# Quantum Walk in Terms of Quantum Bernoulli Noise and Quantum Central Limit Theorem for Quantum Bernoulli Noise.

1. IntroductionQuantum walks are quantum analogs of the classical random walk, which have found wide application in quantum information, quantum computing, and many other fields [13]. In the past fifteen years, quantum walks with a finite number of internal degrees of freedom have been intensively studied and many deep results have been obtained of them (see, e.g., [2, 4-9] and references therein). One typical result in this aspect is the finding that those walks have quite different asymptotic behavior, compared to their classical counterparts. For example, Konno [10] proved that, for localized initial states, a discrete-time quantum walk on the line with a finite number of internal degrees of freedom usually has a limit distribution with scaling speed n, which is far from being Gaussian. Similar properties have also been found for continuous-time quantum walks with a finite number of internal degrees of freedom [11]. However, little attention has been paid to quantum walks with infinitely many internal degrees of freedom, which are of interest at least from a theoretical point of view.

Quantum Bernoulli noise is the family of annihilation and creation operators acting on Bernoulli functionals, which satisfies a canonical anticommutation relation (CAR) in equal time, and can be viewed as a discrete-time counterpart of the quantum white noise introduced by Huang [12]. Recently, with the help of quantum Bernoulli noise, Wang and Ye [13] have constructed a discrete-time quantum walk with infinitely many internal degrees of freedom, which we call the QBN-based walk below.

The QBN-based walk takes [l.sup.2](Z, H) as its state space, where H denotes the space of square integrable Bernoulli functionals, which is infinitely dimensional. It has been shown [13] that, for some localized initial states, the QBN-based walk has a Gaussian limit distribution with scaling speed [square root of (n)], which is in striking contrast with the case of the usual discrete-time quantum walks with a finite number of internal degrees of freedom. Machida [6] has found that, for a very particular nonlocalized initial state, a discrete-time quantum walk on the line with 2 internal degrees of freedom can generate a Gaussian limit distribution with scaling speed n. And he wondered [6] whether a discrete-time quantum walk can generate a Gaussian limit distribution with scaling speed [square root of (n)]. The QBN-based walk then seems to give an answer to this question in a way.

In this paper, we would like to further examine the structure property of the QBN-based walk and show its application to quantum probability. Our main work is as follows.

Let [{[[partial derivative].sub.k], [[partial derivative].sup.*.sub.k]}.sub.k[greater than or equal to]0] be quantum Bernoulli noise, namely, annihilation and creation operators on H, and X be the position operator in [l.sup.2](Z, H). In the first part of the present paper, by using the Fourier transform on [l.sup.2](Z, H), we obtain a representation of the QBN-based walk in the momentum space. In particular, we obtain the following relations:

[mathematical expression not reproducible] (1)

where [[PHI].sub.n] [member of] [l.sup.2](Z, H) denotes the state of the walk at time n [greater than or equal to] 0 and [phi] = [[PHI].sub.0](0). Since the quantity [mathematical expression not reproducible] is exactly the rth moment of the walk's probability distribution at time n [greater than or equal to] 0, the above relations actually provide a formula that links the moments of the walk's probability distributions directly with the annihilation and creation operators.

Quantum central limit theorems are quantum analogs of the classical central limit theorem, which deal with observables from a quantum probability point of view. Cushen and Hudson [14] established the quantum central limit theorem for a pair of conjugate observables P and Q (i.e., such that [P, Q] = iI), which was later generalized to arbitrary CCR algebras by Quaegebeur [15]. Giri and von Waldenfels [16] proved an algebraic quantum central limit theorem in the setting of *-algebra by using the method of noncommutative moments. Voiculescu has developed a noncommutative probability theory (now known as the free probability theory), which offers the free central limit theorem associated with the free independence [17]. There are many other types of quantum central limit theorems in the literature (see, e.g., [18-22] and references therein).

Obviously, operators [[partial derivative].sup.*.sub.k] + [[partial derivative].sub.k], k [greater than or equal to] 0, are observables on H, which serves as the coin space of the QBN-based walk. In fact, these operators just play the role of quantum bias in the construction of the QBN-based walk. In the second part of the present paper, as application of our results mentioned above, we prove a quantum central limit theorem for observables [[partial derivative].sup.*.sub.k] + [[partial derivative].sub.k], k [greater than or equal to] 0.

The paper is organized as follows. In Section 2, we briefly recall main notions and facts about quantum Bernoulli noise. Section 3 describes the quantum walk introduced by Wang and Ye [13], namely, the QBN-based walk. Our main work then lies in Sections 4 and 5. Finally in Section 6, we make some conclusion remarks.

Notation and Conventions. Throughout, Z always denotes the set of all integers, while N means the set of all nonnegative integers. We denote by r the finite power set of N; namely,

[GAMMA] = {[sigma] | [sigma] [subset] N, #[sigma] < [infinity]}, (2)

where #[sigma] means the cardinality of [sigma]. Unless otherwise stated, letters like j, k, and n stand for nonnegative integers, namely, elements of N.

2. Quantum Bernoulli Noise

In this section, we briefly recall main notions and facts about quantum Bernoulli noise. We refer to [13,23, 24] for details.

Let [OMEGA] = [{-1, 1}.sup.N] be the set of all mappings [omega] : N [??] {-1,1} and [([[zeta].sub.n]).sub.n[greater than or equal to]0] the sequence of canonical projections on [OMEGA] given by

[[zeta].sub.n] ([omega]) = [omega](n), [omega] [member of] [OMEGA]. (3)

Let F be the [sigma]-field on [OMEGA] generated by the sequence [([[zeta].sub.n]).sub.n[greater than or equal to]0] and [([p.sub.n]).sub.n[greater than or equal to]0] a given sequence of positive numbers with the property that 0 < [p.sub.n] < 1 for all n [greater than or equal to] 0. It is known [25] that there exists a unique probability measure P on F such that

[mathematical expression not reproducible] (4)

for [n.sub.j] [member of] N, [[epsilon].sub.j] [member of] {-1,1} (1 [less than or equal to] j [less than or equal to] k) with [n.sub.i] [not equal to] [n.sub.j] when i [not equal to] j and k [member of] N with k [greater than or equal to] 1. Thus one has a probability measure space ([OMEGA], F, P),which is referredto as the Bernoulli space and random variables on it are known as Bernoulli functionals.

Let Z = [([Z.sub.n]).sub.n[greater than or equal to]0] be the sequence of Bernoulli functionals defined by

[Z.sub.n] = [[zeta].sub.n] + [q.sub.n] - [p.sub.n]/2 [square root of ([p.sub.n][q.sub.n])], n [greater than or equal to] 0, (5)

where [q.sub.n] = 1 - [p.sub.n]. Clearly Z = [([Z.sub.n]).sub.n[greater than or equal to]0] is an independent sequence of random variables on the probability measure space ([OMEGA], F, P).

Let H be the space of square integrable complex-valued Bernoulli functionals; namely,

H = [L.sup.2] ([OMEGA], F, P). (6)

We denote by <x, x> the usual inner product of the space H and by [parallel] x [parallel] the corresponding norm. It is known [25] that Z has the chaotic representation property. Thus H has {[Z.sub.[sigma]] | [sigma] [member of] [GAMMA]} as its orthonormal basis, where [Z.sub.0] = 1 and

[mathematical expression not reproducible] (7)

which shows that H is an infinite dimensional complex Hilbert space.

Lemma 1 (see [23]). For each k [member of] N, there exists a bounded operator [[partial derivative].sub.k] on H such that

[[partial derivative].sub.k][Z.sub.[sigma]] = [1.sub.[sigma]] (k) [Z.sub.[sigma]\k], [sigma] [member of] [GAMMA], (8)

where [sigma] \ k = [sigma] \ {k} and [1.sub.[sigma]](k) is the indicator of [sigma] as a subset of N.

Lemma 2 (see [23]). Let k [member of] N. Then [[partial derivative].sup.*.sub.k] the adjoint of operator [[partial derivative].sub.k] has the following property:

[[partial derivative].sup.*.sub.k][Z.sub.[sigma]] = [1 - [1.sub.[sigma]] (k)] [Z.sub.[sigma][union]k] [sigma] [member of] [GAMMA], (9)

where [sigma] [union] k = [sigma] [union] {k}.

The operator [[partial derivative].sub.k] and its adjoint [[partial derivative].sup.*.sub.k] are usually known as the annihilation and creation operators acting on Bernoulli functionals, respectively.

Definition 3 (see [23]). The family [{[[partial derivative].sub.k], [[partial derivative].sup.*.sub.k]}.sub.k[greater than or equal to]0] of annihilation and creation operators is called quantum Bernoulli noise.

The next lemma shows that quantum Bernoulli noise satisfies the canonical anticommutation relations (CAR) in equal time.

Lemma 4 (see [23]). Let k, l [member of] N. Then it holds true that

[mathematical expression not reproducible] (10)

where I is the identity operator on H.

Lemma 5 (see [13]). For n [greater than or equal to] 0, write [R.sub.n] = (1/2)([[partial derivative].sup.*.sub.n] + [[partial derivative].sub.n] + I) and [L.sub.n] = (1/2)([[partial derivative].sup.*.sub.n] + [[partial derivative].sub.n] - I). Then both [R.sub.n] and [L.sub.n] are self-adjoint operators on H, and moreover

[R.sup.2.sub.n] = [R.sub.n],

[R.sub.n][L.sub.n] = [L.sub.n][R.sub.n] = 0,

[L.sup.2.sub.n] = -[L.sub.n]. (11)

It follows easily from Lemma 4 that operators [R.sub.n], [L.sub.n], n [greater than or equal to] 0, form a commutative family; namely,

[R.sub.k][R.sub.l] = [R.sub.l][R.sub.k],

[R.sub.k][L.sub.l] = [Lsub.l][R.sub.k],

[L.sub.k][L.sub.l] = [L.sub.l][L.sub.k] (12)

hold for all k, l [greater than or equal to] 0.

3. QBN-Based Walk

The present section describes the quantum walk introduced in [13], namely, the QBN-based walk mentioned above.

Recall that H = [L.sup.2]([OMEGA], F, P), the space of square integrable complex-valued Bernoulli functionals. Let [lsup.2](Z, H) be the space of square summable functions defined on Z and valued in H; namely,

[mathematical expression not reproducible] (13)

Then [l.sup.2](Z, H) remains a complex Hilbert space, whose inner product [mathematical expression not reproducible] is given by

[mathematical expression not reproducible] (14)

where <x, x> denotes the inner product of H as indicated in Section 2. By convention, we denote by [parallel]-[parallel]p the norm induced by [mathematical expression not reproducible]. Note that [l.sup.2](Z, H) has a countable orthonormal basis {[[phi].sub.z,[phi]] | z [member of] Z, [sigma] [member of] [GAMMA]}, where [[phi].sub.z,[phi]] : Z [right arrow] H is defined by

[mathematical expression not reproducible] (15)

Thus [l.sup.2] (Z, H) is separable.

As usual, a vector [PHI] [member of] [l.sup.2](Z, H) is called a state if it satisfies the normalized condition [mathematical expression not reproducible].

Definition 6 (see [13]). The QBN-based walk is such a quantum walk whose state space is [l.sup.2](Z, H) and whose time evolution is governed by

[[PHI].sub.n+1] (x) = [R.sub.n][[PHI].sub.n] (x - 1) + [L.sub.n] [[PHI].sub.n] (x + 1), x [member of] Z, n [greater than or equal to] 0, (16)

where [[PHI].sub.n] [member of] [l.sup.2](Z, H) denotes the state of the walk at time n [greater than or equal to] 0.

Let [([[PHI].sub.n]).sub.n[greater than or equal to]0] be the state sequence of the QBN-based walk. Then the function x [??] [[parallel][[PHI].sub.n](x)[parallel].sup.2] makes a probability distribution on Z, which is called the probability distribution of the walk at time n [greater than or equal to] 0. In particular, [[parallel][[PHI].sub.n](x)[parallel].sup.2] is the probability that the quantum walker is found at position x [member of] Z at time n [greater than or equal to] 0. As usual, the QBN-based walk is assumed to start at position x = 0, which implies that its initial state [[PHI].sub.0] satisfies [[parallel][[PHI].sub.0](0)[parallel].sup.2] = 1 and [PHI](x) = 0 for x [member of] Z with x [not equal to] 0.

Remark 7. It is well known that [l.sup.2](Z, H) [congruent to] [l.sup.2](Z) [cross product] H. Thus, [l.sup.2](Z) describes the position of the QBN-based walk, while H describes the internal degrees of freedom of the walk. As shown in Section 2, the dimension of H is infinite, which means that the QBN-based walk has infinitely many internal degrees of freedom.

Lemma 8 (see [13]). For each n [greater than or equal to] 0, there exists a unitary operator [U.sub.n] on [l.sup.2] (Z, H) such that

[mathematical expression not reproducible] (17)

where [U.sup.*.sub.n] denotes the adjoint of [U.sub.n].

One can verify that unitary operators [U.sub.n], n [greater than or equal to] 0, commute mutually; namely, [U.sub.m] [U.sub.n] = [U.sub.n] [U.sub.m] for all m, n [greater than or equal to] 0. The next lemma shows that the QBN-based walk belongs to the category of unitary quantum walks.

Lemma 9 (see [13]). The QBN-based walk has a unitary representation; more precisely,

[mathematical expression not reproducible], (18)

where [[PHI].sub.n] is the state of the walk at time n [greater than or equal to] 0.

4. Structure Property of QBN-Based Walk

In this section, we apply the Fourier transform theory to the QBN-based walk and examine its structure property. We continue to use the notation made in previous sections.

4.1. Fourier Transform on State Space. Consider [L.sup.2]([0, 2[pi]], H), the space of all functions f: [0, 2[pi]] [right arrow] H that are Bochner integrable [26] with respect to Lebesgue measure and satisfy condition [[integral].sup.2[pi].sub.0] [[parallel]t(t)[parallel].sup.2]dt < [infinity]. It is known that [L.sup.2]([0, 2[pi]], H) is a Hilbert space with the inner product [mathematical expression not reproducible] given by

[mathematical expression not reproducible] (19)

where <x, x> denotes the inner product of H as indicated in Section 2.

A direct verification shows that the system {[e.sub.z,[sigma]] | z [member of] Z, [sigma] [member of] [GAMMA]} is orthonormal in [L.sup.2]([0, 2[pi]], H), where [e.sub.z,[sigma]]s : [0, 2[pi]] [right arrow] H is defined by

[e.sub.z,[sigma]] (t) = [e.sup.izt][Z.sub.[sigma]], t [member of] [0, 2[pi]]. (20)

We denote by [L.sup.2](S, H) the closed subspace of [L.sup.2]([0, 2[pi]], H) spanned by the system {[e.sub.z,[sigma]] | z [member of] Z, [sigma] [member of] [GAMMA]}. Then [L.sup.2](S, H) together with [mathematical expression not reproducible] forms a separable complex Hilbert space.

Clearly {[e.sub.z,[sigma]] | z [member of] Z, [sigma] [member of] [GAMMA]} is a countable orthonormal basis of [L.sup.2](S, H). This, together with the fact that the family {[[phi].sub.z,[sigma]] | z [member of] Z, [sigma] [member of] [GAMMA]} is a countable orthonormal basis of [l.sup.2](Z, H), yields that there exists an isometric isomorphism F : [l.sup.2](Z, H) [right arrow] [L.sup.2](S, H) such that

F[[phi].sub.z,[sigma]] = [e.sub.z,[sigma]], z [member of] Z, [sigma] [member of] [GAMMA]. (21)

The mapping F is then called the Fourier transform on [l.sup.2](Z, H).

It is easy to see that [F.sup.-1] = [F.sup.*]; namely, F is a unitary operator from [l.sup.2](Z, H) to [L.sup.2](S, H). Let [PHI] [member of] [l.sup.2](Z, H) and [??] = F[PHI]. Then one can prove that

[mathematical expression not reproducible] (22)

which justifies the name of F. As usual, F[PHI] is called the Fourier transform of [PHI]. It can also be proven that the inverse [F.sup.-1] of F admits the following representation:

[mathematical expression not reproducible] (23)

where the integral on the righthand side means the Bochner integral.

Just as in the scalar case, the position operator X in [l.sup.2](Z, H) is defined by

[X[PHI]] (x) = x[PHI] (x), x [member of] Z, [PHI] [member of] dom X, (24)

where dom X, the domain of X, is given by

[mathematical expression not reproducible] (25)

It can be verified that X is self-adjoint, and every integer x [member of] Z is its eigenvalue with

X[[phi].sub.x,[sigma]] = x[[phi].sub.x,[sigma]], [for all][sigma] [member of] [GAMMA]. (26)

Let r [greater than or equal to] 1 be a positive integer. Then, by the theory of spectral resolution for self-adjoint operators [27], [X.sup.r] is well defined and remains a self-adjoint operator in [l.sup.2](Z, H), and moreover, its domain is determined by

[mathematical expression not reproducible] (27)

and its action is given by

[[X.sup.r][PHI]] (x) = [x.sup.r][PHI](x), x [member of] Z, [PHI] [member of] dom [X.sup.r]. (28)

Remark 10. Let r [greater than or equal to] 1 be a positive integer and f : [0, 2[pi]] [right arrow] H a continuous function that has continuous derivatives up to order r. Suppose that [f.sup.(j)] [member of] [L.sup.2](S, H) for all 0 [less than or equal to] j [less than or equal to] r. Then [F.sup.-1] f [member of] dom [X.sup.r], and moreover

F[X.sup.r][F.sup.-1] f = [(-i).sup.r] [f.sup.(r)]. (29)

4.2. Structure Property. This subsection focuses on exploring the structure property of the QBN-based walk Recall that the QBN-based walk takes [l.sup.2](Z, H) as its state space. Thus we may call [L.sup.2](S, H) the momentum space of the walk.

Theorem 11. Let n [greater than or equal to] 0 and [V.sub.n] = F[U.sub.n][F.sup.-1]. Then [V.sub.n] is a unitary operator on [L.sup.2](S, H), and moreover [V.sub.n] admits the following representation:

[mathematical expression not reproducible] (30)

where f [member of] [L.sup.2](S, H).

Proof. It is easy to verify that [V.sub.n] is a unitary operator on [L.sup.2](S, H). Now define

g(t) = [[e.sup.it][R.sub.n] + [e.sup.-it][L.sub.n]] f(t), t [member of] [0, 2[pi]]. (31)

Then g [member of] [L.sup.2](S, H).Thus, to prove (30), we need only to verify [V.sub.n] f = g.

Let z [member of] Z, [sigma] [member of] [GAMMA]. Then, by writing [PHI] = [F.sup.-1] f, we have

[mathematical expression not reproducible] (32)

On the other hand, by using properties of the Bochner integrals as well as the representation of [F.sup.-1], we can work out

[mathematical expression not reproducible] (33)

Thus

[mathematical expression not reproducible] (34)

which implies [V.sub.n]f = g.

Theorem 11 allows us to deal with the QBN-based walk in the momentum space [L.sup.2](S, H). In the following, we set

[V.sub.n] = F[U.sub.n][F.sup.-1], n [greater than or equal to] 0. (35)

Clearly {[V.sub.n] | n [greater than or equal to] 0} is a commutative family of unitary operators. The next theorem then offers a representation of the QBN-based walk in the momentum space [L.sup.2](S, H).

Theorem 12. Let [([[PHI].sub.n]).sub.n[greater than or equal to]0] [subset] [l.sup.2](Z, H) be the state sequence of the QBN-based walk, and [f.sub.n] = F[[PHI].sub.n], n [greater than or equal to] 0. Then [mathematical expression not reproducible] and

[mathematical expression not reproducible] (36)

Proof. This is an immediate consequence of Lemma 9 and Theorem 11 together with properties of the Fourier transform F.

Remark 13. Let n [greater than or equal to] 0 and [f.sub.n] the same as in Theorem 12. Then, on interval [0, 2[pi]], one has

[mathematical expression not reproducible] (37)

Recall that X denotes the position operator in the state space [l.sup.2](Z, H) of the QBN-based walk. The next theorem then interprets the meaning of the quantity [mathematical expression not reproducible].

Theorem 14. Let [([[PHI].sub.n]).sub.n[greater than or equal to]0] [subset] [l.sup.2](Z, H) be the state sequence of the QBN-based walk, where the initial state [[PHI].sub.0] satisfies [[PHI].sub.0](x) = 0, x [not equal to] 0, x [member of] Z. Then

[[PHI].sub.n] [member of] dom [X.sup.r], n [greater than or equal to] 0, r [greater than or equal to] 1, (38)

and moreover, for all n [greater than or equal to] 0, r [greater than or equal to] 1, the quantity [mathematical expression not reproducible] is exactly the rth moment of the probability distribution of the walk at time n; namely

[mathematical expression not reproducible] (39)

Proof. Let n [greater than or equal to] 0, r [greater than or equal to] 1. Since [[PHI].sub.0](%) = 0, x [not equal to] 0, x [member of] Z, it follows from Definition 6 that

[[PHI].sub.n] (x) = 0, [absolute value of (x)] > n, x [member of] Z, (40)

which gives

[mathematical expression not reproducible] (41)

which together with (27) implies [[PHI].sub.n] [member of] dom [X.sup.r]. By using (28), we immediately get

[mathematical expression not reproducible] (42)

This competes the proof.

It is easy to verify that {[e.sup.it][R.sub.n] + [e.sup.-it][L.sub.n] | t [member of] R, n [greater than or equal to] 0} makes a commutative family of unitary operators on H. Here, by convention, R denotes the set of all real numbers. In the following, for n [greater than or equal to] 0, we define operator-valued function [V.sub.n](t) as

[V.sub.n] (t) = [e.sup.it][R.sub.n] + [e.sup.-it][L.sub.n], t [member of] R, (43)

which is continuous and has continuous derivatives up to order r with the operator norm for any positive integer r.

Proposition 15. Let n [greater than or equal to] 0 and r [greater than or equal to] 1 be integers. Then the rth derivative of the operator-valued function [V.sub.n](t) satisfies

[d.sup.r][V.subn](t)/[dt.sup.r] = [i.sup.r] [[[V.sub.n] (0)].sup.r] [V.sub.n] (t), (44)

where i denotes the imaginary unit.

Proof. A direct calculation gives

d[V.sub.n] (t)/dt = ([e.sup.it] [R.sub.n] + [e.sup.-it] [L.sub.n])' = i([e.sup.it] [R.sub.n] - [e.sup.-it] [L.sub.n]). (45)

On the other hand, by using Lemma 5, we find

[V.sub.n] (0) [V.sub.n] (t) = ([R.sub.n] + [L.sub.n]) ([e.sup.it] [R.sub.n] + [e.sup.it] [L.sub.n]) = [e.sup.it] [R.sub.n] - [e.sup.it] [L.sub.n]. (46)

Thus d[V.sub.n](t)/dt = i[V.sub.n](0)[V.sub.n](t). By induction, formula (44) follows.

Proposition 16. Let n [greater than or equal to] 0 and r [greater than or equal to] 1 be integers. Then it holds that

[mathematical expression not reproducible] (47)

Proof. By using Proposition 15 and induction, we can get the desired result easily.

As an immediate consequence of Propositions 15 and 16, we have the next proposition.

Proposition 17. Let [phi] [member of] H, and n [greater than or equal to] 0. Define function f : [0, 2[pi]] [right arrow] H as

[mathematical expression not reproducible] (48)

Then, for any integer r [greater than or equal to] 1, f is a continuous function on [0, [pi]] that has continuous derivatives up to order r, and moreover

[mathematical expression not reproducible] (49)

Proposition 18. Let [phi] [member of] H, and n [greater than or equal to] 0. Let f be the same as in Proposition 17. Then [f.sup.(r)] [member of] [L.sup.2](S, H) for all integers r [greater than or equal to] 0.

Proof. We first fix some notation. Let [GAMMA].sub.n]] = {[sigma] [member of] [GAMMA]] | max [sigma] [less than or equal to] n}. For [sigma] [member of] [[GAMMA].sub.n]], we put

[mathematical expression not reproducible] (50)

where [R.sub.[sigma]] = I when [sigma] = 0. Similarly, we use [L.sub.[sigma]] for [sigma] [member of] [[GAMMA].sub.n]].

Now, by a direct calculation, we can get

[mathematical expression not reproducible] (51)

where [[sigma].sup.c] = {0, 1, ..., n} \ [sigma]. Clearly, for each [sigma] [member of] [[GAMMA].sub.n]], the function

[mathematical expression not reproducible] (52)

belongs to [L.sup.2](S, H), which implies f [member of] [L.sup.2](S, H). Using formula (49) and formula (51), we can similarly show that [f.sup.(r)] [member of] [L.sup.2] (S, H) holds for all integers r [greater than or equal to] 0.

Recall that [[partial derivative].sub.k] and [[partial derivative].sup.*.sub.k] denote the annihilation and creation operators, respectively, which are members of quantum Bernoulli noise. The next theorem then offers a formula that links the moments of the QBN-based walk's probability distributions directly with the annihilation and creation operators.

Theorem 19. Let [([[PHI].sub.n]).sub.n[greater than or equal to]0] [subset] [l,sup.2](Z, H) be the state sequence of the QBN-based walk. Suppose the initial state [[PHI].sub.0] takes the following form:

[mathematical expression not reproducible] (53)

where [phi] [member of] H with [parallel][phi][parallel] = 1. Then, the rth moment [mathematical expression not reproducible] of the walks probability distribution at time n [greater than or equal to] 1 satisfies

[mathematical expression not reproducible], (54)

where r [greater than or equal to] 1 is a positive integer.

Proof. By Theorem 14, we know [[PHI].sub.n] [member of] dom [X.sup.r] for all n [greater than or equal to] 0 and r [greater than or equal to] 1. Let [f.sub.n] = F[[PHI].sub.n] for n [greater than or equal to] 0. Then [f.sub.0](t) = [phi], t [member of] [0, 2[pi]], and by Theorems 12 and 11,

[mathematical expression not reproducible] (55)

Now let n, r [greater than or equal to] 1. Then, by Remark 10, we have

[mathematical expression not reproducible] (56)

which together with Proposition 17 gives

[mathematical expression not reproducible] (57)

On the other hand, by using the commutativity of the unitary operator family {[V.sub.k](t)} as well as (55), we have

[mathematical expression not reproducible] (58)

where t [member of] [0, 2[pi]]. Thus

[mathematical expression not reproducible] (59)

which, together with the fact that [V.sub.k](0) = [[partial derivative].sup.*.sub.k] + [[partial derivative].sub.k], k [greater than or equal to] 0, yields the desired formula.

5. Quantum Central Limit Theorem

Quantum central limit theorems are quantum analogs of the classical central limit theorem, which deal with observables from a quantum probability point of view. In the present section, we use the results obtained in the previous section to prove a quantum central limit theorem for quantum Bernoulli noise itself.

In what follows, we denote by B(R) the Borel [sigma]-filed over the real line R. For a Borel set A [member of] B(R), we sue [1.sub.A] to mean its indicator as usual.

Recall that H denotes the space of square integrable Bernoulli functionals, which serves as the coin space of the QBN-based walk. Now consider the following observables:

[[XI].sub.k] = [[partial derivative].sup.*.sub.k] + [[partial derivative].sub.k], k [greater than or equal to] 0, (60)

where [[partial derivative].sub.k] and [[partial derivative].sup.*.sub.k] are the annihilation and creation operators on H, which are members of quantum Bernoulli noise. Let [phi] [member of] H be a unit vector. Then, by the well-known von Neumann's spectral theorem [27], there exists a sequence [([[mu].sup.([phi]).sub.n]).sub.n[greater than or equal to]1] of Borel probability measures on the real line R such that

[mathematical expression not reproducible] (61)

where <x, x> denotes the inner product of the space H.

Theorem 20. Let [([[mu].sup.([phi]).sub.n]).sub.n[greater than or equal to]1] be the probability measure sequence described in (61). Suppose the unit vector [phi] [member of] H takes the form [phi] = [alpha][Z.sub.0] + [beta][Z.sub.0], where [alpha], [beta] [member of] C with [[absolute value of ([alpha])].sup.2] + [[absolute value of ([beta])].sup.2] = 1. Then, for each n [greater than or equal to] 1, the probability measure [[mu].sup.([phi]).sub.n] has the following representation:

[mathematical expression not reproducible] (62)

where [mathematical expression not reproducible] and [mathematical expression not reproducible].

Proof. Consider the QBN-based walk with the initial state [[PHI].sub.0] given by

[mathematical expression not reproducible] (63)

Let [[PHI].sub.n] be the state of the walk at time n and [Y.sub.n] a random variable with the following probability distribution:

P{[Y.sub.n] = x} = [[parallel][[PHI].sub.n] (x)[parallel].sup.2], x [member of] Z. (64)

Then it follows from the proof of Theorem 4.5 of [13] that

[mathematical expression not reproducible] (65)

On the other hand, by using (61) as well as Theorems 19 and 14, we can get

[mathematical expression not reproducible] (66)

which, together with the fact that both [[mu].sup.([phi]).sub.n] and the distribution of [Y.sub.n]/[square root of (n)] have compact supports, yields

[mathematical expression not reproducible] (67)

which implies that [[mu].sup.([phi]).sub.n] is exactly the same as the distribution of [Y.sub.n]/[square root of (n)]. Thus, by (65), we get (62).

Theorem 21. Let [([[mu].sup.([phi]).sub.n]).sub.n[greater than or equal to]1] be the probability measure sequence described in (61). Suppose the unit vector [phi] [member of] H takes the form [phi] = [alpha][Z.sub.0] + [beta][Z.sub.0], where [alpha], [beta] [member of] C with [[absolute value of ([alpha])].sup.2] + [[absolute value of ([beta])].sup.2] = 1. Then

[[mu].sup.([phi]).sub.n] [??] N (0,1) as n [right arrow] [infinity]; (68)

namely, the probability measure sequence [([[mu].sup.([phi]).sub.n]).sub.n[greater than or equal to]1] converges weakly to the standard Gaussian distribution N(0,1) on the real line R.

Proof. Again consider the QBN-based walk with the initial state [[PHI].sub.0] given by

[mathematical expression not reproducible] (69)

For each n [greater than or equal to] 1, let On be the state of the walk at time n and [Y.sub.n] a random variable with the following probability distribution:

P {[Y.sub.n] = x} = [[parallel][[PHI].sub.n] (x)[parallel].sup.2], x [member of] Z. (70)

Then, by Theorem 4.5 of [13], [Y.sub.n]/[square root of (n)] [??] N(0, 1), which, together with (67) in the proof of Theorem 20, yields

[mathematical expression not reproducible] (71)

which then implies that [[mu].sup.([phi]).sub.n] [??] N(0,1) as n [right arrow] [infinity].

Remark 22. As can be seen, observables [[XI].sub.k] = [[partial derivative].sup.*.sub.k] + [[partial derivative].sub.k], k [greater than or equal to] 0, actually play the role of quantum bias in the construction of the QBN-based walk. Theorem 21 then establishes a quantum central limit theorem for these observables.

6. Conclusion Remarks

In the final section, we would like to make some further remarks about the QBN-based walk.

As is known, the QBN-based walk belongs to the category of discrete-time unitary quantum walks on the line. However, it is still different from a usual discrete-time unitary quantum walk on the line. In fact, the QBN-based walk is of the following form:

[mathematical expression not reproducible], (72)

where <x | [[PHI].sub.n]> belongs to a Hilbert coin space, [[??].sub.n](t) is its Fourier transform, and

[[??].sub.n] = 1/2 [[e.sup.it] ([B.sub.n] + I) + [e.sup.-it] ([B.sub.n] - I)], (73)

where [B.sub.n] is a self-adjoint unitary operator acting on the coin space. However, a usual discrete-time unitary quantum walk on the line (take the one with a 2-dimensional coin space as example) reads

[mathematical expression not reproducible], (74)

where [C.sub.n], called the coin operator, acts the coin space, and

[mathematical expression not reproducible], (75)

is known as the spin-dependent shift operator.

As shown in Theorem 21, the limi tprobability distribution of the QBN-based walk for a localized initial state can lead to a quantum central limit theorem for observables [[partial derivative].sup.*.sub.k] + [[partial derivative].sub.k], k [greater than or equal to] 0. However, only for some special localized initial states, have we obtained the walk's limit probability distributions. It is still unclear whether or not the walk has a limit probability distribution for a general localized initial state. On the other hand, as its name suggests, the QBN-based walk might be viewed as such a quantum walk in an open environment that its evolution will be affected by the effects of environment described by quantum Bernoulli noise. In other words, decoherence might happen in the evolution of the walk. Thus we conjecture that, for a general localized initial state, the QBN-based walk might still have the same limit probability distribution as the classical random walk.

The Bernoulli-type random variables [([Z.sub.n]).sub.n[member of]N] described in Section 2 play an important role in understanding the structure of the coin space H of the QBN-based walk. However, those parameters [p.sub.n], [q.sub.n] in their distributions actually have nothing to do with the properties of the walk although an explicit n-dependence is indicated [p.sub.n] by and [q.sub.n]. More precisely, the properties of the QBN-based walk are independent of the choice of those parameters [p.sub.n], [q.sub.n] in the distributions of random variables [([Z.sub.n]).sub.n[member of]N], and indicating explicitly the n-dependence of and is only for the sake of generality. In fact, one can weaken the conditions on [([Z.sub.n]).sub.n[member of]N] without invalidating the main theorems presented in this paper. For instance, one can take (Zn)neN such that it is a "discrete-time (correlated) noise" with the chaotic representation property (see [25, 28]).

https://doi.org/10.1155/2018/2507265

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work is supported by National Natural Science Foundation of China (Grant no. 11461061).

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Caishi Wang (iD), (1) Ce Wang (iD), (2) Yuling Tang (iD), (1) and Suling Ren (iD) (1)

(1) School of Mathematics and Statistics, Northwest Normal University, Lanzhou, Gansu 730070, China

(2) School of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, China

Correspondence should be addressed to Caishi Wang; wangcs@nwnu.edu.cn

Received 24 August 2017; Revised 10 December 2017; Accepted 28 December 2017; Published 24 January 2018

Academic Editor: Antonio Scarfone

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Title Annotation: | Research Article |
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Author: | Wang, Caishi; Wang, Ce; Tang, Yuling; Ren, Suling |

Publication: | Advances in Mathematical Physics |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Jan 1, 2018 |

Words: | 6691 |

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