The time-averaged limit measure of the Wojcik model

0.25 MB·

by  Takako Endo

The time-averaged limit measure of the Wojcik model
4
1 TAKAKOENDO
0 Department of Physics,
2 Ochanomizu University,2-1-1 Ohtsuka, Bunkyo, Tokyo, 112-0012, Japan

l
u NORIOKONNO
J
Department of Applied Mathematics, Faculty of Engineering,

0 Yokohama National University,Hodogaya, Yokohama, 240-8501, Japan
1

]
h
p
Abstract
-
h
Weinvestigate“theWojcikmodel”introducedandstudiedbyWojciketal. [1],whichisaone-defectquantum
t
a walk (QW) having a single phase at the origin. They reported that giving a phase at one point causes an
m astonishing effect for localization. There are three types of measures having important roles in the study of
QWs: time-averaged limit measure, weak limit measure, and stationary measure. The first two measures imply
[
a coexistence of localized behavior and the ballistic spreading in the QW. As Konno et al. [3] suggested, the

2 time-averaged limit and stationary measures are closely related to each other for some models. In this paper,we
v focusonarelationbetweenthetwomeasuresfortheWojcikmodel. Thestationarymeasurewasalreadyobtained
0
by our previous work [2]. Here, we get the time-averaged limit measure by several methods. Our results show
7
that the stationary measure is a special case of thetime-averaged limit measure.
0
3
.
1
0 1 Introduction
4
1 Asaquantumcounterpartoftherandomwalk,quantumwalks(QWs)describemanykindsofphenomenainquantum
: scale [11, 12]. There are two distinct types of QWs, one is the discrete time walk and the other is the continuous
v
i one. Discrete time QWs have been intensively studied in [8, 18]. Here, we focus on a two-state discrete time QW
X
in one dimension. The two-state corresponds to left and right chiralities, respectively [4]. It has been reported that
r
a one-dimensional discrete time QWs have characteristic properties, that is, localization and the ballistic spreading.
TherearetwokindsoflimittheoremstoshowtheasymptoticbehavioroftheQWs: thetime-averagedlimittheorem
correspondingto localization,andthe weaklimittheoremcorrespondingto the ballisticspreading. Inthis paper,we
say that the walk starting from the origin exhibits localization if and only if its time-averaged limit measure at the
origin is strictly positive. As Konno et al. [3] reported, the time-averaged limit and stationary measures are closely
related to eachother. Therefore,we clarify the relationbetween the two measuresfor a suitable QWmodel. Wojcik
et al. [1] showed that giving a phase at a single point in the QW on the line exhibits an astonishing localization
effect. In this paper, we callthe model “the Wojcik model”. Our previous work [2] gavea stationarymeasure of the
1
model, andthis paper isa sequentialworkof[2]. We presentthe time-averagedlimit measure,derivedfromthe pass
counting method [5, 7], the CGMV method [15], and the generating function method [3] explained in Sect. 5. Our
result implies that the stationary measure is a special case of the time-averagedlimit measure.
The rest of this paper is organized as follows. Section 2 gives the definition of the time-averaged limit measure
and localization for the discrete time QW starting at the origin. In Sect. 3, we introduce the Wojcik model and
presentourmainresults,Theorems1and2. Section4isdevotedtotheresultbasedontheCGMVmethod. Wegive
the proofs of Lemma 1 in Sect. 5 and Theorem 2 in Sect. 6, respectively. Appendix A gives the proof of Theorem
1, and Appendix B presents the proof of Lemma 7.
2 The time-averaged limit measure and localization
Inthissection,weintroducethetime-averagedlimitmeasureanddefinelocalizationfortheQWstartingattheorigin.
First, we givethe notationof the space-inhomogeneousQWs onthe line. The walkerhas a coinstate described by a
two-dimensionalvector which is called “the probability amplitude”. We define the coin state at position x and time
n by
ΨL(x)
Ψ (x)= n .
n ΨR(x)
(cid:20) n (cid:21)
The upper and lower elements express left and right chiralities, respectively. Let
Ψ =T[...,ΨL( 1),ΨR( 1),ΨL(0),ΨR(0),ΨL(1),ΨR(1),...],
n n − n − n n n n
where T means the transposedoperation. The time evolutionis defined byits initialcoinstate Ψ and2 2unitary
0
×
matrices U (x Z) :
x
∈
a b
U = x x ,
x c d
x x
(cid:20) (cid:21)
where subscript x Z denotes the location. Then the evolution is determined by the following recurrence formula:
∈
Ψ (x)=P Ψ (x+1)+Q Ψ (x 1),
n+1 x+1 n x−1 n
−
where
a b 0 0
P = x x , Q = .
x 0 0 x c d
x x
(cid:20) (cid:21) (cid:20) (cid:21)
Note that P (resp. Q ) expresses that the walker moves to the left (resp. right) at position x in each time step.
x x
Let R =[0, ). Then for
+
∞
Ψn =T ..., ΨΨRLn((−11)) , ΨΨRLn((00)) , ΨΨRLn((11)) ,... ∈(C2)Z,
(cid:20) (cid:20) n − (cid:21) (cid:20) n (cid:21) (cid:20) n (cid:21) (cid:21)
we define a map µ :Z [0, ] as
n
→ ∞
µ (x)= ΨL(x)2+ ΨR(x)2 (x Z).
n | n | | n | ∈
Our interest in this paper is the sequence of measures:
µ ,µ ,µ ,... .
0 1 2
{ }
2
If µ is a probability measure, let X be a random variable defined by µ , that is, for x Z,
n n n
∈
P(X =x)=µ (x).
n n
Now we introduce the time average of µ (x) and its limit. The time average of µ (x) is defined by
n n
T−1
1
µ (x)= µ (x),
T T n
n=0
X
and if the limit exists, we define the limit of µ (x) by
T
T−1
1
µ (x)= lim µ (x)= lim P(X =x). (1)
∞ T→∞ T T→∞T n
n=0
X
Here we put
= µ =µΨ0 ZZ 0 :Ψ CZ , (2)
M∞ { ∞ ∞ ∈ +\{ } 0 ∈ }
whereµΨ∞0 representsthe dependence onthe initialstateΨ0 and{0}=T[...,0,0,0,...]. We callthe elementofM∞
the time-averagedlimit measure of the QW. Then, localization for discrete time QW is defined as follows.
Definition 1 We say that localization for the QW starting at the origin happens if
µ (0)>0.
∞
3 Model and main results
3.1 Model
In this paper, we treat a space-inhomogeneous QW, “the Wojcik model”, introduced by Wojcik et al. [1], whose
time evolution is defined by the unitary matrices U (x Z) as follows.
x
∈
H (x Z 0 ),
Ux = ωH (x∈=0)\, { } (3)
(cid:26)
where ω =e2πiφ with φ (0,1). The model has a weight e2πiφ at the origin. Here, H is “the Hadamard matrix”:
∈
1 1 1
H = . (4)
√2 1 1
(cid:20) − (cid:21)
In particular, if φ 0, then the Wojcik model becomes space-homogeneous and is equivalent to the well-known
→
Hadamardwalk whichis one ofthe most intensively studied QWs. We shouldnote thatKonno et al. [3] treatedthe
QW in which det(U ) does not depend on the position x Z. However,the Wojcik model has
x
∈
det(U )= 1, det(U )= ω2 (= 1if x Z 0 ).
0 x
− − 6 − ∈ \{ }
In this paper, we assume that the walk starts at the origin with the initial coin state ϕ = T[α,β], where α,β C
∈
and α2+ β 2 =1.
| | | |
3
3.2 Main result 1: Time-averaged limit measure at the origin
Inthissubsection,wegivethetime-averagedlimitmeasureattheorigin. Weshouldremarkthatwetreatthemeasure
for x 1 case in subsection 3.3. Let us consider the initial coin states ϕ = T[1/√2,i/√2], or ϕ = T[1/√2, i/√2]
| | ≥ −
for a while. At first, we focus on the Hadamard walk (φ 0 case). For the initial coin state, the probability
→
distributionofthewalkissymmetricfortheoriginatanytime. Ifµ isaprobabilitymeasure,letX betherandom
n n
variable of the walk for the position x at time n. We compute the return probability at time n, which we denote it
as r(H)(0)=P(X =0). We should note that r(H) (0)=0(n 0). By a brief calculation, we have
n n 2n+1 ≥
r(H)(0)=0.5, r(H)(0)=0.125, r(H)(0)=0.125, r(H)(0)=0.07031,
2 4 6 8
r(H)(0)=0.07031, r(H)(0)=0.04882, r(H)(0)=0.04882,....
10 12 14
In fact, we see
lim r(H)(0)=0, (5)
2n
n→∞
for example, see [5]. Equation (5) suggests that the Hadamard walk (φ 0 case) does not show localization
→
. From now on, we consider a space-inhomogeneous case, that is, φ (0,1) case. By a simple calculation, we
∈
have the same probability measure as that of the Hadamard walk at time n = 1,2,3 for the initial coin states
ϕ = T[1/√2,ηi/√2] (η = 1, 1). However, we see that the probability measure at time n = 4 depends on the
−
parameter φ. Actually, we have
1 2(2+E)
P(X = 4)=P(X =4) = , P(X = 2)=P(X =2)= ,
4 4 4 4
− 16 − 16
2(3 2E)
P(X =0) = − ,
4
16
where E = C +ηS (η = 1, 1), C = cos(2πφ), and S = sin(2πφ). Hereafter, we present the time-averaged limit
−
measure at the origin for the Wojcik model. Let
ΨL (0)
Ψ (0)= 2n
2n ΨR (0)
(cid:20) 2n (cid:21)
be the probability amplitude at time 2n at the origin. Then, we obtain an explicit expression for Ψ (0) as follows.
2n
Lemma 1 Let ϕ=ϕ(η)=T[1/√2,ηi/√2](η =1, 1) be the initial coin state. Then, we have
−
n k k
1 ω( 1+ηi) 1
Ψ (0)= r∗ − ,
2n √2  2aj−1 2 ηi
kX=1(a1,...,aXk)∈(Z>)k: jY=1 (cid:18) (cid:19) (cid:20) (cid:21)
a1+···+ak=n  
for n 1, where Z = 1,2, , and
>
≥ { ···}
∞ 1 z2+√1+z4
r∗zn = − − .
n z
n=1
X
We prove Lemma 1 in Sect. 5. Noting that the return probability is defined as P(X = 0) = Ψ (0) 2 =
2n 2n
k k
ΨL (0)2+ ΨR (0)2, we have
| 2n | | 2n |
4
Lemma 2 Let ϕ = ϕ(η) = T[1/√2,ηi/√2] (η = 1, 1) be the initial coin state. Then the limit of the return
−
probability at time 2n for the parameter φ is given as follows.
c(φ) = lim r (0)
2n
n→∞
2 2
1 √2C 1 √2C
− +
= 4 − I (φ)I (η)+4 − I (φ)I (η),
3 2√2C−! (1/4,1) {1} 3 2√2C+! (0,3/4) {−1}
− −
where I (x)=1(x A), I (x)=0(x A), and
A A
∈ 6∈
π √2
C =cos 2πφ = cos(2πφ)+sin(2πφ) ,
−
− 4 2 { }
(cid:16) π(cid:17) √2
C =cos 2πφ+ = cos(2πφ) sin(2πφ) .
+
4 2 { − }
(cid:16) (cid:17)
Interestingly, when φ (0,1), we see that the inequality
∈
c(φ)>0
holds except for η = 1 with φ (0,1/4] or η = 1 with φ [3/4,1). On the other hand, when φ 0, we have
∈ − ∈ →
c(φ)=0 which implies that the Hadamard walk does not exhibit localization.
As for the generalcase,that is,for the initial coinstate ϕ=T[α,β](α,β C, α2+ β 2 =1), we canobtainthe
∈ | | | |
probability amplitudes and the time-averagedlimit measure by Lemma 1. Now, we should note that for the general
initial coin state ϕ, we have
n k
1
Ψ (0)= Ξ∗ ϕ
2n √2  2aj
Xk=1(a1,...,aXk)∈(Z>)k: jY=1
a1+···+ak=n  
1 n k ω k 1 1 k α
= r∗ −
√2  2aj−1 2 1 1 β
Xk=1(a1,...,aXk)∈(Z>)k: jY=1 (cid:16) (cid:17) (cid:20) − − (cid:21) (cid:20) (cid:21)
a1+···+ak=n  
for n 1, where Z = 1,2,... and
>
≥ { }
∞ 1 z2+√1+z4
r∗zn = − − .
n z
n=1
X
Here we should note
α iβ α+iβ
k ( 1+i)k − +( 1 i)k
1 1 α − 2 − − 2
− = (cid:18) (cid:19) (cid:18) (cid:19) .
1 1 β α iβ α+iβ
(cid:20) − − (cid:21) (cid:20) (cid:21) ( 1+i)ki − +( 1 i)k( i)
 − 2 − − − 2 
 (cid:18) (cid:19) (cid:18) (cid:19) 
 
Therefore, we obtain the concrete expression of Ψ (0) for the general initial coin state ϕ as follows.
2n
5
Lemma 3 For the initial coin state ϕ=T[α,β](α,β C, α2+ β 2 =1), we have
∈ | | | |
n k k
α iβ ω( 1+i) 1
Ψ (0)= − r∗ −
2n 2  2aj−1 2 i
(cid:18) (cid:19)Xk=1(a1,...,aXk)∈(Z>)k: jY=1 (cid:18) (cid:19) (cid:20) (cid:21)
a1+···+ak=n  
n k k
α+iβ ω( 1 i) 1
+ r∗ − − ,
2  2aj−1 2 i
(cid:18) (cid:19)kX=1(a1,...,aXk)∈(Z>)k: jY=1 (cid:18) (cid:19) (cid:20) − (cid:21)
a1+···+ak=n  
for n 1.
≥
Thus,multiplyingtheresultsby(α ηiβ)/√2(η =1, 1)foreachinitialcoinstateϕ=ϕ(η)=T[1/√2,ηi/√2](η =
− −
1, 1) and then summing the results each other gives the time-averagedlimit measure. By Lemma 3, we obtain the
−
following one of our main results.
Theorem 1 1. For the initial coin state ϕ=T[α,β](α,β C, α2+ β 2 =1), we have
∈ | | | |
1 E 1 E
Ψ(L,ℜ)(0) (α iβ) − + cos(nθ )I (φ)+(α+iβ) − − cos(nθ )I (φ),
2n ∼ − 3 2E 0 (1/4,1) 3 2E 0 (0,3/4)
+ −
− −
1 E S C 1 E S+C
Ψ(L,ℑ)(0) (α iβ) − + − sin(nθ )I (φ)+(α+iβ) − − sin(nθ )I (φ),
2n ∼ − 3 2E S C 0 (1/4,1) 3 2E S+C 0 (0,3/4)
+ −
− | − | − | |
1 E S C 1 E S+C
Ψ(R,ℜ)(0) (α iβ) − + − sin(nθ )I (φ).+(α+iβ) − − sin(nθ )I (φ),
2n ∼− − 3 2E S C 0 (1/4,1) 3 2E S+C 0 (0,3/4)
+ −
− | − | − | |
1 E 1 E
Ψ(R,ℑ)(0) (α iβ) − + cos(nθ )I (φ) (α+iβ) − − cos(nθ )I (φ),
2n ∼ − 3 2E 0 (1/4,1) − 3 2E 0 (0,3/4)
+ −
− −
for n 1, where
≥
(j,ℜ) (j) (j,ℑ) (j)
Ψ (0) = Ψ (0) , Ψ (0)= Ψ (0) (j =L,R),
2n ℜ 2n 2n ℑ 2n
(cid:16) (cid:17) (cid:16) (cid:17)
E = C S =cos(2πφ) sin(2πφ),
±
± ±
2(1 E)2 (2 E)S C
cosθ − , sinθ = − | − |.
0 0
− 3 2E 3 2E
− −
Here, we should note that (z) is the real part and (z) is the imaginary part of z (z C). Moreover, we have
ℜ ℑ ∈
6
2.
r (0)
2n
µ (0) = lim
∞ n→∞ 2
2 2
1 E 1 E
= − + α iβ 2I (φ)+ − − α+iβ 2I (φ).
3 2E | − | (1/4,1) 3 2E | | (0,3/4)
(cid:18) − +(cid:19) (cid:18) − −(cid:19)
2 2
1 √2C 1 √2C
= − − α iβ 2I (φ)+ − + α+iβ 2I (φ),
3 2√2C−! | − | (1/4,1) 3 2√2C+! | | (0,3/4)
− −
where
π 1
C =cos 2πφ = cos(2πφ) sin(2πφ) .
±
± 4 √2{ ∓ }
(cid:16) (cid:17)
The proof of Theorem 1 appears in Appendix A.
3.3 Main result 2: Time-averaged limit measure for general x Z
∈
Let us consider the Wojcik model starting at the origin with the initial coin state ϕ = T[α,β], where α,β C with
∈
α2+ β 2 =1. In this subsection, we present the time-averagedlimit measure µ (x)(x Z).
| | | | ∞ ∈
Theorem 2 1. x=0.
µ (0)=µ(1)(0)+µ(2)(0).
∞
2. x=0.
6
1 |x| 1 |x|
µ (x)=(2 √2C ) µ(1)(0)+(2 √2C ) µ(2)(0),
∞ − + 3 2√2C − − 3 2√2C
(cid:18) − +(cid:19) (cid:18) − −(cid:19)
where
(1 √2C )2
µ(1)(0)= − + α+iβ 2I (φ),
(3 2√2C )2| | (0,3/4)
+
−
and
(1 √2C )2
µ(2)(0)= − − α iβ 2I (φ),
(3 2√2C )2| − | (1/4,1)
−
−
with
π 1
C =cos 2πφ+ = cos(2πφ) sin(2πφ) ,
+
4 √2{ − }
 (cid:16) π(cid:17) 1
 C− =cos 2πφ− 4 = √2{cos(2πφ)+sin(2πφ)}.
(cid:16) (cid:17)
We emphasize that the time-averagedlimit measure is symmetric for the originand localizationheavily depends
on the choice of the initial coin state ϕ and parameter φ. For instance, when α = iβ and φ (3/4,1), we see
∈
µ (x) = 0(x Z) holds. When α= iβ and φ (0,1/4), we also have µ (x) =0 (x Z). Moreover, our results
∞ ∈ − ∈ ∞ ∈
imply that the stationary measure given by [2] stated below is a special case for the time-averaged limit measure.
The proof of Theorem 2 is given in Sect. 6.
Hereweconsidertherelationbetweenthetime-averagedandstationarymeasures. First,wepresentthestatioary
measure for the Wojcik model in Theorem 2 of Ref. [2] as follows:
7
Theorem 3
Γ(φ) (x=0),
µ(x)=kΨ(x)k2 =2|α|2|θs|2|x|× 1 (x=6 0),
(cid:26)
where
2 cos(2πφ) sin(2πφ) (β =iα),
Γ(φ)= − −
2 cos(2πφ)+sin(2πφ) (β = iα),
(cid:26) − −
and
1
(β =iα),
3 2cos(2πφ) 2sin(2πφ)
θ 2 = − − (6)
| s|  1 (β = iα).
3 2cos(2πφ)+2sin(2πφ) −
 −
We should note that
1 1
=
3 2√2C 3 2cos(2πφ) sin(2πφ)
− + − −
and
1 1
=
3 2√2C 3 2cos(2πφ)+2sin(2πφ)
− − −
in Theorem 2 agree with θ 2:
s
| |
1
(β =iα),
3 2cos(2πφ) 2sin(2πφ)
θ 2 = − −
| s|  1 (β = iα).
3 2cos(2πφ)+2sin(2πφ) −
 −
From now on, we consider the two cases. When α = 1/√2 and β = i/√2 for instance, we have α+iβ = 0 and
α iβ =√2. Then we have the time-averagedlimit measure as follows.
−
2(1 √2C )2
−
− I (φ) (x=0),
(3 2√2C )2 (1/4,1)
µ∞(x)= (2−−√2C−−) 3 21√2C |x|µ∞(0) (x6=0). (7)
(cid:18) − −(cid:19)
On the other hand, Theorem 2 in Ref. [2] gives the stationary measure as
c2 (x=0),
| |
µ (x)= 1 |x| (8)
∞  (2 √2C )c2 (x=0).
−
 − | | 3 2√2C 6
(cid:18) − −(cid:19)

Equations (7) and (8) suggest that when c2 =2(1 √2C )2/(3 2√2C )2, then the time-averagedlimit measure
− −
| | − −
coincides with the stationary measure.
8
Next, when α=1/√2 and β = i/√2, we have α+iβ =√2 and α iβ =0. Then we obtain the time-averaged
− −
limit measure as follows.
2(1 √2C )2
+
− I (φ) (x=0),
(3 2√2C )2 (0,3/4)
µ∞(x)= (2−−√2C++) 3 21√2C |x|µ∞(0) (x6=0). (9)
(cid:18) − +(cid:19)
On the other hand, Theorem 2 in Ref. [2] gives the stationary measure.
c2 (x=0),
| |
µ (x)= 1 |x| (10)
∞  (2 √2C )c2 (x=0).
+
 − | | (cid:18)3−2√2C+(cid:19) 6
Equations(9)and(10)suggestthatwhen c2 =2(1 √2C )2/(3 2√2C )2,then the time-averagedlimit measure
+ +
| | − −
also coincides with the stationary measure.
4 Result via the CGMV method
We can derive the time-averaged limit measure at the origin µ (0) also from the CGMV method [15]. From now
∞
on, we use the same expressions as in Ref. [15]. Applying the CGMV method to the Wojcik model, we have
i i 1 i
a= e−2πφi, b= , w =e2πφi, ζ (b)= + .
±
√2 √2 ±√2 √2
As condition , we see that the following inequality holds.
+
M
1
<φ<1.
4
On the other hand, as condition , we obtain
−
M
3
0<φ< .
4
Moreover,we have
σ π
σ =0, σ =π, σ =σ +σ =π, θ = = ,
1 2 1 2
2 2
τ =2πφ, τ =2πφ+π, τ =τ +τ =4πφ+π,
1 2 1 2
and
1 1
C = (C+S), C = (C S),
+ −
√2 √2 −
with
C =cos(2πφ), S =sin(2πφ).
Conditions and imply 1/4<φ<1 and 0<φ<3/4, respectively. According to the CGMV method, we get
+
M M
1 ρ2 2 (αˆ 2 βˆ2) b+2ρ (ωαˆβˆ)
lim P(0)(2n)= 1 a 1 | | −| | ℜ bℜ , (11)
n→∞ α,β 2(cid:18) − |ζ±(b)−a|2(cid:19) ( ∓ √1−ℑ2b )
9
whereP(0)(2n)istheprobabilitythatthewalkerreturntotheoriginattime2nwiththeinitialcoinstateϕ=T[α,β]
α,β
where α,β C and α2+ β 2 =1. Here,
∈ | | | |
1 1 1
ρ = , ζ (b) a2 = (3 2(C S))= (3 2√2C ),
a ± ±
√2 | − | 2 − ± 2 −
1 1
b= , b=0, ρ = , αˆ =λˆ(1)α=α,
ℑ √2 ℜ b √2 0
βˆ=λˆ(2)β =ei((σ2−σ1)/2+τ2−σ2)β =eπi/2ω =iω.
1
Therefore, Eq. (11) becomes
2(1 √2C )2
− ± α iβ 2 (φ (1/4,1)),
(3 2√2C )2| − | ∈
lim P(0)(2n)= − ±
n→∞ α,β  2(1−√2C±)2 α+iβ 2 (φ (0,3/4)).
(3 2√2C )2| | ∈
Thus, we obtain the time-averagedlimit measur−e µ (0)±as follows.
∞
1
µ (0)= lim P(0)(2n)
∞ 2n→∞ α,β
2 2
1 √2C 1 √2C
= − + α iβ 2I (φ)+ − − α+iβ 2I (φ),
3 2√2C+! | − | (1/4,1) 3 2√2C−! | | (0,3/4)
− −
which agrees with our result.
5 Proof of Lemma 1
At first, we consider the case of the Hadamard walk on Z = 0,1,2, starting at m( 1).
≥
{ ···} ≥
1 1 1
U =H = (x 0), (12)
x √2 1 1 ≥
(cid:20) − (cid:21)
which can be devided into P and Q as
x x
U =P +Q (x 1), (13)
x x x
≥
where
1 1 1 1 0 0
P =P = , Q =Q= .
x √2 0 0 x √2 1 1
(cid:20) (cid:21) (cid:20) − (cid:21)
Next let Ξ(∞,m) be the sum of all the passages starting at m ( 1) and arrive at the origin at time n for the first
n
≥
time. For instance, we have
Ξ(∞,1) =P2QPQ+P3Q2.
5
Here we introduce R and S as
1 1 1 1 0 0
R= − , S = .
√2 0 0 √2 1 1
(cid:20) (cid:21) (cid:20) (cid:21)
10