E(d \mathbf {x}) \sim \int _{\mathbf {x}\in \mathbb {R}^d} f \left( \frac{\mathbf {x}}{t} \right) E(d \mathbf {x}), \end{aligned}$$
is a bounded operator with small operator norm. Therefore, \(U^t \xi \) is in the domain of \(\int _{\mathbf {x}\in \mathbb {R}^d} f \left( \frac{\mathbf {x}}{t} \right) E(d \mathbf {x})\). For such f, it is easy to show that
$$\begin{aligned} \int _{\mathbf {v}\in \mathbb {R}^d} f(\mathbf {v}) p_t(d \mathbf {v}) = \left\langle \int _{\mathbf {x}\in \mathbb {R}^d} f \left( \frac{\mathbf {x}}{t} \right) E(d \mathbf {x}) U^t \xi , U^t \xi \right\rangle . \end{aligned}$$
Since f is uniformly close to g, the integral \(\int _{\mathbf {v}\in \mathbb {R}^d} f(\mathbf {v}) p_t(d \mathbf {v})\) is close to \(\int _{\mathbf {v}\in \mathbb {R}^d}\) \(g(\mathbf {v})\) \(p_t(d \mathbf {v})\). \(\square \)
Theorem 3.10
Suppose that the quantum walk \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \) is smooth and that the initial unit vector \(\xi \) is smooth. There exists a bounded subset K of \(\mathbb {R}^d\) such that \(\lim _{t \rightarrow \infty } p_t(K) = 1\).
Proof
Let i be an arbitrary element of \(\{1, \cdots , d\}\). Let m be an arbitrary natural number. Note that the equation
$$\begin{aligned} \int _{(v_1, \cdots , v_d) \in \mathbb {R}^d} v_i^m p_t(d v_1 \cdots d v_i \cdots v_d) = \left\langle \frac{D_i^m}{t^m} U^t \xi , U^t \xi \right\rangle \end{aligned}$$
holds, by Lemma 3.9.
The triplet \((\mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, D_i)\) is a one-dimensional quantum walk defined in [ 8, Definition 2.1]. Therefore, we can employ the theory of one-dimensional quantum walks developed in [ 8]. As proved in [ 8, Definition 2.27], for every \(m \in \mathbb {N}\),
$$\begin{aligned} \limsup _t \left| \left\langle \frac{D_i^m}{t^m} U^t \xi , U^t \xi \right\rangle \right| \le \Vert [D_i, U]\Vert ^m. \end{aligned}$$
Take and fix a positive number \(L_i\) larger than \(\Vert [D_i, U]\Vert \). The inequality
$$\begin{aligned} \limsup _t \left| \int _{(v_1, \cdots , v_d) \in \mathbb {R}^d} v_i^m p_t(d v_1 \cdots d v_i \cdots v_d) \right| \le \Vert [D_i, U]\Vert ^m. \end{aligned}$$
implies that \(\lim _t p_t(\mathbb {R}^{i - 1} \times [- L_i, L_i] \times \mathbb {R}^{d - i}) = 1\). We conclude that \(\lim _t p_t([- L_1, L_1] \times \cdots \times [- L_d, L_d]) = 1\). \(\square \)
Corollary 3.11
Suppose that the quantum walk \(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \) is smooth and that the initial unit vector \(\xi \) is smooth. Then, the following conditions are equivalent:
(1)
(Convergence in Law). There exists a Borel probability measure \(p_\infty \) on \(\mathbb {R}_d\) such that for every polynomial function g on \(\mathbb {R}^d\)
$$\begin{aligned} \lim _t \int _{\mathbf {v}\in \mathbb {R}^d} g(\mathbf {v}) p_t(d \mathbf {v}) = \int _{\mathbf {v}\in \mathbb {R}^d} g(\mathbf {v}) p_\infty (d \mathbf {v}). \end{aligned}$$
 
(2)
(Weak convergence). There exists a Borel probability measure \(p_\infty \) on \(\mathbb {R}_d\) such that for every bounded continuous function f on \(\mathbb {R}^d\)
$$\begin{aligned} \lim _t \int _{\mathbf {v}\in \mathbb {R}^d} f(\mathbf {v}) p_t(d \mathbf {v}) = \int _{\mathbf {v}\in \mathbb {R}^d} f(\mathbf {v}) p_\infty (d \mathbf {v}). \end{aligned}$$
 
If the above conditions hold, then these limit distributions coincide and their support is compact.
Proof
Let K be a compact subset of \(\mathbb {R}^d\) in Theorem 3.10. Let \(\epsilon \) be an arbitrary positive real number. For every polynomial function g on \(\mathbb {R}^d\), there exists a bounded continuous function f on \(\mathbb {R}^d\) such that \(|f(\mathbf {x}) - g(\mathbf {x})| < \epsilon \) for arbitrary \(\mathbf {x}\in K\). For every bounded continuous function f on \(\mathbb {R}^d\), there exists a polynomial function g on \(\mathbb {R}^d\) such that \(|g(\mathbf {x}) - f(\mathbf {x})| < \epsilon \) for arbitrary \(\mathbf {x}\in K\) by the theorem of Stone–Weierstrass. \(\square \)
Proposition 3.12
Let \((\mathcal {H}_1, (U_1^t)_{t \in \mathbb {Z}}, E_1)\), \((\mathcal {H}_2, (U_2^t)_{t \in \mathbb {Z}}, E_2)\) be two d-dimensional smooth quantum walks. Suppose that there exists a smooth unitary operator \(V :\mathcal {H}_1 \rightarrow \mathcal {H}_2\) which intertwines \(U_1\) and \(U_2\). Let \(\xi \) be a unit vector in \(\mathcal {H}_1\) which is smooth with respect to \(E_1\). Let \(p_t^{(1)}\) be the sequence of probability measures on \(\mathbb {R}^d\) given by \((\mathcal {H}_1, (U_1^t)_{t \in \mathbb {Z}}, E_1)\) and \(\xi \). Let \(p_t^{(2)}\) be the sequence of probability measures on \(\mathbb {R}^d\) given by \((\mathcal {H}_2, (U_2^t)_{t \in \mathbb {Z}}, E_2)\) and \(V \xi \). If \(p_t^{(1)}\) converges in law, then \(p_t^{(2)}\) also converges in law. Furthermore, these limit distributions coincide.
Proof
The proof is substantially the same as that of [ 8, Theorem 2.31]. \(\square \)

4 Convergence theorems on homogeneous QWs

4.1 Definition of homogeneous QWs

Definition 4.1
A quadruple \((\mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E, \rho )\) is called a d-dimensional homogeneous quantum walk, if the following conditions hold:
(1)
\(\left( \mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E \right) \) is a d-dimensional quantum walk.
 
(2)
\(\rho = (\rho (\mathbf {x}))_{\mathbf {x}\in \mathbb {Z}^d}\) is a unitary representation of the additive group \(\mathbb {Z}^d\) on \(\mathcal {H}\).
 
(3)
For every Borel subset \(\Omega \subset \mathbb {R}^d\), \(\rho (\mathbf {x})^{-1} E(\Omega ) \rho (\mathbf {x}) = E(\Omega + \mathbf {x})\).
 
(4)
For every \(\mathbf {x}\in \mathbb {Z}^d\), \(U \rho (\mathbf {x}) = \rho (\mathbf {x}) U\).
 
(5)
The rank of \(E \left( [0, 1)^d \right) \) is finite.
 
The rank of \(E \left( [0, 1)^d \right) \) is called the degree of freedom.
We note that the image of \(E \left( [0, 1)^d \right) \) is the fundamental domain of the action by \((\rho (\mathbf {x}))_{\mathbf {x}\in \mathbb {Z}^d}\). More precisely, \(\{\rho (\mathbf {x}) (\mathrm {image}(E([0, 1)^d))) \}_{\mathbf {x}\in \mathbb {Z}^d}\) is mutually orthogonal and generates \(\mathcal {H}\). We may consider a quadruple satisfying conditions (1), (2), (3), (4) which does not satisfy condition (5). In such a case, we call the quadruple a homogeneous quantum walk with infinite degree of freedom.
Let \((\mathcal {H}, \left( U^t \right) _{t \in \mathbb {Z}}, E, \rho )\) be a homogeneous quantum walk with finite degree of freedom. We denote by n the degree of freedom. There exists a unitary operator
$$\begin{aligned} V :\mathcal {H}\rightarrow \ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^n \end{aligned}$$
satisfying that V maps \(\mathrm {image}(E([0, 1)^n))\) onto \(\delta _{\mathbf {0}} \otimes \mathbb {C}^n\) and that V is compatible with \(\rho \) and with the right regular representation \(\widetilde{\rho }\) on \(\ell _2(\mathbb {Z}^d)\). Let \(\widetilde{E}\) be the projection-valued measure on \(\mathbb {R}^d\) defined by
$$\begin{aligned} \widetilde{E}(\Omega ) = (\text {the orthogonal projection } \ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^n \rightarrow \ell _2(\mathbb {Z}^d \cap \Omega ) \otimes \mathbb {C}^n ). \end{aligned}$$
It is easy to show that V is smooth with respect to E and \(\widetilde{E}\). In fact, V is analytic with respect to E and \(\widetilde{E}\). Thus, we obtain a new homogeneous quantum walk
$$\begin{aligned} \left( \ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^n, (V U^t V^{-1})_{t \in \mathbb {Z}}, \widetilde{E}, \widetilde{\rho } \right) , \end{aligned}$$
which is similar to the original walk in the sense of Definition 2.13. If the original walk is smooth, then new one is also smooth. If the original walk is analytic, then new one is also analytic. By Proposition 3.12, if the latter walk has limit distribution (as in Theorem 4.3), the original walk has the same limit distribution.
For the rest of this paper, we study d-dimensional analytic homogeneous quantum walks. Without loss of generality, we may concentrate on the homogeneous walks of the form \((\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^n, \left( U^t \right) _{t \in \mathbb {Z}}, \widetilde{E}, \widetilde{\rho })\).

4.2 The proof of the convergence theorem for homogeneous QWs

We often use the inverse Fourier transform \(\mathcal {F}^{-1} :\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^n \rightarrow L^2 \left( \mathbb {T}_{2 \pi }^d \right) \otimes \mathbb {C}^n\). We express the Pontryagin dual \(\mathbb {T}_{2 \pi }^d\) of \(\mathbb {Z}^d\) by \(\{(k_1, \cdots , k_d) \ |\ k_1, \cdots , k_d \in \mathbb {R}/ (2 \pi \mathbb {Z})\}\). Via the inverse Fourier transform \(\mathcal {F}^{-1}\),
  • The analytic unitary operator U corresponds to a \((d \times d)-\)matrix \(\widehat{U}\) whose entries are analytic functions on \(\mathbb {T}_{2 \pi }^d\),
  • and the diagonal operator
    $$\begin{aligned} D_i = \int _{\mathbf {x}\in \mathbb {R}^d} x_i \widetilde{E}(d x_1 \cdots d x_i \cdots d x_d) :\delta _{\mathbf {x}} \otimes \delta _y \mapsto x_i \delta _{\mathbf {x}} \otimes \delta _y \end{aligned}$$
    corresponds to the partial differential operator \(- \mathbf {i}\frac{\partial }{\partial k_i}\).
Our main result (Theorem 4.3) relies on the following proposition.
Proposition 4.2
Let \(\left( U^t \right) _{t \in \mathbb {Z}}\) be a homogeneous analytic quantum walk acting on \(\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^n\). Let \(w_1, \cdots , w_d\) be real numbers. Denote by D the operator \(\sum _i w_i D_i\).
(1)
As t tends to infinity, the average \(c_t / t = (U^{-t} D U^t - D) / t\) of the logarithmic derivative of \(\left( U^t \right) _{t \in \mathbb {Z}}\) converges to some self-adjoint operator H in the strong operator topology.
 
(2)
Then, as t tends to infinity, the unitary operators defined by the commutators
$$\begin{aligned} U^{-t} \exp \left( \mathbf {i}\frac{D}{t} \right) U^t \exp \left( - \mathbf {i}\frac{D}{t} \right) \end{aligned}$$
converge to the unitary operator \(\exp (\mathbf {i}H)\) in the strong operator topology.
 
For the case that the walk is one-dimensional \((d = 1)\), the above two sequences converge in norm, and the limits are analytic operators.
To employ the analytic perturbation theory by Tosio Kato, we suppose analyticity. We denote by \(\mathrm {diag}(\alpha _i)_i\) the diagonal matrix whose diagonal entries are \(\alpha _1\), \(\cdots \), \(\alpha _n\).
Proof
We make use of the Fourier transform \(\widehat{U}(\mathbf {k}), \mathbf {k}\in \mathbb {T}_{2 \pi }^d\) of the walk. The operator D corresponds to the operator \(\widehat{D} = - \mathbf {i}\sum _i w_i \frac{\partial }{\partial k_i}\). Denote by \(\mathbf {w}\) the real vector \((w_i)\). By Lemma 2.7, the average of the logarithmic derivative \(c_t / t = U^{-t} [D, U^t] / t\) corresponds to
$$\begin{aligned} \widehat{c_t} / t= \frac{1}{t} \widehat{U}^{-t} \left[ \widehat{D}, \widehat{U}^t \right] = \frac{1}{\mathbf {i}t} \widehat{U}^{-t} \sum _i w_i \frac{\partial \widehat{U}^t}{\partial k_i}. \end{aligned}$$
(8)
We make use of the directional derivative \(\frac{d}{d s} = \sum _i w_i \frac{\partial }{\partial k_i}\) along the curve of the form \(\mathbb {R}\ni s \mapsto \mathbf {k}_0 + s \mathbf {w}\in \mathbb {T}_{2 \pi }^d\). Choose an arbitrary base point \(\mathbf {k}_0 \in \mathbb {T}_{2 \pi }^d\). Let I be a closed interval in \(\mathbb {R}\) whose interior part includes 0. For a while, we study the behavior of \(\widehat{U}\) on the segment \(\mathbf {k}_0 + I \mathbf {w}\subset \mathbb {T}_{2 \pi }^d\). Making the interval I shorter, we may assume that the map
$$\begin{aligned} I \ni s \mapsto \mathbf {k}_0 + s \mathbf {w}\in \mathbb {T}_{2 \pi }^d. \end{aligned}$$
is injective. Denote by \(\widehat{u}(s)\) the unitary \(\widehat{U}(\mathbf {k}_0 + s \mathbf {w})\). Note that the map \(I \ni s \mapsto \widehat{u}(s)\) can be extended to a holomorphic map defined on a complex domain including I. By Eq. ( 8), The average of the logarithmic derivative \(\widehat{c_t} / t\) is equal to
$$\begin{aligned} \frac{\widehat{c_t}(\mathbf {k}_0 + s \mathbf {w})}{t} = \frac{1}{\mathbf {i}t} \widehat{u}^{-t}(s) \frac{d }{d s} (\widehat{u}^t(s)). \end{aligned}$$
(9)
Let us make use of the analytic perturbation theory by T. Kato. The unitary \(\widehat{u}(s)\) can be decomposed as follows
$$\begin{aligned} \widehat{u}(s) = \widehat{v}(s) \cdot \mathrm {diag}(\lambda _1(s), \cdots , \lambda _n(s)) \cdot \widehat{v}(s)^{-1}, \end{aligned}$$
(10)
by [ 9, Theorem II.1.8 and II.1.10]. See also [ 9, Section II.4.6]. Here, \(\widehat{v}\) is an analytic map to invertible matrices, and the eigenvalue functions \(\lambda _1\), \(\cdots \), \(\lambda _n\) are holomorphic functions defined on a complex domain including I.
A direct calculation yields that
$$\begin{aligned} \frac{d}{d s} (\widehat{u}(s)^t)= & {} \frac{d \widehat{v}}{d s}(s) \cdot \mathrm {diag} \left( \lambda _i(s)^t \right) \cdot \widehat{v}(s)^{-1} \\&+\, \widehat{v}(s) \cdot \mathrm {diag} \left( \dfrac{d}{d s} (\lambda _i(s)^t) \right) \cdot \widehat{v}(s)^{-1} \\&-\, \widehat{v}(s) \cdot \mathrm {diag}(\lambda _i(s)^t) \cdot \widehat{v}(s)^{-1} \cdot \frac{d \widehat{v}}{d s}(s) \cdot \widehat{v}(s)^{-1}. \end{aligned}$$
The second term is equal to
$$\begin{aligned} t \cdot \widehat{v}(s) \cdot \mathrm {diag} \left( \lambda _i(s)^{t - 1} \cdot \dfrac{d \lambda _i}{d s}(s) \right) \cdot \widehat{v}(s)^{-1}. \end{aligned}$$
As t tends to infinity, the norm increases linearly. The norms of the first and the third terms are bounded. It follows that for the calculation of the average with respect to time, it suffices to see the second term. By Eq. ( 9), we have
$$\begin{aligned}&\lim _{t \rightarrow \infty } \frac{\widehat{c_t}(\mathbf {k}_0 + s \mathbf {w})}{t}\nonumber \\&\quad = \widehat{v}(s) \cdot \mathrm {diag}(\lambda _i(s)^t) \cdot \widehat{v}(s)^{-1} \cdot \widehat{v}(s) \cdot \mathrm {diag} \left( \lambda _i(s)^{t - 1} \cdot \dfrac{d \lambda _i}{d s}(s) \right) \cdot \widehat{v}(s)^{-1}\nonumber \\&\quad =\widehat{v}(s) \cdot \mathrm {diag} \left( \lambda _i(s)^{- 1} \cdot \dfrac{d \lambda _i}{d s}(s) \right) \cdot \widehat{v}(s)^{-1}. \end{aligned}$$
(11)
The convergence is uniform on the closed interval I. Because \(|\lambda _i(s)|^2 = 1\), \(\lambda _i(s)^{- 1} \cdot \dfrac{d \lambda _i}{d s}(s)\) is in \(\mathbf {i}\mathbb {R}\), as t tends to infinity, the average of logarithmic derivatives \(\widehat{c_t}(\mathbf {k}_0) / t\) converges to a skew self-adjoint matrix \(\mathbf {i}\widehat{H}(\mathbf {k}_0)\), at each point \(\mathbf {k}_0 \in \mathbb {T}_{2 \pi }^d\). The operator norm is uniformly bounded by \(\left\| \sum _i w_i \frac{\partial \widehat{U}}{\partial k_i} \right\| \) due to Lemma 3.2. This upper bound is independent of \(\mathbf {k}_0\). Therefore, the convergence of \(\widehat{c_t}(\mathbf {k}_0) / t\) at each point \(\mathbf {k}_0\) yields that in the strong operator topology. Applying the Fourier transform, we obtain the first assertion.
Consider the case that d is one. Then, the convergence on each closed interval in \(\mathbb {T}_{2 \pi }\) is uniform, and the limit is given by a matrix-valued analytic function as in Eq. ( 11). Since the one-dimensional torus \(\mathbb {T}_{2 \pi }\) is a union of two segments, the average of logarithmic derivative converges in norm and the limit is analytic.
Let us consider general d again. The inverse Fourier transform of \(U^{-t} \exp \left( \mathbf {i}\frac{D}{t} \right) U^t\) \(\exp \left( - \mathbf {i}\frac{D}{t} \right) \) is
$$\begin{aligned} \widehat{U}^{-t} \exp \left( t^{-1} \sum _i w_i \frac{\partial }{\partial k_i} \right) \widehat{U}^t \exp \left( - t^{-1} \sum _i w_i \frac{\partial }{\partial k_i} \right) . \end{aligned}$$
For every \(\mathbb {C}^n\)-valued analytic functions \(\widehat{\xi }(\mathbf {k})\) defined on \(\mathbb {T}_{2 \pi }^d\), we calculate
$$\begin{aligned} \widehat{U}^{-t} \exp \left( t^{-1} \sum _i w_i \frac{\partial }{\partial k_i} \right) \widehat{U}^t \exp \left( - t^{-1} \sum _i w_i \frac{\partial }{\partial k_i} \right) \widehat{\xi } \end{aligned}$$
as follows. The \(\mathbb {C}^n\)-valued function \(\exp \left( - t^{-1} \sum _i w_i \frac{\partial }{\partial k_i} \right) \widehat{\xi }\) is given by
$$\begin{aligned} \left[ \exp \left( - t^{-1} \sum _i w_i \frac{\partial }{\partial k_i} \right) \widehat{\xi } \right] (\mathbf {k}) = \sum _{m = 0}^\infty \frac{1}{m !} t^{-m} \left[ \left( \sum _i w_i \frac{\partial }{\partial k_i} \right) ^m \widehat{\xi } \right] (\mathbf {k}). \end{aligned}$$
This is the Taylor expansion of \(\xi (\mathbf {k}- t^{-1} \mathbf {w})\). Therefore, the unitary \(\exp \left( - t^{-1} \sum _i \frac{\partial }{\partial k_i} \right) \) acts on analytic vectors by the translation of \(t^{-1} \mathbf {w}\). Since analytic vectors are dense in the Hilbert space, the unitary \(\exp \left( - t^{-1} \sum _i \frac{\partial }{\partial k_i} \right) \) acts on every vector in \(L^2(\mathbb {T}_{2 \pi }^d) \otimes \mathbb {C}^n\) by the translation of \(t^{-1} \mathbf {w}\). By the same reason, the unitary \(\exp \left( t^{-1} \sum _i \frac{\partial }{\partial k_i} \right) \) means the translation by \(- t^{-1} \mathbf {w}\). It follows that for every \(\widehat{\xi } \in L^2 \left( \mathbb {T}_{2 \pi }^d \right) \otimes \mathbb {C}^n\), the following equation holds:
$$\begin{aligned}&\left[ \widehat{U}^{-t} \exp \left( t^{-1} \sum _i \frac{\partial }{\partial k_i} \right) \widehat{U}^t \exp \left( - t^{-1} \sum _i \frac{\partial }{\partial k_i} \right) \widehat{\xi } \right] (\mathbf {k})\\&\quad = \widehat{U}(\mathbf {k})^{-t} \cdot \widehat{U}(\mathbf {k}+ t^{-1} \mathbf {w})^t \cdot \widehat{\xi }(\mathbf {k}). \end{aligned}$$
Since the vector \(\widehat{\xi }\) in \(L^2(\mathbb {T}_{2 \pi }^d) \otimes \mathbb {C}^n\) is arbitrary, we obtain the following equation between two operators:
$$\begin{aligned}&\left[ \widehat{U}^{-t} \exp \left( t^{-1} \sum _i \frac{\partial }{\partial k_i} \right) \widehat{U}^t \exp \left( - t^{-1} \sum _i \frac{\partial }{\partial k_i} \right) \right] (\mathbf {k})\nonumber \\&\quad = \widehat{U}(\mathbf {k})^{-t} \cdot \widehat{U}(\mathbf {k}+ t^{-1} \mathbf {w})^t. \end{aligned}$$
(12)
We make use of the restriction \(\widehat{u}(s) = \widehat{U}(\mathbf {k}+ s \mathbf {w})\) on the segment \(\mathbf {k}_0 + I \mathbf {w}\subset \mathbb {T}_{2 \pi }^d\) and its diagonal decomposition \(\widehat{v}(s) \cdot \mathrm {diag}(\lambda _i(s))_i \cdot \widehat{v}(s)^{-1}\) as in Eq. ( 10). Assume that s is in I. Since I is simply connected, there exist real-valued analytic functions \(l_1\), \(l_2\), \(\cdots \), \(l_n\) such that \(\exp ( \mathbf {i}l_i(s)) = \lambda _i(s)\).
Formula ( 12) is equal to \(\widehat{u}(0)^{-t} \widehat{u}(t^{-1})^t\). If t is large, then the matrix \(\widehat{v}(t^{-1})\) is close to \(\widehat{v}(0)\) in norm. The unitary matrix \(\widehat{u}(0)^{-t} \widehat{u}(t^{-1})^t\) is uniformly close to
$$\begin{aligned}&\widehat{v}(0) \cdot \mathrm {diag}(\lambda _i(0)^{-t})_i \cdot \widehat{v}(0)^{-1} \cdot \widehat{v}(t^{-1}) \cdot \mathrm {diag}(\lambda _i(t^{-1})^t)_i \cdot \widehat{v}(t^{-1})^{-1}\\&\quad \sim \widehat{v}(0) \cdot \mathrm {diag}(\lambda _i(0)^{-t})_i \cdot \mathrm {diag}(\lambda _i(t^{-1})^t)_i \cdot \widehat{v}(t^{-1})^{-1}\\&\quad = \widehat{v}(0) \mathrm {diag}\left( \exp \left( \mathbf {i}\frac{l_i(t^{-1}) - l_i(0)}{t^{-1}} \right) \right) _i \widehat{v}(0)^{-1}. \end{aligned}$$
Note that the logarithms \(l_1(s)\), \(\cdots \), \(l_n(s)\) are differentiable, since \(\lambda _i(s)\) are analytic.
Therefore, we have
$$\begin{aligned} \lim _{t \rightarrow \infty } \widehat{u}(0)^{-t} \widehat{u}(t^{-1})^t= & {} \widehat{v}(0) \mathrm {diag}\left( \exp \left( \mathbf {i}\dfrac{d l_i}{d s}(0) \right) \right) _i \widehat{v}(0)^{-1} \\= & {} \widehat{v}(0) \mathrm {diag}\left( \exp \left( \lambda _i(0)^{-1} \dfrac{d \lambda _i}{d s}(0) \right) \right) _i \widehat{v}(0)^{-1}\\= & {} \exp \left( \widehat{v}(0) \mathrm {diag}\left( \lambda _i(0)^{-1} \dfrac{d \lambda _i}{d s}(0) \right) _i \widehat{v}(0)^{-1} \right) . \end{aligned}$$
By Eq. ( 11) and by the definition of the skew self-adjoint matrix \(\mathbf {i}\widehat{H}(\mathbf {k}_0)\), we have the following convergence at each point \(\mathbf {k}_0\):
$$\begin{aligned} \lim _{t \rightarrow \infty } \widehat{U}(\mathbf {k}_0)^{-t} \cdot \widehat{U}(\mathbf {k}_0 + t^{-1} \mathbf {w})^t= & {} \exp \left( \mathbf {i}\widehat{H}(\mathbf {k}_0) \right) . \end{aligned}$$
Because the sequence consists of unitary operators, the convergence at each point implies that in the strong operator topology. Applying the Fourier transform, we obtain the second assertion.
In the case that d is one, the convergence is that in the operator norm topology, since the convergence is uniform on each segment included in the torus. \(\square \)
In case that d is one, the following theorem is proved in [ 10].
Theorem 4.3
For any d-dimensional homogeneous analytic quantum walk with finite degree of freedom, and for any initial unit vector, the weak limit of probability measures \(\{p_t\}\) defined in Sect. 3.2 exists.
The limit distribution is described as follows. Let \((U^t)_{t \in \mathbb {Z}}\) be a homogeneous quantum walk acting on \(\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^n\). Let \(\xi \in \ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^n\) be an initial unit vector. By Proposition 4.2, the limit of the average of logarithmic derivatives
$$\begin{aligned}H_i = \lim _{t \rightarrow \infty } \frac{1}{\mathbf {i}t} U^{-t} \partial _i(U^t) \end{aligned}$$
exists and is a bounded self-adjoint operator. Since the self-adjoint operators \((D_1\), \(\cdots \), \(D_d)\) mutually commute, the self-adjoint operators \((H_1\), \(\cdots \), \(H_d)\) also mutually commute. There exists a unique projection-valued Borel probability measure \(\mathcal {E}\) on a compact subset of \(\mathbb {R}^d\) satisfying that for every integers \(m(1), \cdots , m(d) \in \mathbb {Z}_{\le 0}\),
$$\begin{aligned} \prod _i H_i^{m(i)} = \int _{\mathbf {v}\in \mathbb {R}^d} \prod _i v_i^{m(i)} \mathcal {E}(d \mathbf {v}). \end{aligned}$$
The following argument shows that the limit distribution of the walk is equal to \(p_\infty ( \ \cdot \ ) = \langle \mathcal {E}( \ \cdot \ ) \xi , \xi \rangle \).
Proof
We first assume that \(\xi \) is smooth. Via the inverse Fourier transform, the walk U corresponds to an \((n \times n)\)-matrix \(\widehat{U}\) whose entries are analytic functions on \(\mathbb {T}_{2 \pi }^d\). The initial unit vector \(\xi \) corresponds to a smooth element \(\widehat{\xi } \in L^2 \left( \mathbb {T}_{2 \pi }^d \right) \otimes \mathbb {C}^n\). For every \(i = 1, \cdots , d\), the self-adjoint operator \({\widehat{D}}_i\) is given by \(- \mathbf {i}\frac{\partial }{ \partial k_i}\). Define \({\widehat{D}}\) by \(- \mathbf {i}\sum _i w_i \frac{\partial }{\partial k_i}\). By Eq. ( 7) in Lemma 3.7, the mean of the function \(\exp (\mathbf {i}(\cdot , \mathbf {w})_{\mathbb {R}^d})\) on \(\mathbb {R}^d\) with respect to \(p_t\) is equal to the following inner product:
$$\begin{aligned}&\int _{\mathbf {v}\in \mathbb {R}^d} \exp (\mathbf {i}(\mathbf {v}, \mathbf {w})_{\mathbb {R}^d})p_t(d \mathbf {v})\\&\quad = \left\langle \exp \left( \mathbf {i}t^{-1} \widehat{D} \right) \widehat{U}^t \widehat{\xi }, \widehat{U}^t \widehat{\xi } \right\rangle \\&\quad = \left\langle \widehat{U}^{-t} \exp \left( \mathbf {i}t^{-1} \widehat{D} \right) \widehat{U}^t \exp \left( -\mathbf {i}t^{-1} \widehat{D} \right) \exp \left( \mathbf {i}t^{-1} \widehat{D} \right) \widehat{\xi }, \widehat{\xi } \right\rangle . \end{aligned}$$
Let \(\widehat{H}\) be the limit of the averages of logarithmic derivatives of \(\widehat{U}^t\) with respect to the differential operator \(- \mathbf {i}\sum w_i \frac{\partial }{\partial k_i}\). As in the proof of Proposition 4.2, define self-adjoint operators \(\widehat{H_i}\) by
$$\begin{aligned} - \mathbf {i}\lim _{t \rightarrow \infty } \widehat{U}(\mathbf {k})^{-t} \frac{\partial \widehat{U}^t}{\partial k_i}(\mathbf {k}), \end{aligned}$$
and \(\widehat{H}\) by \(\sum _{i=1}^d w_i \widehat{H_i}\). By Proposition 4.2, the operator
$$\begin{aligned} \widehat{U}^{-t} \exp \left( \mathbf {i}t^{-1} \widehat{D} \right) \widehat{U}^t \exp \left( -\mathbf {i}t^{-1} \widehat{D} \right) \end{aligned}$$
converges to \(\exp \left( \mathbf {i}\widehat{H} \right) \) in the strong operator topology. The vector \(\exp \left( \mathbf {i}t^{-1} \widehat{D} \right) \widehat{\xi }\) is given by \(\left[ \exp \left( \mathbf {i}t^{-1} \widehat{D} \right) \widehat{\xi } \right] (\mathbf {k}) = \widehat{\xi }(\mathbf {k}+ t^{-1} \mathbf {w})\). It is uniformly close to \(\widehat{\xi }(\mathbf {k})\). As t tends to infinity, the above integral converges to
$$\begin{aligned} \left\langle \exp \left( \mathbf {i}\widehat{H} \right) \widehat{\xi }, \widehat{\xi } \right\rangle _{L^2 \left( \mathbb {T}_{2 \pi }^d \right) \otimes \mathbb {C}^d} = \left\langle \prod _i \exp (\mathbf {i}w_i H_i) \xi , \xi \right\rangle . \end{aligned}$$
We obtain
$$\begin{aligned} \lim _{t \rightarrow \infty } \int _{\mathbf {v}\in \mathbb {R}^d} \exp (\mathbf {i}(\mathbf {v}, \mathbf {w})_{\mathbb {R}^d})p_t(d \mathbf {v})= & {} \left\langle \prod _i \exp (\mathbf {i}w_i H_i) \xi , \xi \right\rangle \\= & {} \left\langle \int _{\mathbf {v}\in \mathbb {R}^d} \exp (\mathbf {i}(\mathbf {v}, \mathbf {w})_{\mathbb {R}^d}) \mathcal {E}(d \mathbf {v}) \xi , \xi \right\rangle . \end{aligned}$$
We obtain that for every linear combination g of \(\{ \exp (\mathbf {i}(\cdot , \mathbf {w})_{\mathbb {R}^d}) \ |\ \mathbf {w}\in \mathbb {R}^d \}\),
$$\begin{aligned}\lim _{t \rightarrow \infty } \int _{\mathbf {v}\in \mathbb {R}^d} g(\mathbf {v}) p_t(d \mathbf {v}) = \left\langle \int _{\mathbf {v}\in \mathbb {R}^d} g(\mathbf {v}) \mathcal {E}(d \mathbf {v}) \xi , \xi \right\rangle .\end{aligned}$$
By Theorem 3.10, there exists a compact subset K of \(\mathbb {R}^d\) such that
$$\begin{aligned} \lim _{t \rightarrow \infty } p_t(K) = 1. \end{aligned}$$
The linear span \(\mathrm {span}\{ \exp (\mathbf {i}(\cdot , \mathbf {w})_{\mathbb {R}^d}) \ |\ \mathbf {w}\in \mathbb {R}^d \}\) is the space of trigonometric functions. By the theorem of Stone–Weierstrass, the linear span is dense in C( K) with respect to the supremum norm. It follows that for every bounded continuous function f on \(\mathbb {R}^d\),
$$\begin{aligned} \lim _{t \rightarrow \infty } \int _{\mathbf {v}\in \mathbb {R}^d} f(\mathbf {v}) p_t(d \mathbf {v}) = \left\langle \int _{\mathbf {v}\in \mathbb {R}^d} f(\mathbf {v}) \mathcal {E}(d \mathbf {v}) \xi , \xi \right\rangle . \end{aligned}$$
In the special case that \(\xi \) is smooth, we finish the proof.
Let \(\widetilde{\xi }\) be an initial unit vector in \(\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^n = \ell _2(\mathbb {Z}^d \rightarrow \mathbb {C}^n)\) which is not necessarily smooth. Let \(\widetilde{p_t}\) be the sequence of probability measures defined by \(U^t\) and \(\widetilde{\xi }\). Let \(\epsilon \) be an arbitrary positive number. Choose a smooth unit vector \(\xi \in \ell _2(\mathbb {Z}^d \rightarrow \mathbb {C}^n)\) satisfying that \(\left\| \xi - \widetilde{\xi } \right\| < \epsilon \). For every element \(\eta \) of \(\ell _2(\mathbb {Z}^d \rightarrow \mathbb {C}^n)\), define an \(\ell _1\) function \(|\eta |^2\) on \(\mathbb {Z}^d\) by
$$\begin{aligned} \left| \eta \right| ^2(\mathbf {x}) = \left\| \eta (\mathbf {x}) \right\| _{\mathbb {C}^n}^2, \end{aligned}$$
By the inequality \(\left\| U^t \xi - U^t \widetilde{\xi } \right\| < \epsilon \), we have \(\left\| |U^t \xi |^2 - \left| U^t \widetilde{\xi } \right| ^2 \right\| _{\ell _1} < 2 \epsilon \). By Eq. ( 7), for every bounded Borel function f on \(\mathbb {R}^d\), we have
$$\begin{aligned} \left| \int _{\mathbf {v}\in \mathbb {R}^d} f(\mathbf {v}) p_t(d \mathbf {v}) - \int _{\mathbf {v}\in \mathbb {R}^d} f(\mathbf {v}) \widetilde{p_t} (d \mathbf {v}) \right|\le & {} \sup _{\mathbf {v}\in \mathbb {R}^d} |f(\mathbf {v})| \cdot \left\| |U^t \xi |^2 - \left| U^t \widetilde{\xi } \right| ^2 \right\| _{\ell _1} \\\le & {} 2 \epsilon \sup _{\mathbf {v}\in \mathbb {R}^d} |f(\mathbf {v})|. \end{aligned}$$
We also obtain
$$\begin{aligned}&\left| \left\langle \int _{\mathbf {v}\in \mathbb {R}^d} f(\mathbf {v}) \mathcal {E}(d \mathbf {v}) \xi , \xi \right\rangle - \left\langle \int _{\mathbf {v}\in \mathbb {R}^d} f(\mathbf {v}) \mathcal {E}(d \mathbf {v}) \widetilde{\xi }, \widetilde{\xi } \right\rangle \right| \\&\quad \le 2 \left\| \int _{\mathbf {v}\in \mathbb {R}^d} f(\mathbf {v}) \mathcal {E}(d \mathbf {v}) \right\| \left\| \xi - \widetilde{\xi } \right\| \le 2 \epsilon \sup _{\mathbf {v}\in \mathbb {R}^d} |f(\mathbf {v})|. \end{aligned}$$
It follows that for every bounded continuous function f on \(\mathbb {R}^d\),
$$\begin{aligned} \lim _{t \rightarrow \infty } \int _{\mathbf {v}\in \mathbb {R}^d} f(\mathbf {v}) \widetilde{p_t}(d \mathbf {v}) = \left\langle \int _{\mathbf {v}\in \mathbb {R}^d} f(\mathbf {v}) \mathcal {E}(d \mathbf {v}) \widetilde{\xi }, \widetilde{\xi } \right\rangle = \int _{\mathbf {v}\in \mathbb {R}^d} f(\mathbf {v}) \left\langle \mathcal {E}(d \mathbf {v}) \widetilde{\xi }, \widetilde{\xi } \right\rangle . \end{aligned}$$
It follows that the sequence of probability measures \(\left\{ \widetilde{p_t} \right\} \) weakly converges. \(\square \)

4.3 A quantum walk with an initial unit vector whose support is not localized

The convergence theorem (Theorem 4.3) holds true for an arbitrary initial unit vector. Let us consider an example, in which the support of the initial unit vector \(\xi \) is the whole space \(\mathbb {Z}\). We define U by the 3-state Grover walk
$$\begin{aligned} U = \left( \begin{array}{ccc} S &{}\quad 0 &{}\quad 0\\ 0 &{}\quad 1 &{}\quad 0\\ 0 &{}\quad 0 &{}\quad S^{-1} \end{array} \right) \cdot \frac{1}{3} \left( \begin{array}{rrr} -1 &{} 2 &{} 2\\ 2 &{} -1 &{} 2\\ 2 &{} 2 &{} -1 \end{array} \right) \end{aligned}$$
acting on \(\ell _2(\mathbb {Z}) \otimes \mathbb {C}^3\). We set the initial unit vector \(\xi \) by the infinite sum
$$\begin{aligned} \xi = \frac{1}{\sqrt{3}} \sum _{x \in \mathbb {Z}} \delta _x \otimes \left( \begin{array}{c} 0\\ 2^{- |x| / 2} \\ 0 \end{array} \right) . \end{aligned}$$
When we regard \(\xi \) as a map from \(\mathbb {Z}\) to \(\mathbb {C}^3\), the support of \(\xi \) is \(\mathbb {Z}\). However, according to Theorem 3.10, the limit distribution of velocity \(\{p_t\}_{t = 1}^{\infty }\) defined in 3.2 should have compact support. To see what happens, let us look at the following results of numerical calculations.
Figure  1 shows the probability of position in \(\mathbb {Z}\) according to the quantum state \(\xi \). The left end stands for \(x = -4\), and the right end stands for \(x = 4\). Since the support of \(\xi :\mathbb {Z}\rightarrow \mathbb {C}^3\) is \(\mathbb {Z}\), the probability never vanishes.
Figure  2 shows the probability distribution on \(\frac{1}{5}\mathbb {Z}\) defined by the unit vector \(U^5 \xi \). The left end stands for \(x = -3\), and the right end stands for \(x = 3\). The tails on the both sides become closer to the level of 0.
Figure  3 shows the probability distribution on \(\frac{1}{100}\mathbb {Z}\) defined by the unit vector \(U^{100} \xi \). The left end stands for \(x = -1\), and the left end stands for \(x = 1\). Although the support of \(U^{100} \xi :\mathbb {Z}\rightarrow \mathbb {C}^3\) is not supported on a compact set, the distribution is very close to that with compact support and the difference is invisible. The probability on the interval \([- 0.7, 0.7]\) is more than 0.999.

5 Interpretation of this paper from the view point of quantum physics

Many researchers have intensively studied quantum walks acting on the Hilbert space \(\ell _2(\mathbb {Z}^d) \otimes \mathbb {C}^n\). The natural number d stands for the dimension of the space. The natural number n stands for the local degree of freedom at a point in \(\mathbb {Z}^d\). This paper enables us to generalize the framework of the theory of quantum walks. In the following, we see how our definitions and theorems in this paper enlarge the theory of quantum walks.
General n In the present study of quantum walks, the local degree of freedom is restricted to small natural numbers n such as 2. This is because researchers consider concrete physical observable such as the quantum spin. This observable can be described as a self-adjoint matrix acting on \(\mathbb {C}^2\). In our new framework described in Definition 2.1, we can consider large finite-dimensional local Hilbert space \(\mathbb {C}^n\). The results in Sect. 3 can be applied to the case that the local Hilbert spaces are infinite dimensional.
General d The space \(\mathbb {Z}^d\) has a physical meaning in the case of \(d = 1, 2, 3\). We no longer have to divide our studies into these three cases.
Quantum walks on more general spaces It is no longer necessary to stick to the concrete lattice \(\mathbb {Z}^d\) inside \(\mathbb {R}^3\). Let us consider the case that some devices or atoms are located on some discrete subset \(X \subset \mathbb {R}^d\). In this paragraph, we put no requirement related to symmetry on X. Let us consider some Hilbert spaces \(\mathcal {H}_x\) are attached to every point \(x \in X\), where \(\mathcal {H}_x\) may be infinite dimensional. Here, unit vectors in \(\mathcal {H}_x\) describe quantum states at \(x \in X\). The whole Hilbert space \(\mathcal {H}\) is defined by the direct sum of Hilbert space \(\oplus _{x \in X} \mathcal {H}_x\), and the spectral measure E is given by the orthogonal projections \(E(\Omega )\) from \(\mathcal {H}\) onto \(\oplus _{x \in \Omega \cap X} \mathcal {H}_x\). Therefore, our new framework encompasses dynamical systems on arbitrary solid structures given by atoms and on those by devices.
Quantum walks without finite propagation Almost all the known quantum walks have finite propagation. The term “ finite propagation” is defined as follows: The operator U on \(\mathcal {H}= \oplus _{x \in X} \mathcal {H}_x\) is said to have finite propagation, if there exists a positive number R such that for every \(x, y \in X\), if \(\mathrm {dist}(x, y) > R\), then \(U \mathcal {H}_x\) and \(\mathcal {H}_y\) are perpendicular. The meaning of this condition is that if xy are distant, and \(\xi \in \mathcal {H}\) is located at \(\mathcal {H}_x\) and \(\eta \in \mathcal {H}\) is located at \(\mathcal {H}_y\), then the transition probability \(|\langle U \xi , \eta \rangle |^2\) is zero. It is easy to show that having finite propagation implies analyticity defined in Definition 2.10. The converse does not hold. The theory in this paper uses analyticity or more mild conditions. Thus, we obtain a wider framework.
Quantum walks on arbitrary crystal lattices Preparing a general framework is not the only goal of this paper. Let us apply our mathematical argument to quantum walks on crystal lattices. For every crystal lattice \(X \subset \mathbb {R}^d\), there exists an additive subgroup \(G \subset \mathbb {R}^d\) which is isomorphic to \(\mathbb {Z}^d\) such that X is invariant under the addition by G. Here, d is 2 or 3. The Hilbert space \(\mathcal {H}= \oplus _{x \in X} \mathcal {H}_x\) also has translation symmetry under the action of G; in other words, for every \(g \in G\) and \(x \in X\), \(g + x \in X\) and \(\mathcal {H}_{g + x} = \mathcal {H}_x\). Let U be a unitary operator on \(\mathcal {H}\). The unitary operator satisfies a convergence theorem, if the following assumption holds:
  • All the local Hilbert spaces \(\mathcal {H}_x\) are finite dimensional.
  • The unitary operator U has the translation symmetry with respect to G. More precisely, if we denote by \(\rho (g)\) the unitary operator on \(\mathcal {H}\) given by the shift \(g \in G\), then \(U \rho (g) = \rho (g) U\).
  • The unitary operator U has finite propagation.
Almost all the known quantum walks with translation symmetry on crystal lattices satisfy these conditions.
Corollary 5.1
For any initial unit vector \(\xi \in \mathcal {H}\), and for the quantum walk \((\mathcal {H}, (U^t)_{t \in \mathbb {Z}}, E)\) on arbitrary crystal lattice satisfying above conditions, the weak limit of the distribution of velocity \(\{p_t\}\) defined in Sect. 3.2 exists.
Proof
A quantum walk with finite propagation is analytic. The quantum walk U is space-homogeneous with respect to the additive group \(G \cong \mathbb {Z}^d\). This corollary is a direct conclusion of Theorem 4.3. \(\square \)
Physical meaning of 1- cocycles and logarithmic derivatives of quantum walks Let A be an observable described by a self-adjoint operator acting on a dense subspace of \(\mathcal {H}= \oplus _{x \in X} \mathcal {H}_x\). Let \((U^t)_{t \in \mathbb {Z}}\) be a dynamical system acting on \(\mathcal {H}\). According to the Heisenberg representation of time evolution, the sequence \((U^{-t} A U^t)_{t \in \mathbb {Z}}\) stands for the evolution of the observable with time. The \(U^{-t} A U^t - A\) stands for the difference between two observables A and \(U^{-t} A U^t\), where the latter represents the observable after t. Let us consider the case that \(U^{-1} A U - A\) is bounded. In this case, the possible values of the observable \(U^{-1} A U - A\) are restricted to some closed interval. The family of bounded self-adjoint operators \((c_t)_{t \in \mathbb {Z}} = (U^{-t} A U^t - A)_{t \in \mathbb {Z}}\) forms a 1-cocycle introduced in Definition 3.1. Thus, using 1-cocycle, we can treat the difference of the observable A and that after t. Proposition 4.2 means that the average of the 1-cocycle \(\frac{1}{t} (U^{-t} A U^t - A)\) converges if the quantum walk is analytic and homogeneous. This can be applied to analytic quantum walks on crystal lattices with translation symmetry.
In our mathematical argument for the convergence theorem (Theorem 4.3), a 1-cocycle called the logarithmic derivative plays a key role.
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Footnotes
1
In fact, it is possible to construct a quantum walk whose eigenvalue function is not smooth. Because we need many pages for the construction, we omit. Our convergence theorem also works for such an example.
 
2
In this definition, we require that V is unitary. This condition can be relaxed. It is possible to replace the smooth unitary intertwiner V with a smooth invertible intertwiner. These definitions are equivalent. Because its proof is long, we omit the explanation.
 
3
The equation is similar to the right-hand side of the following formula in calculus: \((\log f(t))' = f(t)^{-1} f'(t)\). This is the motivation of the definition of the logarithmic derivative.
 
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