LLIMIT THEOREMS FOR QUANTUM WALKS ON THE UNION OF PLANES
CLEMENT AMPADU
31 Carrolton Road Boston, Massachusetts, 02132 U.S.A. e-mail: [email protected]
Abstract
We extend the construction given by [Chisaki et.al, arXiv:1009.1306v1] from lines to planes, and obtain the associated limit theorems for quantum walks on such a graph.
KEY WORDS : limit theorems, quantum walk, localization, Grover walk
PACS: I. Introduction
This paper adds to the growing literature on limit theorems for quantum walks. These theorems seek to explain observations in numerical simulations of quantum random walks whereby the probability distribution of the walkers position is seen to exhibit a persistent major spike at the initial position and two other minor spikes which drift to infinity in either direction. Under the pseudonym “localization” theoretical explanations for this phenomenon has been given by many authors in various contexts, see [1-22] for examples. In this paper we consider a discrete time quantum walk kt P , at time t on a graph with joined quarter planes t P , which is composed of k quarter planes with the same origin. Employing the reduction technique of Chisaki et.al [5], we obtain two types of limit theorems. The first is an asymptotic behavior of kt P , which corresponds to localization (Theorem 1). The second is a weak convergence statement for kt P , (Theorem 2). This paper is organized as follows. In Sections II, III, IV, and V we give the definitions of the discrete-time quantum walk on the plane, the quarter plane, a graph with joined quarter planes, and a new type of homogeneous tree. Noting that the plane is a regular graph, in the introduction of Section III we express a general definition of the discrete-time quantum walk on the undirected graph. Sections V and VI is mainly concerned with introducing a quantum walk with an enlarged basis and the reduction of kt P , to the walk on a quarter plane. Lemma 1 gives the reduction of yx kkt T ,, to kt P , whilst Lemma 2 extends the state of kt P , to kt P , . In Section VII we obtain the generating function of he walk, and use it in Section VIII to prove Theorem 1. In Section IX we prove Theorem 2 using the Fourier transform of the generating function. Section X is devoted to the conclusions. II.
Discrete-time quantum walk on the plane
Let Z be the set of integers, and let ZyxyxZZ ,:),( . The quantum walk on ZZ is defined by ZZCZZP
HHH , where ZyxyxspanH
ZZP ,:, and DownUpRightLeftspanH
ZZC ,,, . Put T Left , T Right , T Down , T Up , where T is the transposed operator. Recall part of the unitary operator is given by ZZPZZ
CIF where P I is the identity operator on ZZP H , and )4(
22 22
Udcdcdc bdadbcac bdbcadac bababaC ZZ with abcd , where )4( U is the set of unitary matrices. The shift operator ZZ S is defined by uplupyx downldownyx rightlrightyx leftlleftyxlyxS ZZ ,,1, ,,1, ,,,1 ,,,1,, III.
Discrete-time quantum walk on the quarter plane
Note that the plane is a regular graph. In general given an undirected connected graph G . Let )( GV be the set of all sites in G . Let )()( , GEGE yx be the set of all bonds which connect the site )(),( GVyx . Let the position subpace be given by )(),(:, GVyxyxspanH P . For )(),( GVyx let the coin subspace be given by )(: , ),( GEllspanH yxC yx . A discrete-time quantum walk on G is given by ),( )()(),(:,, ),( yx Cpyx
HHGElandGVyxlyxspanH . If G is a regular graph, then )0,0( CP HHH . In general for the quantum walk on the undirected connected graph G , the evolution operator on the space ),( yx CP HHH is determined by SFU , where HHS : is a shift operator and HHF : is defined by )(),( ),( ., GVyx yx
CyxyxF , where ),( yx C is the coin operator on ),( yx C H . If G is a regular graph and for every )(),( GVyx , ),( yx C is ndependent of ),( yx , then we can write CIF P , where P I is the identity operator on P H . In general for the quantum walk on the undirected connected graph G , SFU is a unitary operator if and only if S is a permutation on ),( yx CP HHH and all ),( yx C are unitary operators. We should remark that the state of the walker is given by )( ),( ,,),,(),( GEl tt yx lyxlyxyx , where Clyx t ),,( is the amplitude of the base lyx ,, at time t and C is the set of complex numbers. The probability of finding the walker at ),( yx at time t is given by )( 22 , ),,(),( GEl tt yx lyxyx . Let us consider initial states starting from the origin, in particular let )0,0( be such that . Concerning the quarter plane which we denote by Q , it is a graph with sites ,2,1,,:),(0,0)( ZZyZxyxQV . Note that ,2,1,,:),( ZZyZxyx is the Cartesian product ZZ . We say sites ),( yx and ),( yx connect if and only if yyxx . Let t Q be the quantum walk on Q whose Hilbert space ,0,0,,,,,:,, DownUpLeftRightlZyxlyxspanH Q . As mentioned earlier, we put T Left , T Right , T Down , T Up , where T is the transposed operator. We should note that the unitary operator is given by QQ FSU , where ZZyxQ
CyxyxcF ,,~0,00,0 , , where Cc ~ is such that c and ZZ C is as given earlier on. The shift operator Q S is given by UpRightS Q ,1,0,0,1,0,0 , RightlRight leftllS Q ,,0,2 ,,0,0,0,1 , UplUp DownllS Q ,,2,0 ,,0,0,1,0 , yxUplUpyx DownlDownyx RightlRightyx LeftlleftyxlyxS Q V. Discrete-time quantum walk on the “join” of quarter planes
Let the graph with joined quarter planes be denoted by k P and let kt P , be the quantum walk on k P . Define kK k . For k Kr with k , put k quarter planes r Q by rrr ZZhQV )0,0()( , where ,2,1,,:),( ZZyZxyxhZZ rr . We say sites ),( yxh r and ),( yxh r connect if and only if yyxx . We define k P as j jk QVPV )()( , and j jk QEPE )()( with r h for any r . Let )0,0( be the origin of k P so that the quarter planes join at )0,0( . Let ,,:,0,0,,,,)0,0(\)(),(:,, kkP llLeftRightDownUplPVyxlyxspanH k be Hilbert space of the quantum walk on k P , where rkrTr . The unitary operator is given by PP FSU ; where )0,0(/)(, ,,~0,00,0 k PVyx ZZkP
CyxyxGcF , where k G will be taken to be the Grover operator, Cc ~ , with c , and ZZ C is as defined earlier. The shift operator P S is given by UphRighthS rrrP ),1,0(),0,1(,0,0 , RightlRighth LeftllhS r rrP ,),0,2( ,,0,0),0,1( , UplUph DownllhS r rrP ,),2,0( ,,0,0),1,0( , yxuplupyxh downldownyxh rightlrightyxh leftlleftyxhlyxhS rrrrrP V. Discrete-time quantum walk on homogeneous trees
Recall the goal of this paper is to obtain the associated limit theorems for quantum walks on the structure defined in the previous section. To adapt the reduction technique of Chisaki et.al [5], we introduce a new type of homogeneous tree. We define a new type of homogeneous tree yx k V , , and a quantum walk yx kt T , , on yx k V , . Fix k , and let x xx k )1(0 , , and let y yy k )1(0 , be the set of generators subject to xx e j and yy e j for k Kj , where k K is as defined earlier on, and x e and y e are the identity of the groups respectively. Let njforiiandnTV jjxxxxk jiinix , njforiiandnTV jjxyyyk jiiniy , and the Cartesian roduct )()( yx kk TVTV be the set of all sites on yx k V , . We will say that sites g and h connect if and only if x y gh . Let )()(),(:, , yxyxk kkVP TVTVyxyxspanH be the Hilbert space of the position subspace, and let yyxxyxVC jjjjyxk spanH ,:, , be the Hilbert space of the coin subspace. The unitary operator is defined by VV FSU where yxykxk eeTVTVyx kkyxyxV GyxyxGceeeeF ,\)()(),( ,,~,, , where Cc ~ is such that c , and yyxxyxV yxyxS ,,,,,, , where r x or r y is given by rkrT . We see immediately that we have the following lemma which requires no proof. Lemma 1 (Reduction of yx kt T , , to kt P , ): The quantum walk on yx k V , is reduced to kt P , with coin operator ,,~0,00,0 k PVyx kkTJ
CyxyxGcF , where
22 22 22 22 )1(11 1)1(1 1)1(1 11)1( kkkkkk kkkkkk kkkkkk kkkkkkk bkkbakbaa kbabkakba kbaabkkba akbakbabkC
VI.
The Reduction Technique
In order to give the limit theorems for kt P , we consider a reduction of kt P , on the quarter plane. We first introduce kt P , which is an enlarged basis of kt P , . Let ),( ** tt YX be associated with the event “ yYxX tt ** , ”, we construct ),( ** tt YX as a reduction of kt P , on a quarter plane. To analyze ),( ** tt YX we give the generating function of the states. By using it we shall obtain the limit states and the characteristic function of kt P , . Let ),( yx t be the state of the quantum walk kt P , at time t and position ),( yx . Define the initial state by k Kj jj ,0,0)0,0( with . Now we consider the state at first step. For any k Kr , Righthjkch rjKj rr k ),0,1()(2~))0,1(( and Uphjkch rjKj rr k ),0,1()(2~))1,0(( . Let jKj rr k jkc )(2~ , then we can rite Righthh rrr ),0,1())0,1(( , and Uphh rrr ),1,0())1,0(( . Using the orthonormal basis kj Kj : , we can write RighthRightheIh rrrrKj Pjjr k ),0,1(),0,1())0,1(( , where P I is the identity operator in k P H and we have let k Kj Pjj I )( . Similarly we can write Uphh rrr ),1,0())1,0(( . Now let H be a Hilbert space spanned by an by an orthonormal basis kj Kj : , then we can define kt P , as a quantum walk on k P HH starting at t with the evolution operator PPkP
FSIU and the initial state Is either
Righth jKj j k ),0,1( or Uph jKj j k ),1,0( where k I is the identity operator on H . Let ,, k l and let DownUpRightLeftl yx ,,, , for )0,0(\)(),( k PVyx . Then the state of the quantum walk kt P , at time t and position ),( yx is written as uyxuyxyx jUuKj jtt yxk ,,),,,(),( , . , where ),,,( dcba t is the amplitude of the base dcba ,,, at time t . In particular we have the following. Lemma 2 (Extending the state of kt P , to kt P , ): For any t and )(),( k PVyx , ),()(),( yxyx tt . Proof:
We show by induction on t . If t , there is nothing to prove. Fix t , and assume ),()(),( yxyx tt , then for any )(),( k PVyx , uyxFSuyxuyxuyxFSIyxU PPluKj jjtluKj jjtPPktP yxkyxk ,,),,,(,,),,,(),( ,, ,, On the other hand ),()( ,,),,,()(,,),,,( ,,),,,()(),()(),( ,, , ,,: ,: yxU uyxFSuyxuyxuyxFS uyxuyxFSyxUyxU tP PPluKj jjtluKj jjtjPP luKj jjtPPtPtP yxkyxk yxk
Since this holds for any yx , , then ),()(),( yxyx tt , and the proof is finished. For any )(),( k PVyx , associate the event “ yPxP ktkt ,, , ” by ),()(),( yxyPxPP tktkt . Lemma 2 implies ,(),( ,,,, yPxPPyPxPP ktktktkt for any )(),( k PVyx . In particular for any initial state of kt P , it is enough to consider the initial state Righth jKj j k ),0,1( or Uph jKj j k ),1,0( for kt P , . Now we introduce ),( ** tt YX as a reduction of kt P , on a quarter plane. Here ),( ** tt YX is defined on a Hilbert space generated by the following new basis. For all DownUpLeftRightl ,,, and Zyx , , k Kj jjL
Own ,0,0,,0,0, , k Kj jjD
Own ,0,0,,0,0, , k k Kj jKk jkR
Other \ ,0,0,,0,0, , k k Kj jKk jkU
Other \ ,0,0,,0,0, , k Kj jjL lyxhlyxOwn ),,(,,,, , k Kj jjD lyxhlyxOwn ),,(,,,, , k k Kj jKk jkR lyxhlyxOther \ ),,(,,,, , k k Kj jKk jkU lyxhlyxOther \ ),,(,,,, . On this basis a step of the time evolution is given by, for Zyx , , RightOtherbc RightOwnbacRightOtherbacRightOwnacOwn
Uk DkkRkkLkL ,1,0,~ ,1,0,~,1,0,~,1,0,~,0,0, RightOtherbac RightOwnacRightOtherkbcRightOwnkbacOwn
Ukk DkRkLkkD ,1,0,~ ,1,0,~,1,0,)1(~,1,0,)1(~,0,0, RightOtherbac RightOwnkbcRightOtheracRightOwnkbacOther
Ukk DkRkLkkR ,1,0,~ ,1,0,)1(~,1,0,~,1,0,)1(~,0,0, RightOtheracRightOwnkbac RightOtherkbacRightOwnkbcOther
UkDkk RkkLkU ,1,0,~,1,0,)1(~ ,1,0,)1(~,1,0,)1(~,0,0, and for each URDL
OtherOtherOwnOwnm ,,, , we have Upyxmc DownyxmacRightyxmacLeftyxmaLeftyxm ,1,, ,1,,,,1,,,1,,,, Upyxmcd DownyxmbcRightyxmadLeftyxmabRightyxm ,1,, ,1,,,,1,,,1,,,, pyxmcd DownyxmadRightyxmbcLeftyxmabDownyxm ,1,, ,1,,,,1,,,1,,,, Upyxmd DownyxmbdRightyxmbdLeftyxmbUpyxm ,1,, ,1,,,,1,,,1,,,, Note that we can write the evolution operator of ** , tt YX as say *** HHH
SFU since the subspace generated by this basis is invariant under the operation P U and we can write the initial state of kt P , as RightOwn L ,1,0, . Note that ZZZyxHH
CyxyxCcF ,** ,,~0,00,0 , where ZZ C is as defined earlier on,
22 22 22 222* )1()1( )1()1( )1()1()1( kkkkkk kkkkkk kkkkkk kkkkkkH ababab kbaabkba kbakbaba kbkbakbaaC , and Cc ~ is such that c . For URDL
OtherOtherOwnOwnm ,,, and DownUpLeftRightl ,,, , the shift operator * H S is given by UpmRightmmS H ,1,0,,0,1,,0,0, * , RightlRightm LeftlmlmS H ,,0,2, ,,0,0,,0,1, * , UplUpm DownlmlmS H ,,2,0, ,,0,0,,1,0, * , UplUpyxm DownlDownyxm RightlRightyxm LeftlLeftyxmlyxmS H ,,1,, ,,1,, ,,,1, ,,,1,,,, * . Let ),( * yx t be the state of the quantum walk ),( ** tt YX . Note that we can write the initial state as RightOwn L ,1,0,)1,0( *1 . Now define for k Kr , PrKj URjDLr
IOtherOtherOwnOwn k }\{ . Let ),( ,, rtrt YX be associated with the event "," ,, yYxX rtrt , where ),(),( yxyYxXP trtrt . For t , the probability is determined by ),( yxh rt in the following way otherwiseyxh yxyYxXP rt rtrtrt ,),(( 0;0,),0,0(),( . The relation between the events )",(" , yxhP rkt and "," ,, yYxX rtrt is given by otherwiseyYxXP yxYXPyxhPP rtrtKj jtjtrkt k ),,( 0;0),0,0()),(( ,, ,,, . It should be noted that k Kj Zyx jtjt
YXP }0{, ,, . VII.
The Generating Function of ),( * yx t Let
URDL
OtherOtherOwnOwnm ,,, . Denote ,0,0 m as Leftm ,0,0 and construct
Rightm ,0,0 as the dummy base. Diagrammatically we can express the evolution operator of the walk using weights and arrows, the weights are given by
222 22 22 2220 cababab kbaabkba kbakbaba kbkbakbaaQQ kkkkkk kkkkkk kkkkkk kkkkkk , bababaIP L , bdbcadacIP R , bdadbcacIQ D dcdcdcIQ U , cbc bac baca k kk kk k . Note that ** , tt YX is a walk starting at t and )0,1( , in particular )0,0(~10000001)0,1( *0*1 Q . We define the generating function for the state by tt t zyxzyx ),(),,(~ . In order to compute the generating function we first define ),,()0,0(~ yx as the weight of all paths starting from )0,0( ending at ),( yx after steps, and ),,()0,0( yx as the weight of all paths on another walk defined by R PQ . We now calculate the generating function for ),,()0,0( yx . Since the first operator could be R P or U Q , the weights of path form RR PP or UU QQ or UR QP or RU PQ . So we express ),,()0,0( yx as a linear combination of RRUR
QPQP ,,, giving )(),();,()0,0( );,()0,0();,()0,0();,()0,0(),,()0,0( IIyxQyxb PyxbQyxbPyxbyx
Uq RpUqRp
U RUR where bdbcadacIP R and cdcdcdcIQ U , and for each URUR qpqpj ,,, we define yxb j . The generating function for ),,()0,0( yx is defined by ),();,()0,0( );,()0,0();,()0,0();,()0,0();,()0,0( IIyxQzyxB PzyxBQzyxBPzyxBzyx
Uq RpUqRp
U RUR here for each URUR qpqpj ,,, we have defined zyxbzyxB jj );,()0,0();,()0,0( . Since the left-hand tensor product of UDRL
QQPP ,,, is I , the generating function for );,()0,0( yx corresponds to an analogue of the result in [23] on the square lattice. We should remark that the result in Oka et.al is largely determined by equation (24) in Chisaki et.al [5], namely )( z . However this requires solving the characteristic equation associated with a system of difference equations in reference [14] of Oka et.al [23] . Since we can map the result of Oka et.al [23] onto the square lattice, we mainly need to determine the roots of the characteristic equation from the corresponding system of difference equations for the walk on the square lattice. In particular for sufficiently small z , we have yxda zdzyxB yxp R , yxda zdzyxB yxq U , , zB UR qp
222 2222 yxzca zaza zdzyxB yxp R
222 2222 yxzca zaza zdzyxB yxq U where zc zczzz . Since da , we can take r such that z for rz . Next we calculate the generating function for ),0,0()0,0(~ . To do so we introduce n ;),0,0()0,0(~ as the weight of all paths starting from the origin n times before ending at the origin at time . Consider , for we obtain UDRL UUqDRRpL
QPPP QQyxbPPPyxbP UR ~~)( ~)2);,()0,0(~)2);,()0,0())(1(0;),0,0()0,0(~ where
00 0~)1(~ cab bkaP kk kkR , and U Q ~ is defined in a similar way. For , we define . So the generating function for is given by ~~ );0,0()0,0(~);0,0()0,0(~0,);0,0()0,0( zQPPP zzBQQPzzBPPPz UDRL qUUDpRRL UR We should remark that the above can be written in simplified form involving few operators via multiplication rules for the
DURLDURL
QQPPQQPP matrices, an analogue of the multiplication rules for the
PQRS matrices in reference [14] of Oka et.al [23]. In our paper we do not consider the simplification, since the size of the matrices makes this time consuming. Similarly, for , we have UDRL qUUDpRRL UDRL qUUDpRRL
QPPP bQQPbPPP QPPP bQQPbPPP
Ur Ur ~~)( );0,0()0,0(~);0,0()0,0(~)(1( ~~)( );0,0()0,0(~);0,0()0,0(~)(1(1;);0,0()0,0(~
22 2222 4 12 1112 and for we define . Thus the generating function for is obtained as ~~);0,0()0,0(~);0,0()0,0(~~1;);0,0()0,0(~ zQPPPzzBQQPzzBPPPcz UDRLqUUDpRRL UR Recursively we the following formula for n , ~ ),0,0()0,0(~~2 )1(1;);0,0()0,0(~ c zzBPPPcnzB npRRLnp RR ~ ),0,0()0,0(~~2 )1(1;);0,0()0,0(~ c zzBQQPcnzB nqUUDnq UU From the two equations immediately above we get the generating function for );0,0()0,0(~ by summing over n . We should remark that for ),min( rcrz we have zzzBQQP U qUUD and zzzBPPP R pRRL . So for z such that rz , ~),0,0()0,0(~);0,0()0,0();0,0()0,0( IIQzBPzBz
UqRp UR , where ;);0,0()0,0();0,0()0,0( n pp nzBzB rr and zB u q );0,0()0,0( ~ is defined in a similar way. Note that );,()0,0(~ yx is written by );,()0,0( yx and );,()0,0(~ yx as ),();0,0()0,0(~~);1,()0,0( );0,0()0,0(~~);,1()0,0();,()0,0(~ IIyxPyx Pyxyx
L R From the generating function for );,()0,0( yx and );0,0()0,0(~ we can compute the generating function for );,()0,0(~ yx as follows: Let },,,{ URUR qpqpj , and let the corresponding operator be given by },,,{
URUR
QPQPJ . For example for R pj , the corresponding operator is R PJ , so that the summation over Jj , below is understood to be taken in only one way, then we can write zQIIyxJzyxBJzyxBzyx Jj jj ~),());1,()0,0());,1()0,0();,()0,0( , 4410 So we obtain the generating function as ),();,()0,0(~);,(~ *00* yxzyxzyx VIII.
Towards a Localization Criterion
We compute the limit state of ),( ** tt YX from the generating function for the state which we defined very early on in the previous section as tt t zyxzyx ),(),,(~ . Note that this generating function can be written as lyxmzlyxmzyx DownUpRightLeftl OtherOtherOwnOwnm
URDL ,,,;,,,),,(~ ,,, ,,,* . Now let zc zczzz , )()( c zzdz , zazzv , zccz , ~~~ ~ ccacc cccw , bcad , then there exists r , so that for any z with rz , we have the following, assuming a one-to-one correspondence between URDL
OtherOtherOwnOwnm ,,, and ,,, kkkkkk bbabaas , yxzyxzscz yxzyxzsczazcadzLeftyxm yx yxzyxzsczzRightyxm yxzyxzscz yxzyxzsczbzdbczDownyxm yx yxzyxzsczzUpyxm ~4 )()()()(~)();,( wzwzcc zvzzvzwwca zdzyx yx . From Cauchy’s Theorem for rr , );,(~21),( trzt zdzzyxiyx . Let ,,, , then as t , )1(** ),;,(~Re~),( tgt ggzyxsyx , where wzfs ),(Re is the residue for )( zf for wz . Now taking the residues of the generating functions, we have the limit states as follows: Let
221 22102222 222 ityx , where cos, ]1,(2 c IcI and cos, ),1[2 c IcI , and assume that there is a 1-1 correspondence between the sets hbaaOwnOwn kkkDL ,, and qbbaOtherOther kkkUR ,, . If DL OwnOwnm , and DownLeftl , , then yxtyxhwccItyxhwccI yxtyxhwca cccwcItyxhwca cccwcIlyxm tyxt f UR OtherOtherm , and DownLeftl , , then yxtyxqwccItyxqwccI yxtyxqwca cccwcItyxqwca cccwcIlyxm tyxt If DL OwnOwnm , and UpRightl , , then yx yxtyxhcItyxhcIlyxm tyxt If UR OtherOtherm , and UpRightl , , then yx yxtyxqcItyxqcIlyxm tyxt After some calculations with ),(),( yxyYxXP trtrt and the above limit states we arrive at the localization criterion. Let ~arg cc , i ecK , ii ececK . Recall that ),( yx t is the state of the quantum walk kt P , at time t and position ),( yx . For k Kr , define K Kj jj ,0,0)0,0( , jKj rr K jkc )(2~ , rr , }\{2 rKj jrr r , }\{ 2}\{,3 rKj jkjkj rKkj jkr kr , where )( j r is the indicator function at r . The localization criterion is contained in the following theorem. Theorem 1(Localization):
For k , }0{, Zyx , k Kr , ),,(cos),(cos),(cos2)1(1~),( ),(]1,(),1[2,, tyxLIyxLIyxLIyYxXP rcccrpcmctyxrtrt where )(~)( tgtf means tg tf t , and ),(),( kj jm yxyxL , ),(),( kj rjrp yxyxL , q rqh rqh rh rrc tyxqtyxhqtyxhqtyxhtyxL ),,(Re2),,()(Re)1(2),,(Re)1(2),,(Re)1(2),,(
32, 2, 212 KaKayxyxK ccqyx yxq
KaKKayxKKyxK ccKKtyx yx We should remark that ),,( tyxL rc plays a similar role to the measure corresponding to localization in the following papers [6,24] for example. In particular ),,( tyxL rc is an oscillatory term corresponding to localization. IX.
Towards a Weak Limit Theorem
We calculate the Fourier transform of the generating function as yx yisxisyx yx ezyxzss , ** );,(~);,(ˆ~ from the following relations yxzyxzscz yxzyxzsczazcadzLeftyxm yx yxzyxzsczzRightyxm yxzyxzscz yxzyxzsczbzdbczDownyxm yx yxzyxzsczzUpyxm , then we obtain the characteristic function from the following relation
20 20 **20 20 ,,, )()(** 20 20 )()(,,, **, ** ,, yxyxtryxtr yxZyxyx yixiyyikxxiktrtr yxyyikxxikyixiZyxyx trtr yixiZyx trtrYiXi dsdsssss dkdkeeyxyx dkdkeeyxyx eyxyxeE yx yxrtrt where vu , is the inner product of the vectors vu , . Now we write },,,{ ,,, ** ,);,,,(ˆ~);,(ˆ~ UpDownRightLeftl OtherOtherOwnOwnm yxyx
URDL lmzsslmzss , then it follows from the relations yxzyxzscz yxzyxzsczazcadzLeftyxm yx yxzyxzsczzRightyxm yxzyxzscz yxzyxzsczbzdbczDownyxm yx yxzyxzsczzUpyxm that we have the following, );,(1)(~)()();,,,(ˆ~ zsszsczazcadzzssLeftm yxyx );,();,()(~1);,,,(ˆ~ zsszsszczzssRightm yxyxyx );,(1)(~)()();,,,(ˆ~ zsszscbzzdbczzssDownm yxyx );,();,()(~1);,,,(ˆ~ zsszsszczzssUpm yxyxyx , where ~4 )()()()(~);,( wzwzcc zvzzvzwwczss yx , )()(4 )();,();,( svzsvz zvzssezss yxikikyx yx , zzeazss yx isisyx , yxyxyxyx isisisisisisisis eaeaeaeasv Next we get the Fourier transform of the state, ),(ˆ * yxt ss . Since );,(ˆ~ zss yx is finite for rz , we can write ),(ˆ);,(ˆ~ t tyxtyx zsszss . In particular for rz , rz tyxyxt zdzzssiss );,(ˆ~21),(ˆ , thus we have the following relation for ),(ˆ * yxt ss : Put ,,, and vvr , , then )1()1(* ),;,(ˆ~Re),;,(ˆ~Re~),(ˆ tr yxtg yxyxt rrzsssggzsssss . Finally using the expression immediately above and the relation for rtrt YiXi eE ,, , we obtain the following equations rpkj rjqcyxwwg yxr CK ccqIdsdsgsss mkj jqcyxwwg yxr CK ccqIdsdsgsss and if wwg , we obtain
20 20 yxyxryxryxryxr dsdsgsssgsssgsssgsss
From the relation involving rtrt
YiXi eE ,, and the Riemann-Lebesgue Lemma we have
20 20 ),(),(20 20 ),(),( ,, yxyxsshisshiyxyxsshisshimrptYitXit dsdsssqedsdssspeCCeE yxyxyxyxrtrt where ),,(ˆ~Re),( vsssssp yxryx and ),,(ˆ~Re),( vsssssq yxryx , and )()(),( yxyx shshssh , where )(cos1 )(sin)( xxx sa sash and )( y sh is defined in a similar way. Choosing suitable contours C and C we can write the last two terms on the right hand side of tYitXit rtrt eE ,, lim as dxdyyxfyxwe dsdsssqsspsspsspe dsdsssqsspsspsspe dsdsssqedsdssspe Hsshisshi yxyxyxyxyxC sshisshi yxyxyxyxyxC sshisshi yxyxsshisshiyxyxsshisshi yxyx yxyx yxyx yxyxyxyx ),(),( 22),(),(),(),( 22),(),(),(),( 22),(22),( where )()(),( yfxfyxf HHH ,
222 2),0[ )1( 1)()( xax axIxf aH and )( yf H is defined in a similar way, )( x shx , )( y shy . After some computations with ),( yxw from ),( yx ssp and ),( yx ssq we get the weak limit theorem as follows. Theorem 2(Weak Convergence):
For k , k Kr , as t we have vu vu rwrtrt dxdyyxvtYuvtXuP ),(),( ,, , where ),(),(),()(cos)(cos),( yxfyxCyxCICIyx Hrdrpcmcrw , m C , rp C , and ),( yxf H are as defined earlier on, and )()(),( yCxCyxC rdrdrd , where sin)1(sin)1( )()(Re2)()( xKxK xxxxC rrrrd , where xcccaaxacax kkk xccecabiexacbx ikkik sin)1(cos21)( xccbx k , and )( yC rd is defined similarly. X. Concluding Remarks
In this paper we have obtained an explicit expression for the limit probability of kt P , , a result corresponding to localization- there exists a site of the graph v such that , vPP ktt . We have also obtained the weak convergence of kt P , . Moreover, the limit measure has a density function )()(),( yfxfyxf HHH , where )( xf H and )( yf H are both half-line versions of a typical function in the weak convergence of the quantum walk [23-29]. The function is defined by