aa r X i v : . [ m a t h - ph ] O c t Weak convergence of the Wojcik model
Takako Endo, ∗ Norio Konno, † Department of Physics, Ochanomizu University,Bunkyo, Tokyo, 112-0012, Japan Department of Applied Mathematics, Faculty of Engineering, Yokohama National University,Hodogaya, Yokohama, 240-8501, Japan
Abstract
We study “the Wojcik model” which is a discrete-time quantum walk (QW) with one defect in one dimension,introduced by Wojcik et al.. For the Wojcik model, we give the weak convergence theorem describing the ballisticbehavior of the walker in the probability distribution in a rescaled position-space. In our previous studies, we obtainedthe time-averaged limit and stationary measures concerning localization for the Wojcik model. As a result, we getthe mathematical expression of the whole picture of the behavior of the walker for the Wojcik model. Here thecoexistence of localization and the ballistic spreading is one of the peculiar properties of one-dimensional QWs withone defect. Due to the coexistence, it has been strongly expected to utilize QWs to quantum search algorithms. Inorder to derive the weak convergence theorem, we take advantage of the generating function method. We emphasizethat the time-averaged limit measure is symmetric for the origin, however, the weight function in the weak limitmeasure is asymmetric in general, which implies that the weak convergence theorem represents the asymmetry of theprobability distribution. Furthermore, the weak limit measure heavily depends on the phase of the defect and initialstate of the walker. Comparing with our previous studies, we also show some numerical results of the probabilitydistribution to confirm that our result is relevant mathematically, and consider the effect of changing the phase andinitial coin state on the probability distribution, or the ballistic spreading, which is one of the motivations of ourstudy. ∗ [email protected] † [email protected] Key words.quantum walk, ballistic behavior, limit measure Introduction
This paper is the sequential to [3, 4]. Quantum walks (QWs) are quantum counter parts of classical randomwalks, and have been intensively studied in various fields, such as computer science [1, 2] and quantumphysics [7,12]. Owing to the rich applications, it is worth to study QWs both theoretically and experimentally.Especially, it is very important to study the behavior of the walker in the long-time limit. Here, the timeevolution of QWs are defined by unitary evolutions of probability amplitudes. On the other hand, classicalrandom walks are obtained by evolutions of probabilities by transition matrices. Recently, QWs have beenalso implemented experimentally by various materials, such as trapped ions [14] and photons [11]. However,we have not been able to experimentally grasp the behavior in the long-time limit, since it is too difficultfor the QWs to implement the state after many steps. Moreover, because of its quantumness, it is hard tointuitively understand the properties of QWs.As recent studies of QWs suggested, the QWs in one dimension, show two characteristic behaviors in thelong-time limit, that is, “localization”and “the ballistic spreading”. In detail, some of the quantum walkersmay localize and return to the starting point even in the long-time limit, which is in marked contrast tothe classical random walks which do not show such localized behaviors. Furthermore, the quantum walkerspreads much faster than the classical one. Indeed, for the QWs, the width of the probability distributiondiverges with the order of time t . On the other hand, the classical random walker diverges with the orderof √ t . Due to the coexistence of localized and the ballistic behavior, the QWs are believed to be far moreefficient for search algorithms than the classical random walks, since the quantum walker spreads much fasterthan the classical one and can localize at the target.Up to this day, two kinds of limit theorems describing the behavior of discrete-time QWs in one dimensionhave been constructed [8]. One is the limit theorem concerning localization, and the other is the limit theoremconcerning the ballistic behavior of the quantum walker. For the mathematical aspects of the QWs, Konnoet al. [8] introduced three kinds of measures for one-dimensional QWs: time-averaged limit measure, weaklimit measure, and stationary measure. The first two measures describe a coexistence of localized and theballistic behavior in the QW, respectively. In this paper, we focus on the weak limit measure. Now weassume that X t is a discrete-time QW at time t . Then, the weak limit measure of X t /t is described in2eneral as follows; There exist a rational polynomial w ( x ), and C ∈ [0 , , a ∈ (0 ,
1) such that µ ( dx ) = Cδ ( dx ) + w ( x ) f K ( x ; a ) dx (1.1)where f K ( x ; a ) = √ − a π (1 − x ) √ a − x I ( − a,a ) ( x ) (1.2)with I A ( x ) = (cid:26) x ∈ A )0 ( x / ∈ A ) . We should note that f K ( x ; a ) dx is the weak measure for the Hadamard walk [10]. By recent studies of QWs,it is strongly expected that most of the discrete-time QWs have the weak limit measure written by the convexcombination of the term of delta function δ ( dx ) and the absolutely continuous part w ( x ) f K ( x ; n ) dx . Wetherefore, expect the existence of the universality class for the QWs having the weak limit measure expressedby Eq. (1) [8].In this paper, we study “the Wojcik model”, introduced by Wojcik et al. [13]. By the numerical resultsin main, they reported that changing a phase at a single point gives astonishing localization effect. ThenEndo et al. derived the mathematical forms of localization, that is, the stationary and time-averaged limitmeasures, and they showed the astonishing localization effect mathematically [3, 4]. Based on the previousresults, we focus on the ballistic behavior of the Wojcik model to clarify the whole picture of the asymptoticbehavior. As a result, we obtain the weak convergence theorem, the mathematical expression of the ballisticspreading. We also give numerical results of the probability distributions for some phase parameters of thedefect and initial coin states, and then consider what the weak convergence theorem suggests.The rest of this paper is organized as follows. In Section 2, we define the Wojcik model which is themain target in this paper, and present our main result, the weak convergence theorem of the Wojcik model.Then in Section 3, we show the numerical results of the probability distribution for some phase parametersand initial coin states, and consider what our analytical result implies. Appendix A is devoted to the proofof Theorem 1. 3 Model and main result
Let us introduce the total space of discrete-time QW, H , which is a Hilbert space consisting of two Hilbertspaces H C and H P , that is, H = H P ⊗ H C , where H P = Span {| x i ; x ∈ Z } , H C = Span {| J i ; J ∈ { L, R }} , with | L i = (cid:20) (cid:21) , | R i = (cid:20) (cid:21) . We should note that H C and H P represent the position and the direction of the motion of the walker,respectively. In general, discrete-time QW has a state at each time t and position x , called “qubit” writtenby a two-dimensional vector Ψ t ( x ) = (cid:20) Ψ Lt ( x )Ψ Rt ( x ) (cid:21) ∈ C , and we can define the state of the system at each time t byΨ t = [ · · · , Ψ t ( − , Ψ t (0) , Ψ t (1) , · · · ] ∈ ( C ) Z . In this paper, we focus on a discrete-time QW with one defect on the line, whose time evolution is definedby the unitary matrices on H C as follows; U x = √ (cid:20) − (cid:21) ( x = 0) ,e πiφ √ (cid:20) − (cid:21) ( x = 0) , (2.3)where φ ∈ [0 , U x is called “the quantum coin”. To consider the time evolution, we divide theunitary matrices into P x and Q x as P x = √ (cid:20) (cid:21) ( x = 0) ,e πiφ √ (cid:20) (cid:21) ( x = 0) , Q x = √ (cid:20) − (cid:21) ( x = 0) ,e πiφ √ (cid:20) − (cid:21) ( x = 0) , U x = P x + Q x . Here, P x and Q x are equivalent to the left and right movements, respectively. Usingoperators P x and Q x , the time evolution is determined by the recurrence formula;Ψ t +1 ( x ) = P x +1 Ψ t ( x + 1) + Q x − Ψ t ( x −
1) ( x ∈ Z ) . In this paper, we call the QW “the Wojcik model”. We studied localization for our Wojcik model in [3, 4],that is, we derived the time-averaged limit and stationary measures for the Wojcik model which describelocalization mathematically. Therefore, we obtain the mathematical description of the whole picture of themotion of the Wojcik model in the long-time limit by the weak convergence theorem describing the ballisticbehavior.
Let X t be the quantum walker at time t , and we introduce the characteristic function of X t /t , E h e iξ Xtt i = X j ∈ Z P ( X t = j ) e iξ Xtt , where P ( X t = j ) is the probability that X t = j holds. In this subsection, we consider the expression of E [ e iξX t /t ] in the long-time limit t → ∞ . According to [8], we see1 = (cid:16) lim t →∞ E h e iξ Xtt i(cid:17)(cid:12)(cid:12)(cid:12) ξ =0 = C + Z ∞−∞ w ( x ) f K ( x ; 1 / √ dx, with C = X x µ ∞ ( x ) . Here, we should note that µ ∞ ( x ) is the time-averaged limit measure describing localization.From now on, we give the weak convergence theorem for the missing part 1 − C with 0 ≤ C < , whichdescribes in general the ballistic behavior of QW [10]. Theorem 1
Assume that the Wojcik model starts from the origin with the initial coin state Ψ (0) = T [ α, β ] ,where α, β ∈ C . Put α = ae φ , β = be φ with a, b ≥ , a + b = 1 and φ , φ ∈ R , where R is the set of realnumbers. Let ˜ φ = φ − φ . For the Wojcik model, X t /t converges weakly to the random variable Z whichhas the following probability density function; µ ( x ) = Cδ ( x ) + w ( x ) f K ( x ; 1 / √ , (2.4)5 here f K ( x ; 1 / √
2) = 1 π (1 − x ) √ − x I ( − / √ , / √ ( x ) with I D ( x ) = (cid:26) x ∈ D )0 ( x / ∈ D ) . Here, the weight function w ( x ) is given by w ( x ) = t x + t x + t x + t x s x + s x + s , (2.5) where s = cos (4 πφ ) , s =8 sin ( πφ )(cos(4 πφ ) + 4 sin ( πφ ) sin (2 πφ )) , s =16 sin ( πφ ) cos (2 πφ ) ,t = (cid:26) a cos(4 πφ ) ( x ≥ − b cos(4 πφ ) ( x < , t = (cid:26) a cos(4 πφ ) + 8 a sin ( πφ ) sin(2 πφ ) ( x ≥ b cos(4 πφ ) + 8 b sin ( πφ ) sin(2 πφ ) ( x < ,t = (cid:26) a sin ( πφ ) ( x ≥ − b sin ( πφ ) ( x < , t = (cid:26) − ( πφ )( a sin(2 πφ ) − a ) ( x ≥ − ( πφ )( b sin(2 πφ ) − b ) ( x < , with a = 1 + 2 a − ab cos ˜ φ − a cos(2 πφ ) + 2 ab cos( ˜ φ + 2 πφ ) ,a = 1 − a − ab cos ˜ φ ,a = 2 a ( a sin(2 πφ ) − b sin( ˜ φ + 2 πφ )) , and b = 1 + 2 b + 2 ab cos ˜ φ − ab cos( ˜ φ − πφ ) − b cos(2 πφ ) ,b = 1 − b + 2 ab cos ˜ φ ,b = 2 b ( − a sin( ˜ φ − πφ ) + b sin(2 πφ )) . Note that φ is defined by Eq. (2.3) . We should remark that f K ( x ; 1 / √
2) is the density function of the Hadamard walk in a rescaled position-space [10]. Moreover, the second term of Eq. (2.4), w ( x ) f K ( x ; 1 / √ We consider the Wojcik model for some phase parameters of the defect and initial coin states as follows;(1)
Case of the Hadamard walk:
First of all, we see the Hadamard walk whose quantum coin is given by U x = 1 √ (cid:20) − (cid:21) . (3.6)The Hadamard walk can be obtained by putting φ = 0 in Eq. (2.3).(a) Put the initial coin state Ψ (0) = T [1 , w ( x ) in Eq. (1.1)by w ( x ) = 1 − x. Hence, we have Z √ − √ w ( x ) f K ( x ; 1 / √ dx = 1 . (3.7)(b) Let the initial coin state be Ψ (0) = T [ i/ √ , / √ w ( x ) by w ( x ) = 1 . Therefore, we get Z √ − √ w ( x ) f K ( x ; 1 / √ dx = 1 . (3.8)From the above, we obtain the same weight functions as the previous studies [9, 10] from ourresult. Here we should note that the Hadamard walk does not localize in long time limit [9, 10],and we see that C = 0 in Eq. (1.1). 72) QW with one defect: φ = 1 / case. We consider the QW whose quantum coin is given by U x = √ (cid:20) − (cid:21) ( x = ± , ± , · · · ) , √ (cid:20) − − − (cid:21) ( x = 0) , (3.9)which is obtained by putting φ = 1 / (0) = T [1 , w ( x ) in Eq.(2.4) by w ( x ) = − x + 5 x x + 4 ( x ≥ , − x + x x + 4 ( x < . Hence, we see Z √ − √ w ( x ) f K ( x ; 1 / √ dx = 15 . (3.10)Now, we should note that we obtained the time-averaged limit measure µ ∞ ( x ) by Theorem 2in [4], and as a result, we obtain the coefficient of the delta function δ ( x ) in Eq. (2.4) by C = X x µ ∞ ( x ) = 825 + 2 × ∞ X y =1 (cid:18) (cid:19) y = 45 , (3.11)since µ ∞ (0) = 825 ,µ ∞ ( x ) = 1225 (cid:18) (cid:19) | x | . Therefore, we have C + Z √ − √ w ( x ) f K ( x ; 1 / √ dx = 1 . Here, we give the numerical results of the probability distribution at time t = 100 , x/t, tP t ( x )) ( t = 100 , , x represents the positionof the walker and P t ( x ) is the probability that the walker exists on position x at time t . Weshould remark that x/t corresponds to the real axis, and tP t ( x ) corresponds to the imaginaryaxis, respectively. Also, we put the graph of w ( x ) f K ( x ; 1 / √ µ ( dx ), on the picture at each time. We see that thegraph of w ( x ) f K ( x ; 1 / √
2) is right on the middle of the probability distribution for each position ateach time, which suggests that our result is mathematically proper. We also emphasize that µ ∞ ( x )is symmetric for the origin [4], however, w ( x ) f K ( x ; 1 / √
2) does not have an origin symmetry (Figs.1,3,5) , which suggests that the weak limit measure represents the asymmetry of the probabilitydistribution (Figs. 1,3,5).Figure 1: Case of Ψ (0) = T [1 , x/ , P ( x )) at time 100,Black line: w ( x ) f K ( x ; 1 / √
2) Figure 2: Case of Ψ (0) = T [ i/ √ , / √ x/ , P ( x )) at time 100,Black line: w ( x ) f K ( x ; 1 / √ (0) = T [1 , x/ , P ( x )) at time 1000,Black line: w ( x ) f K ( x ; 1 / √
2) Figure 4: Case of Ψ (0) = T [ i/ √ , / √ x/ , P ( x )) at time 1000,Black line: w ( x ) f K ( x ; 1 / √ (0) = T [ i/ √ , / √ w ( x ) by w ( x ) = 3 x x . (3.12)9igure 5: Case of Ψ (0) = T [1 , x/ , P ( x )) at time 10000,Black line: w ( x ) f K ( x ; 1 / √
2) Figure 6: Case of Ψ (0) = T [ i/ √ , / √ x/ , P ( x )) at time 10000,Black line: w ( x ) f K ( x ; 1 / √ δ ( x ) in Eq. (1.1) by C = X x µ ∞ ( x ) = 825 + 2 × ∞ X y =1 (cid:18) (cid:19) y = 45 , (3.13)where µ ∞ (0) = 825 ,µ ∞ ( x ) = 2425 (cid:18) (cid:19) | x | . Accordingly, we have Z √ − √ w ( x ) f K ( x ; 1 / √ dx = 15 , (3.14)which suggests C + Z √ − √ w ( x ) f K ( x ; 1 / √ dx = 1 . Now, we show the numerical results of the probability distribution at time t = 100 , x/t, tP t ( x )) ( t = 100 , , w ( x ) f K ( x ; 1 / √ w ( x ) f K ( x ; 1 / √
2) is just on the middleof the probability distribution for each position at each time, which suggests that our result isappropriate mathematically. We also emphasize that w ( x ) f K ( x ; 1 / √
2) has an origin symmetry(Figs. 2,4,6) , which indicates that the weak limit measure represents the symmetry of theprobability distribution (Figs. 2,4,6), and the symmetric initial coin state gives the symmetry.103)
QW with one defect: φ = 1 / case. Let us treat the QW whose time evolution is defined by the unitary matrices U x = √ (cid:20) − (cid:21) ( x = ± , ± , · · · ) ,i √ (cid:20) − (cid:21) ( x = 0) . (3.15)The QW is obtained by putting φ = 1 / (0) = T [1 , w ( x ) in Eq. (2.4) fromTheorem 1 by w ( x ) = x + 5 x − x + 2 x + 4 ( x ≥ ,x − x − x + 2 x + 4 ( x < . Hence, we have Z √ − √ w ( x ) f K ( x ; 1 / √ dx = 35 . (3.16)Now, we should note that we obtained the time-averaged limit measure µ ∞ ( x ) by Theorem 2in [4], and as a result, we obtain the coefficient of the delta function δ ( x ) in Eq. (1.1) by C = X x µ ∞ ( x ) = 425 + 2 × ∞ X y =1 (cid:18) (cid:19) y = 25 , (3.17)since µ ∞ (0) = 425 ,µ ∞ ( x ) = 1225 (cid:18) (cid:19) | x | . Therefore, we see C + Z √ − √ w ( x ) f K ( x ; 1 / √ dx = 1 . Here, we give the numerical results of the probability distribution at time t = 100 , x/t, tP t ( x )) ( t = 100 , , x expresses the positionof the walker and P t ( x ) is the probability that the walker exists on position x at time t . Weshould note that x/t corresponds to the real axis, and tP t ( x ) corresponds to the imaginary axis,respectively. Also, we put the graph of w ( x ) f K ( x ; 1 / √
2) on the picture at each time. We seethat the graph of w ( x ) f K ( x ; 1 / √
2) is right on the middle of the probability distribution for each11osition at each time, which suggests that our result is mathematically proper. We also notethat µ ∞ ( x ) is symmetric for the origin [4], however, w ( x ) f K ( x ; 1 / √
2) does not have an originsymmetry (Figs. 7,9,11) , which suggests that the weak limit measure represents the asymmetryof the probability distribution (Figs. 7,9,11). Moreover, comparing with the results of φ = 1 / φ = 1 / φ = 1 / (0) = T [1 , x/ , P ( x )) at time 100,Black line: w ( x ) f K ( x ; 1 / √
2) Figure 8: Case of Ψ (0) = T [ i/ √ , / √ x/ , P ( x )) at time 100,Black line: w ( x ) f K ( x ; 1 / √ (0) = T [1 , x/ , P ( x )) at time 1000,Black line: w ( x ) f K ( x ; 1 / √
2) Figure 10: Case of Ψ (0) = T [ i/ √ , / √ x/ , P ( x )) at time 1000,Black line: w ( x ) f K ( x ; 1 / √ (0) = T [ i/ √ , / √ w ( x ) is given fromTheorem 1 by w ( x ) = 3 x x . (3.18)12igure 11: Case of Ψ (0) = T [1 , x/ , P ( x )) at time 10000,Black line: w ( x ) f K ( x ; 1 / √
2) Figure 12: Case of Ψ (0) = T [ i/ √ , / √ x/ , P ( x )) at time 10000,Black line: w ( x ) f K ( x ; 1 / √ δ ( x ) in Eq. (1.1) by C = X x µ ∞ ( x ) = 825 + 2 × ∞ X y =1 (cid:18) (cid:19) y = 45 , (3.19)where µ ∞ (0) = 825 ,µ ∞ ( x ) = 2425 (cid:18) (cid:19) | x | . Thereby, we have Z √ − √ w ( x ) f K ( x ; 1 / √ dx = 15 , (3.20)which suggests C + Z √ − √ w ( x ) f K ( x ; 1 / √ dx = 1 . Now, we show the numerical results of the probability distribution at time t = 100 , x/t, tP t ( x )) ( t = 100 , , w ( x ) f K ( x ; 1 / √ w ( x ) f K ( x ; 1 / √
2) is just on the middleof the probability distribution for each position at each time, which suggests that our result isappropriate mathematically. We also emphasize that w ( x ) f K ( x ; 1 / √
2) is symmetric for the origin(Figs. 8,10,12) , which indicates that the weak limit measure represents the symmetry of theprobability distribution (Figs. 8,10,12), and the symmetric initial coin state contributes to thesymmetry. 13
Summary
In this paper, we obtained the weak convergence theorem for the Wojcik model, and give numerical resultsof the probability distribution for some conclete phase parameters of the defect and initial coin states. Thepurpose of this work is to clarify mathematically the whole picture of the asymptotic behavior of the Wojcikmodel. Especially, one of the motivations is to investigate the effect of the phase and initial coin state ofthe walker on the probability distribution. Our analytical result, that is the weight function in the weakconvergence theorem suggests in general the asymmetry of the probability distribution. Throughout ourstudy including the numerical results, it can be expected that both the phase and initial coin state heavilyinfluence on the aymptotic behavior of the walker. Indeed, as we saw in Section 3, the symmetry of theinitial coin states give the symmetric distributions, and the asymmetric initial coin states contribute to theasymmetric distributions. Moreover, if the phase gradually approximates to 0, then the probability distri-bution seems also close to that of the Hadamard walk [6].
Acknowledgments.
We would like to thank Shimpei Endo for giving useful advices to the numericalsimulations. NK acknowledges financial support of the Grant-in-Aid for Scientific Research (C) of JapanSociety for the Promotion of Science (Grant No. 24540116).
References [1] A. Ambainis: Quantum walks and their algorithmic applications, Springer, International Journal of QuantumInformation , 507-518 (2003)[2] Salvador El´ i as Venegas-Andraca: Quantum walks for computer scientists, Morgan and claypool publishers (2008)[3] T. Endo and N. Konno: The stationary measure of a space-inhomogeneous quantum walk on the line, YokohamaMathematical Journal , 33-47 (2015)[4] T. Endo and N. Konno: The time-averaged limit measure of the Wojcik model, Quantum Information andComputation , 0105-0133 (2015)[5] S. Endo, T. Endo, N. Konno, E. Segawa, and M. Takei: Limit theorems of a two-phase quantum walk with onedefect, Quantum Information and Computation , pp.1373-1396 (2015)[6] J. Kempe: Quantum random walks: An introductory overview, Contemporary Physics , 307-327, (2003)[7] T. Kitagawa, Topological phenomena in quantum walks: elementary introduction to the physics of topologicalphases, Quantum Information Processing pp 1107-1148 (2012)[8] N. Konno, T. Luczak, and E. Segawa: Limit measures of inhomogeneous discrete-time quantum walks in onedimension, Quantum Information Processing , 33-53 (2013)[9] N. Konno: Quantum random walks in one dimension, Quantum Information Processing , pp. 345-354 (2002)[10] N. Konno: A new type of limit theorems for the one-dimensional quantum random walk, Journal of the Mathe-matical Society of Japan , 1179-1195 (2005)[11] H. B. Perets, Y. Lahini, F. Pozzi, M. Sorel, R. Morandotti, and Y. Silberberg: Realization of Quantum Walkswith Negligible Decoherence in Waveguide Lattices, Physical Review Letters , 170506 (2008)
12] Y. Shikano: From discrete time quantum walk to continuous time quantum walk in limit distribution, Journalof Computational and Theoretical Nanoscience , pp. 1558-1570 (2013)[13] A. Wojcik, T. Luczak, P. Kurzynski, A. Grudka, T. Gdala, and M. Bednarska-Bzdega: Trapping a particle of aquantum walk on the line, Physical Review A , 012329 (2012)[14] F. Z¨ a hringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt, and C. F. Roos: Realization of a quantum walkwith one and two trapped ions, Physical Review Letters , 100503 (2010) Appendix A
In Appendix A, we give the proof of Theorem 1. Throughout the proof, we focus on the characteristicfunction of the Wojcik model in the long-time limit, that is, E h e iξ Xtt i = Z x ∈ Z g X t /t ( x ) e iξx dx ( t → ∞ ) (4.21)where g X t /t ( x ) is the density function of X t /t . Hereafter, we derive E (cid:2) e iξX t /t (cid:3) in the long-time limit.We should remark that to get the second term of Eq. (2.4), w ( x ) f K ( x ; 1 / √ g X t /t ( x ) ( t → ∞ ).Now, we introduce the weight of all the paths of the walker, which moves left l times and moves right m times till time t [8]; Ξ t ( x ) = X l j ,m j P l x l Q m x m P l x l Q m x m · · · P l t x lt Q m t x mt , where l + m = t, − l + m = x, P i l i = l, P j m j = m with l i + m i = 1 , l i , m i ∈ { , } , and P γ = l i ,m j | x γ | = x .Here, we consider z ∈ C on a unit circle. According to [8], E [ e iξX t /t ] ( t → ∞ ) is expressed by square normof the residue of ˜Ξ x ( z ) = P t Ξ t ( x ) z t as E h e iξ Xtt i → Z π X θ ∈ B e − iξθ ′ ( k ) k Res (ˆ˜Ξ( k : z ) : z = e iθ ( k ) ) k dk π ( t → ∞ ) , (4.22)where B is the set of the singular points of ˆ˜Ξ( k : z ) ≡ P x ∈ Z ˜Ξ x ( z ) e ikx . Note θ ′ ( k ) = dθ ( k ) /dk . We willexplain how to derive Eq. (4.22) in Appendix A. Using Eq. (4.22) mainly, we prove Theorem 1. Now, we setworthwhile expressions of ˜Ξ x ( z ) which play important roles in the proof. Lemma 1 corresponds to Lemma5 in [4], which we also took advantage of to derive the time-averaged limit measure of the Wojcik model.Let the quantum walker start from the origin with the initial coin state Ψ (0) = T [ α, β ] with α, β ∈ C and | α | + | β | = 1. 15 emma 1 (1) If x = 0 , we have ˜Ξ ( z ) = 11 − √ ω ˜ f ( z ) + ω { ˜ f ( z ) } − e πiφ √ f ( z ) − e πiφ √ f ( z ) e πiφ √ f ( z ) 1 − e πiφ √ f ( z ) . (2) If | x | ≥ , we have ˜Ξ x ( z ) = (˜ λ x ( z )) x − (cid:20) ˜ λ x ( z ) ˜ f x ( z ) z (cid:21) (cid:20) e πiφ √ , − e πiφ √ (cid:21) ˜Ξ ( z ) ( x ≥ , (˜ λ x ( z )) | x |− (cid:20) z ˜ λ x ( z ) ˜ f x ( z ) (cid:21) (cid:20) e πiφ √ , e πiφ √ (cid:21) ˜Ξ ( z ) ( x ≤ − , where ˜ λ x ( z ) = z ˜ f x ( z ) − √ . Here ˜ f x ( z ) satisfies the following quadratic equation. ( ˜ f x ( z )) − √ z ) ˜ f x ( z ) + z = 0 . (4.23)Hereafter, we write ˜ f x ( z ) by ˜ f ( z ), since ˜ f ( ± ) x ( z ) do not depend on the position. Note that ˜ f x ( z ) is originatedfrom ˜ f ( ± ) x ( z ) which satisfy [8] ˜ f ( ± ) x ( z ) = √ z ( − − √ f ( ± ) x ± ( z ) ) . First, we derive the singular points of ˆ˜Ξ( k : z ) and then, calculate the residues of ˆ˜Ξ( k : z ) at the singularpoints. By means of Lemma 1, we can write down ˆ˜Ξ( k : z ) byˆ˜Ξ( k : z ) = e ik − e ik ˜ λ (+) ( z ) (cid:20) ˜ λ (+) ( z ) ˜ f ( z ) z (cid:21) (cid:20) e πiφ √ , − e πiφ √ (cid:21) ˜Ξ ( z )+ e − ik − e − ik ˜ λ ( − ) ( z ) (cid:20) z ˜ λ ( − ) ( z ) ˜ f ( z ) (cid:21) (cid:20) e πiφ √ , e πiφ √ (cid:21) ˜Ξ ( z ) + ˜Ξ ( z ) . (4.24)The first term corresponds to the positive part of ˜Ξ x ( z ), and the second term corresponds to the negativepart of ˜Ξ x ( z ), respectively. We should also note that the singular points derived from the third term, ˜Ξ ( z ),contributes to localization. Note that if | z | <
1, then | ˜ λ ( ± ) ( z ) | < P x (˜ λ (+) ( z )) x − e ikx and P x (˜ λ ( − ) ( z )) | x |− e − ikx converge. Here, we have ˜ λ ( ± ) ( e iθ ) = ∓{ sgn(cos θ ) √ θ − i √ θ } , ˜ f ( e iθ ) = sgn(cos θ ) e iθ {√ | cos θ | − √ θ − } , (4.25)which will be explained how to derive in Appendix B. Now, the principal singular points in this paper comefrom the denominators of the first and second terms of Eq. (4.24), that is,1 − e ik ˜ λ (+) ( z ) = 0 , (4.26)16nd 1 − e − ik ˜ λ ( − ) ( z ) = 0 . (4.27)Equations (4.26) and (4.27) give the two conditions for the solutions. For Eq. (4.26), we seecos k = − sgn(cos θ (+) ( k )) q θ (+) ( k ) − , (4.28)sin k = √ θ (+) ( k ) , (4.29)and for Eq. (4.27), we have cos k = sgn(cos θ ( − ) ( k )) q θ ( − ) ( k ) − , (4.30)sin k = √ θ ( − ) ( k ) . (4.31)Here, we put − dθ ( ± ) ( k ) /dk = x ± to calculate the RHS of Eq. (4.22) and derive the density function g X t /t ( x ) ( t → ∞ ). Then, we derivate Eqs. (4.28) and (4.30) for k , and we obtain sin k, cos k, sin θ ( ± ) ( k ),and cos θ ( ± ) ( k ) as follows; Eqs. (4.28) and (4.29) give cos k = sgn(cos k ) x + q − x , cos θ (+) ( k ) = − sgn(cos k ) 1 q − x ) , sin k = sgn(sin k ) s − x − x , sin θ (+) ( k ) = sgn(sin k ) s − x − x ) . (4.32)Eqs. (4.30) and (4.31) yield cos k = − sgn(cos k ) x − q − x − , cos θ ( − ) ( k ) = sgn(cos k ) 1 q − x − ) , sin k = sgn(sin k ) s − x − − x − , sin θ ( − ) ( k ) = sgn(sin k ) s − x − − x − ) . (4.33)Thus, we obtain the set of the singular points of ˆ˜Ξ( k : z ); B = { e iθ (+) ( k ) , e iθ ( − ) ( k ) } , where e iθ (+) ( k ) = − sgn(cos k ) q − x ) + i sgn(sin k ) s − x − x ) , e iθ ( − ) ( k ) = sgn(cos k ) q − x − ) + i sgn(sin k ) s − x − − x − ) . Next, we derive the residue of ˆ˜Ξ( k ; z ) at e iθ ( ± ) ( k ) . Substituting the singular points to ˜ f ( z ), we have˜ f ( e iθ (+) ( k ) ) = − sgn(cos k ) e iθ (+) ( k ) √ − x | x | , ˜ f ( e iθ ( − ) ( k ) ) = sgn(cos k ) e iθ ( − ) ( k ) √ − x | x | . Owing to Lemma 1, we see e ik − e ik ˜ λ (+) ( z ) (cid:20) ˜ f ( z )˜ λ (+) ( z ) z (cid:21) (cid:20) e πiφ √ , − e πiφ √ (cid:21) ˜Ξ ( z )= e πiφ √ ( z ) e ik − e ik ˜ λ (+) ( z ) (cid:20) ˜ f ( z )˜ λ (+) ( z ) z (cid:21) { α − β − √ ωα ˜ f ( z ) } . By definition, the square norm of residue of the first term of Eq. (4.24) is written by (cid:12)(cid:12)(cid:12)(cid:12)
Res (cid:18) e ik − e ik ˜ λ (+) ( z ) (cid:20) ˜ f ( z )˜ λ (+) ( z ) z (cid:21) (cid:20) e πiφ √ , − e πiφ √ (cid:21) ˜Ξ ( z ) : z = e iθ (+) ( k ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = 12 (cid:12)(cid:12)(cid:12)(cid:12) Res (cid:18) − e ik ˜ λ (+) ( z ) : z = e iθ (+) ( k ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) | ˜Λ ( e iθ (+) ( k ) ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)" ˜ f ( e iθ (+) ( k ) )˜ λ (+) ( e iθ (+) ( k ) ) e iθ (+) ( k ) × | α − β − √ ωα ˜ f ( e iθ (+) ( k ) ) | . In a similar way, we get the second term of Eq. (4.24) by (cid:12)(cid:12)(cid:12)(cid:12)
Res (cid:18) e − ik − e − ik ˜ λ ( − ) ( z ) (cid:20) z ˜ f ( z )˜ λ ( − ) ( z ) (cid:21) (cid:20) e πiφ √ , e πiφ √ (cid:21) ˜Ξ ( z ) : z = e iθ ( − ) ( k ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = 12 (cid:12)(cid:12)(cid:12)(cid:12) Res (cid:18) − e − ik ˜ λ ( − ) ( z ) : z = e iθ ( − ) ( k ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) | ˜Λ ( e iθ ( − ) ( k ) ) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)" e iθ ( − ) ( k ) ˜ f ( e iθ (+) ( k ) )˜ λ ( − ) ( e iθ ( − ) ( k ) ) × | α + β − √ ωβ ˜ f ( e iθ ( − ) ( k ) ) | . Consequently, we obtain k Res (ˆ˜Ξ( k : z ) : z = e iθ ( ± ) ( k ) ) k = 12 (cid:12)(cid:12)(cid:12)(cid:12) Res (cid:18) − e ik ˜ λ (+) ( z ) : z = e iθ (+) ( k ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) | ˜Λ ( e iθ (+) ( k ) ) | × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)" ˜ f ( e iθ (+) ( k ) )˜ λ (+) ( e iθ (+) ( k ) ) e iθ (+) ( k ) | α − β − √ ωα ˜ f ( e iθ (+) ( k ) ) | + 12 (cid:12)(cid:12)(cid:12)(cid:12) Res (cid:18) − e − ik ˜ λ ( − ) ( z ) : z = e iθ ( − ) ( k ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) | ˜Λ ( e iθ ( − ) ( k ) ) | × (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)" e iθ ( − ) ( k ) ˜ f ( e iθ ( − ) ( k ) )˜ λ ( − ) ( e iθ ( − ) ( k ) ) | α + β − √ ωβ ˜ f ( e iθ ( − ) ( k ) ) | . (4.34)18ereafter, we will rewrite the items below in terms of x + or x − , and then substitute the items in Eq. (4.34).(1) (cid:12)(cid:12)(cid:12)(cid:12) Res (cid:18) − e ik ˜ λ (+) ( z ) : z = e iθ (+) ( k ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12) Res (cid:18) − e − ik ˜ λ ( − ) ( z ) : z = e iθ ( − ) ( k ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ,(2) 1 | ˜Λ ( e iθ ( ± ) ( k ) ) | ,(3) 12 | α − β − √ ωα ˜ f ( e iθ (+) ( k ) ) | and 12 | α + β − √ ωβ ˜ f ( e iθ ( − ) ( k ) ) | ,(4) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" ˜ λ (+) ( e iθ (+) ( k ) ) ˜ f ( e iθ (+) ( k ) ) e iθ (+) ( k ) and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" e iθ ( − ) ( k ) ˜ λ ( − ) ( e iθ ( − ) ( k ) ) ˜ f ( e iθ ( − ) ( k ) ) .1 . Calculation of (cid:12)(cid:12)(cid:12)(cid:12)
Res (cid:18) − e ik ˜ λ (+) ( z ) : z = e iθ (+) ( k ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) and (cid:12)(cid:12)(cid:12)(cid:12) Res (cid:18) − e − ik ˜ λ ( − ) ( z ) : z = e iθ ( − ) ( k ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) .Put g ( ± ) ( z ) = 1 − e ± ik ˜ λ ( ± ) ( z ). Expanding g ( ± ) ( z ) around z = e iθ ( ± ) ( k ) , we get Res (cid:18) − e ± ik ˜ λ ( ± ) ( z ) : z = e iθ ( ± ) ( k ) (cid:19) = 1 dg ( ± ) ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = e iθ ( ± )( k ) . Eq. (4.25) gives dg ( ± ) ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) z = e iθ ( ± )( k ) = − sgn(cos k ) q − x ± e − i ( θ ( ± ) ( k ) ∓ k ) − i sgn(cos k sin k ) q − x ± x ± , which suggests (cid:12)(cid:12)(cid:12)(cid:12) Res (cid:18) − e ik ˜ λ (+) ( z ) : z = e iθ (+) ( k ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = x , (cid:12)(cid:12)(cid:12)(cid:12) Res (cid:18) − e − ik ˜ λ ( − ) ( z ) : z = e iθ ( − ) ( k ) (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) = x − . (4.35)2 . Calculation of 1 | ˜Λ ( e iθ ( ± ) ( k ) ) | .From Lemma 1, we have | ˜Λ ( e iθ ) | = 1 + 2 | ˜ f ( e iθ ) | + |{ ˜ f ( e iθ ) } | − √ ℜ ( e πiφ ˜ f ( e iθ )) + 2 ℜ ( e πiφ { ˜ f ( e iθ ) } ) − √ ℜ ( e πiφ ˜ f ( e iθ ) ˜ f ( e iθ ) ) , (4.36)19or any real number θ ∈ R . Therefore, substituting the singular points into Eq. (4.36), we obtain (cid:12)(cid:12)(cid:12)(cid:12) ( e iθ (+) ( k ) ) (cid:12)(cid:12)(cid:12)(cid:12) = (1 + x + ) { − cos(2 πφ ) + x cos(4 πφ ) / − sgn(sin k cos k ) sin(2 πφ ) q − x sin(2 πφ )(1 − cos(2 πφ )) } , (cid:12)(cid:12)(cid:12)(cid:12) ( e iθ ( − ) ( k ) ) (cid:12)(cid:12)(cid:12)(cid:12) = (1 − x − ) { − cos(2 πφ ) + x − cos(4 πφ ) / k cos k ) sin(2 πφ ) q − x − sin(2 πφ )(1 − cos(2 πφ )) } . (4.37)3 . Calculation of | α − β − √ ωα ˜ f ( e iθ (+) ( k ) ) | / | α + β − √ ωβ ˜ f ( e iθ ( − ) ( k ) ) | / (0) = T [ α, β ], where α = ae iφ , β = be iφ with a, b ≥ a + b = 1. Noting12 | α − β − √ ωα ˜ f ( e iθ (+) ( k ) ) | = 12 { | α | | ˜ f ( e iθ (+) ( k ) ) | − ℜ ( αβ ) − √ | α | ℜ ( e πiφ ˜ f ( e iθ (+) ( k ) )) + 2 √ ℜ ( αβe πiφ ˜ f ( e iθ (+) ( k ) )) } , and12 | α + β − √ ωβ ˜ f ( e iθ ( − ) ( k ) ) | = 12 { | β | | ˜ f ( e iθ ( − ) ( k ) ) | + 2 ℜ ( βα ) − √ | β | ℜ ( e πiφ ˜ f ( e iθ ( − ) ( k ) )) − √ ℜ ( αβe πiφ ˜ f ( e iθ ( − ) ( k ) )) } , we obtain | α − β − √ ωα ˜ f ( e iθ (+) ( k ) ) | = 12 + a − x + x + − ab cos ˜ φ − a x + { cos(2 πφ ) + sgn(sin k cos k ) q − x sin(2 πφ ) } + ab x + { cos( ˜ φ + 2 πφ ) + sgn(sin k cos k ) q − x sin( ˜ φ + 2 πφ ) } , | α + β − √ ωβ ˜ f ( e iθ ( − ) ( k ) ) | = 12 + b x − − x − + ab cos ˜ φ − b − x − { cos(2 πφ ) − sgn(sin k cos k ) q − x − sin(2 πφ ) }− ab − x − { cos( ˜ φ + 2 πφ ) − sgn(sin k cos k ) q − x sin( ˜ φ + 2 πφ ) } . (4.38)4 . Calculation of (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" ˜ λ (+) ( e iθ (+) ( k ) ) ˜ f ( e iθ (+) ( k ) ) e iθ (+) ( k ) and (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" e iθ ( − ) ( k ) ˜ λ ( − ) ( e iθ ( − ) ( k ) ) ˜ f ( e iθ ( − ) ( k ) ) .By definition, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" ˜ λ (+) ( e iθ (+) ( k ) ) ˜ f ( e iθ (+) ( k ) ) e iθ (+) ( k ) = | ˜ λ (+) ( e iθ (+) ( k ) ) | | ˜ f ( e iθ (+) ( k ) ) | + 1 = 21 + x + ( x + > , (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)" e iθ ( − ) ( k ) ˜ λ ( − ) ( e iθ ( − ) ( k ) ) ˜ f ( e iθ ( − ) ( k ) ) = 1 + | ˜ λ ( − ) ( e iθ ( − ) ( k ) ) | | ˜ f ( e iθ ( − ) ( k ) ) | = 21 − x − ( x − < . (4.39)20oting − dθ ( ± ) ( k ) dk = x ± , (4.40)we see x + = | cos k |√ k , x − = − | cos k |√ k . (4.41)Thus, we can treat x + and x − as a variable x ; x = (cid:26) x + ( x > ,x − ( x < . Combining Eqs. (4.32) and (4.33) with Eq. (4.41), and noting Eq. (4.40), we have dxdk = ∓ sgn(sin k cos k )(1 − x ) p − x . Hence, we see dk = (cid:26) − sgn(sin k cos k ) f K ( x ; 1 / √ πdx ( x > , sgn(sin k cos k ) f K ( x ; 1 / √ πdx ( x < . (4.42)Substituting the items given in 1 . to 4 . into Eq. (4.34) and combining with Eq. (4.22), the proof of Theorem1 is completed. Appendix B
We give the detailed explanation of Eq. (4.22), which is a key to prove Theorem 1. Put ω l ( k ) = Res ( ˆ˜Ψ t ( k :21 ) : z = e iθ ( l ) ( k ) ) with Ψ t ( x ) = Ξ t ( x ) ϕ and l = + , − . Then, we have by definition E h e iξ Xtt i = X j P ( X t = j ) e iξ jt = X j k Ξ t ( j ) ϕ k e iξ jt = Z π X x,y ϕ ∗ Ξ ∗ t ( y )Ξ t ( x ) ϕ e iξ xt e ik ( x − y ) dk π = Z π X x,y h Ψ t ( y ) , Ψ t ( x ) i e iξ xt e ik ( x − y ) dk π = Z π (cid:28) ˆΨ t ( k ) , ˆΨ t (cid:18) k + ξt (cid:19)(cid:29) dk π = Z π *X l ω l ( k ) e − i ( t +1) θ ( l ) ( k ) , X m ω m (cid:18) k + ξt (cid:19) e − i ( t +1) θ ( m ) ( k + ξt ) + dk π (4.43)= Z π (X l | ω l ( k ) | e − iξ t +1 t dθ ( l )( k ) dk e − i ( t +1) O ( t ) + O (cid:18) t (cid:19)) dk π + Z π (X l X m ω l ( k ) e i ( t +1) θ ( l ) ( k ) ω m ( k ) e − i ( t +1) θ ( m ) ( k ) e − iξ t +1 t dθ ( m )( k ) dk e − i ( t +1) O ( t ) + O (cid:18) t (cid:19)) dk π . (4.44)We took advantage of the residue theorem Z π ˆ˜Ψ t ( k ; z ) dz = 2 πi X i Res ( ˆ˜Ψ t ( k ; z ) , z = θ ( i ) ( k ))and the inverse Fourier transform ˆΨ t ( k ) = 12 πi Z π ˆ˜Ψ t ( k ; z ) dzz t +1 to obtain Eq. (4.43). Using Maclaurin’s expansion for w m ( k + ξ/t ) e − i ( t +1) θ ( m ) ( k + ξ/t ) , that is, w m ( k + ξ/t ) e − i ( t +1) θ ( m ) ( k + ξ/t ) = (cid:18) w ( k ) + ξt dw ( k ) dk + ξ t d w ( k ) d k + · · · (cid:19) e − i ( t +1) (cid:26) θ ( m ) ( k )+ ξt dθ ( m )( k ) dk + ξ t d θ ( m )( k ) d k + ··· (cid:27) , we got Eq. (4.44). According to the Riemann-Lebesgue Theorem, the second term of Eq. (4.44) vanisheswhen t → ∞ , and we obtain the desired relation. Appendix C
Hereafter, we explain in detail, how ˜ f ( z ) and ˜ λ ( ± ) ( z ) are determined when we focus on the ballistic behaviorof the Wojcik model. Owing to [8], we see ˜ λ ( ± ) ( ω ) = ± i √ { ( ω + ω − ) − p ( ω + ω − ) − } , ˜ f ( ω ) = − ω √ { ( ω − ω − ) + p ( ω + ω − ) − } . ω = i (1 − ǫ ) e iθ ( ǫ ∈ R , | ǫ | ≪ ǫ → p ( ω + ω − ) − θ with the range of cos θ or sin θ . Note | ǫ | ≪
1, and we can approximate ˜ λ ( ± ) ( ω ) as [5]˜ λ ( ± ) ( ω ) = ± i √ (cid:26) (1 − ǫ ) ie iθ − (1 − ǫ ) − ie − iθ − q { (1 − ǫ ) ie iθ − (1 − ǫ ) − ie − iθ } − (cid:27) ∼ ∓ i √ n θ + 2 iǫ cos θ + δ p θ − o , (4.45)where we put δ ∈ R with δ = 1. Noting | ˜ λ ( ± ) ( ω ) | <
1, Eq. (4.45) suggests that we need to take into accountthe next two cases.(1) Case of | sin θ | ≥ / √ ( θ + 2 δ r sin θ − ) < . Hence, we have 2 sin θ + 2 sin θδ r sin θ − < . Consequently, we get δ = − sgn(sin θ ).(2) Case of | sin θ | < / √ θ + ( ǫ cos θ + 2 δ r − sin θ ) < . Therefore, we see 4 ǫ cos θ + 8 ǫ cos θδ r − sin θ < . Consequently, we obtain δ = − sgn(cos θ ).Accordingly, the square root is expressed aslim ǫ → p ( ω + ω − ) − − θ ) r sin θ −
12 ( | sin θ | ≥ / √ , − i sgn(cos θ ) r − sin θ ( | sin θ | < / √ . (4.46)Next, we determine in detail ˜ λ ( ± ) ( z ) and ˜ f ( z ). When we consider the weak convergence theorem forour Wojcik model, we choose the square root so that 1 / (1 − e ik ˜ λ (+) ( z )) and 1 / (1 − e − ik ˜ λ ( − ) ( z )) have thesingular points, that is, | ˜ f ( z ) | 6 = 1. Therefore, we see from Eqs. (4.45) and (4.46), ˜ λ ( ± ) ( z ) = ∓{ sgn(cos θ ) √ θ − i √ θ } , ˜ f ( z ) = sgn(cos θ ) e iθ {√ | cos θ | − √ θ − } , ( | sin θ | < / √ z = e iθiθ