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

Wang, Caishi; Wang, Ce; Tang, Yuling; Ren, Suling

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

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2018 | Article ID 2507265 | https://doi.org/10.1155/2018/2507265

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

Caishi Wang,¹Ce Wang,²Yuling Tang,¹and Suling Ren¹

Academic Editor: Antonio Scarfone

Received24 Aug 2017

Revised10 Dec 2017

Accepted28 Dec 2017

Published24 Jan 2018

Abstract

As a unitary quantum walk with infinitely many internal degrees of freedom, the quantum walk in terms of quantum Bernoulli noise (recently introduced by Wang and Ye) shows a rather classical asymptotic behavior, which is quite different from the case of the usual quantum walks with a finite number of internal degrees of freedom. In this paper, we further examine the structure of the walk. By using the Fourier transform on the state space of the walk, we obtain a formula that links the moments of the walk’s probability distributions directly with annihilation and creation operators on Bernoulli functionals. We also prove some other results on the structure of the walk. Finally, as an application of these results, we establish a quantum central limit theorem for the annihilation and creation operators themselves.

1. Introduction

Quantum walks are quantum analogs of the classical random walk, which have found wide application in quantum information, quantum computing, and many other fields [1–3]. 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 , 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 as its state space, where 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 , 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 internal degrees of freedom can generate a Gaussian limit distribution with scaling speed . And he wondered [6] whether a discrete-time quantum walk can generate a Gaussian limit distribution with scaling speed . 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 be quantum Bernoulli noise, namely, annihilation and creation operators on , and be the position operator in . In the first part of the present paper, by using the Fourier transform on , we obtain a representation of the QBN-based walk in the momentum space. In particular, we obtain the following relations:where denotes the state of the walk at time and . Since the quantity is exactly the th moment of the walk’s probability distribution at time , 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 and (i.e., such that ), 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 , , are observables on , 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 , .

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, always denotes the set of all integers, while means the set of all nonnegative integers. We denote by the finite power set of ; namely,where means the cardinality of . Unless otherwise stated, letters like , , and stand for nonnegative integers, namely, elements of .

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 be the set of all mappings and the sequence of canonical projections on given byLet be the -field on generated by the sequence and a given sequence of positive numbers with the property that for all . It is known [25] that there exists a unique probability measure on such thatfor , with when and with . Thus one has a probability measure space , which is referred to as the Bernoulli space and random variables on it are known as Bernoulli functionals.

Let be the sequence of Bernoulli functionals defined bywhere . Clearly is an independent sequence of random variables on the probability measure space .

Let be the space of square integrable complex-valued Bernoulli functionals; namely,We denote by the usual inner product of the space and by the corresponding norm. It is known [25] that has the chaotic representation property. Thus has as its orthonormal basis, where andwhich shows that is an infinite dimensional complex Hilbert space.

Lemma 1 (see [23]). For each , there exists a bounded operator on such thatwhere and is the indicator of as a subset of .

Lemma 2 (see [23]). Let . Then , the adjoint of operator has the following property:where .

The operator and its adjoint are usually known as the annihilation and creation operators acting on Bernoulli functionals, respectively.

Definition 3 (see [23]). The family 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 , . Then it holds true thatwhere is the identity operator on .

Lemma 5 (see [13]). For , write and . Then both and are self-adjoint operators on , and moreover

It follows easily from Lemma 4 that operators , , , form a commutative family; namely,hold for all , .

3. QBN-Based Walk

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

Recall that , the space of square integrable complex-valued Bernoulli functionals. Let be the space of square summable functions defined on and valued in ; namely,

Then remains a complex Hilbert space, whose inner product is given bywhere denotes the inner product of as indicated in Section 2. By convention, we denote by the norm induced by . Note that has a countable orthonormal basis , where is defined byThus is separable.

As usual, a vector is called a state if it satisfies the normalized condition .

Definition 6 (see [13]). The QBN-based walk is such a quantum walk whose state space is and whose time evolution is governed by where denotes the state of the walk at time .

Let be the state sequence of the QBN-based walk. Then the function makes a probability distribution on , which is called the probability distribution of the walk at time . In particular, is the probability that the quantum walker is found at position at time . As usual, the QBN-based walk is assumed to start at position , which implies that its initial state satisfies and for with .

Remark 7. It is well known that . Thus, describes the position of the QBN-based walk, while describes the internal degrees of freedom of the walk. As shown in Section 2, the dimension of is infinite, which means that the QBN-based walk has infinitely many internal degrees of freedom.

Lemma 8 (see [13]). For each , there exists a unitary operator on such thatwhere denotes the adjoint of .

One can verify that unitary operators , , commute mutually; namely, for all , . 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,where is the state of the walk at time .

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 , the space of all functions that are Bochner integrable [26] with respect to Lebesgue measure and satisfy condition . It is known that is a Hilbert space with the inner product given bywhere denotes the inner product of as indicated in Section 2.

A direct verification shows that the system is orthonormal in , where is defined byWe denote by the closed subspace of spanned by the system . Then together with forms a separable complex Hilbert space.

Clearly is a countable orthonormal basis of . This, together with the fact that the family is a countable orthonormal basis of , yields that there exists an isometric isomorphism such thatThe mapping is then called the Fourier transform on .

It is easy to see that ; namely, is a unitary operator from to . Let and . Then one can prove thatwhich justifies the name of . As usual, is called the Fourier transform of . It can also be proven that the inverse of admits the following representation:where the integral on the righthand side means the Bochner integral.

Just as in the scalar case, the position operator in is defined bywhere , the domain of , is given byIt can be verified that is self-adjoint, and every integer is its eigenvalue withLet be a positive integer. Then, by the theory of spectral resolution for self-adjoint operators [27], is well defined and remains a self-adjoint operator in , and moreover, its domain is determined byand its action is given by

Remark 10. Let be a positive integer and a continuous function that has continuous derivatives up to order . Suppose that for all . Then , and moreover

4.2. Structure Property

This subsection focuses on exploring the structure property of the QBN-based walk. Recall that the QBN-based walk takes as its state space. Thus we may call the momentum space of the walk.

Theorem 11. Let and . Then is a unitary operator on , and moreover admits the following representation:where .

Proof. It is easy to verify that is a unitary operator on . Now defineThen . Thus, to prove (30), we need only to verify .
Let , . Then, by writing , we have On the other hand, by using properties of the Bochner integrals as well as the representation of , we can work outThus which implies .

Theorem 11 allows us to deal with the QBN-based walk in the momentum space . In the following, we setClearly is a commutative family of unitary operators. The next theorem then offers a representation of the QBN-based walk in the momentum space .

Theorem 12. Let be the state sequence of the QBN-based walk, and , . Then and

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

Remark 13. Let and the same as in Theorem 12. Then, on interval , one hasRecall that denotes the position operator in the state space of the QBN-based walk. The next theorem then interprets the meaning of the quantity .

Theorem 14. Let be the state sequence of the QBN-based walk, where the initial state satisfies , , . Thenand moreover, for all , , the quantity is exactly the th moment of the probability distribution of the walk at time ; namely,

Proof. Let , . Since , , , it follows from Definition 6 that which giveswhich together with (27) implies . By using (28), we immediately get This competes the proof.

It is easy to verify that makes a commutative family of unitary operators on . Here, by convention, denotes the set of all real numbers. In the following, for , we define operator-valued function aswhich is continuous and has continuous derivatives up to order with the operator norm for any positive integer .

Proposition 15. Let and be integers. Then the th derivative of the operator-valued function satisfieswhere denotes the imaginary unit.

Proof. A direct calculation gives On the other hand, by using Lemma 5, we find Thus . By induction, formula (44) follows.

Proposition 16. Let and be integers. Then it holds that

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 , and . Define function asThen, for any integer , is a continuous function on that has continuous derivatives up to order , and moreover

Proposition 18. Let , and . Let be the same as in Proposition 17. Then for all integers .

Proof. We first fix some notation. Let . For , we put where when . Similarly, we use for .
Now, by a direct calculation, we can getwhere . Clearly, for each , the function belongs to , which implies . Using formula (49) and formula (51), we can similarly show that holds for all integers .

Recall that and 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 be the state sequence of the QBN-based walk. Suppose the initial state takes the following form:where with . Then, the th moment of the walk’s probability distribution at time satisfieswhere is a positive integer.

Proof. By Theorem 14, we know for all and . Let for . Then , , and by Theorems 12 and 11,Now let , . Then, by Remark 10, we havewhich together with Proposition 17 givesOn the other hand, by using the commutativity of the unitary operator family as well as (55), we havewhere . Thus which, together with the fact that , , 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 the Borel -filed over the real line . For a Borel set , we sue to mean its indicator as usual.

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

where and are the annihilation and creation operators on , which are members of quantum Bernoulli noise. Let be a unit vector. Then, by the well-known von Neumann’s spectral theorem [27], there exists a sequence of Borel probability measures on the real line such thatwhere denotes the inner product of the space .

Theorem 20. Let be the probability measure sequence described in (61). Suppose the unit vector takes the form , where , with . Then, for each , the probability measure has the following representation:where and .

Proof. Consider the QBN-based walk with the initial state given byLet be the state of the walk at time and a random variable with the following probability distribution:Then it follows from the proof of Theorem of [13] thatOn the other hand, by using (61) as well as Theorems 19 and 14, we can get which, together with the fact that both and the distribution of have compact supports, yieldswhich implies that is exactly the same as the distribution of . Thus, by (65), we get (62).

Theorem 21. Let be the probability measure sequence described in (61). Suppose the unit vector takes the form , where , with . Then namely, the probability measure sequence converges weakly to the standard Gaussian distribution on the real line .

Proof. Again consider the QBN-based walk with the initial state given by For each , let be the state of the walk at time and a random variable with the following probability distribution: Then, by Theorem of [13], , which, together with (67) in the proof of Theorem 20, yieldswhich then implies that as .

Remark 22. As can be seen, observables , , 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:where belongs to a Hilbert coin space, is its Fourier transform, andwhere 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 -dimensional coin space as example) readswhere , called the coin operator, acts the coin space, andis known as the spin-dependent shift operator.

As shown in Theorem 21, the limit probability distribution of the QBN-based walk for a localized initial state can lead to a quantum central limit theorem for observables , . 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 described in Section 2 play an important role in understanding the structure of the coin space of the QBN-based walk. However, those parameters , in their distributions actually have nothing to do with the properties of the walk although an explicit -dependence is indicated by and . More precisely, the properties of the QBN-based walk are independent of the choice of those parameters , in the distributions of random variables , and indicating explicitly the -dependence of and is only for the sake of generality. In fact, one can weaken the conditions on without invalidating the main theorems presented in this paper. For instance, one can take such that it is a “discrete-time (correlated) noise” with the chaotic representation property (see [25, 28]).

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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Copyright © 2018 Caishi Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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