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PureMathematicsnØêÆ,2022,12(8),1381-1391
PublishedOnlineAugust2022inHans.http://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2022.128151
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þfi§šàg§²-ÿݧ˜—ÿÝ
UniformStationaryMeasureof
Space-InhomogeneousOne-Dimensional
Three-StateQuantumWalks
PengYe,LixiaZhang,CaishiWang
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:Jul.19
th
,2022;accepted:Aug.19
th
,2022;published:Aug.31
st
,2022
©ÙÚ^:“+,Üw_,â¬.˜mšàg˜‘nþfi˜—²-ÿÝ[J].nØêÆ,2022,12(8):
1381-1391.DOI:10.12677/pm.2022.128151
“+
Abstract
Inthispaper,weconsidertheuniformstationarymeasureofspace-inhomogeneous
threestatequantumwalksonthelineandcycles.Firstly,theeigenvalueproblemis
solvedbytransfermatrixandthecorrespondinguniformstationarymeasureisgiven
ontheline.Inaddition,wegivetheperiodicrepresentationoftheevolutionmatrix
underthemodelontheline.Then,weshowtheuniformstationarymeasureofthe
cyclesbyrestrictingthepositionspacetothecycles.
Keywords
Quantum,Inhomogeneous,StationaryMeasure,UniformMeasure
Copyright
c
2022byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/pm.2022.1281511385nØêÆ
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−
12
=
b
y+1
λ
,d
−
13
=
c
y+1
λ
d
−
21
= −
a
y+1
(f
y
g
y
−l
y
d
y
)
λ{l
y
(λ−e
y
)+h
y
f
y
}
,d
−
22
= −
b
y+1
(f
y
g
y
−l
y
d
y
)
λ{l
y
(λ−e
y
)+h
y
f
y
}
d
−
23
=
λ
2
f
y
−c
y+1
(g
y
f
y
−l
y
d
y
)
λ{l
y
(λ−e
y
)+h
y
f
y
}
,d
−
31
= −
a
y+1
{h
y
d
y
+g
y
(λ−e
y
)}
λ{l
y
(λ−e
y
)+h
y
f
y
}
d
−
32
= −
b
y+1
{h
y
d
y
+g
y
(λ−e
y
)}
λ{l
y
(λ−e
y
)+h
y
f
y
}
,d
−
33
= −
(λ−e
y
)(λ
2
−g
y
c
y+1
)−h
y
c
y+1
d
y
λ{l
y
(λ−e
y
)+h
y
f
y
}
.
3.̇(J9Ùy²
!̇ïĘmšàg˜‘nþfi.,ÙüzÝXe
U
x
=




cosθ0e
ω
x
i
sinθ
0e
ω
x
i
0
e
−ω
x
i
sinθ0−cosθ




(ω
x
∈[0,2π),θ∈(0,2π)).
Ù¥xL« ˜,AO•ÄéÝ¥ëêω
x
,•3φ∈[0,2π),÷v^‡ω
x
−ω
x−1
= 2φ,x∈Z.
DOI:10.12677/pm.2022.1281511386nØêÆ
“+
3.1.†‚þi˜—²-ÿÝ9üzÝ±Ï5L«
¼T.AŠ9ƒAA•þ,
½n3.1.1eüzÝ÷vU
(s)
Ψ=λΨ, -λ= e
φi
, lAŠ¯KC†•U
(s)
Ψ=e
φi
Ψ,
Ù¥
Ψ(x) =











Π
x
y=1
D
+
y
Ψ(0)(x≥1),
Ψ(0)(x= 0),
Π
x
y=−1
D
−
y
Ψ(0)(x≤−1),
=£ÝD
±
y
•
D
+
x
=




e
φi
cosθ0e
α
x
i
sinθ
000
e
−α
x
i
sinθ0−e
−φi
cosθ




Ú
D
−
x
=




e
−φi
cosθ0e
α
x
i
sinθ
000
e
−α
x+1
i
sinθ0−e
φi
cosθ




,
Ù¥α
x
= ω
x−1
+φ=ω
x
−φ, ω
x
∈[0,2π),θ∈(0,2π).qφ(Φ) ∈Θ
u
∩Θ, K3†‚þk˜—²-
ÿÝ.
y²:OŽÙ=£Ý•
D
+
x
=




λ
2
−e
−ω
x−1
i
sin
2
θe
ω
x
i
λcosθ
0
e
ω
x
i
sinθ
λ
000
e
−ω
x−1
i
sinθ
λ
0
−cosθ
λ




=





e
2φi
−e
(ω
x
−ω
x−1
)i
sin
2
θ
e
φi
cosθ
0
e
ω
x
i
sinθ
e
φi
000
e
−ω
x−1
i
sinθ
e
φi
0
−cosθ
e
φi





=





e
2φi
−e
2φi
sin
2
θ
e
φi
cosθ
0e
(ω
x
−φ)i
sinθ
000
e
(φ−ω
x−1
)i
sinθ0−e
−φi
cosθ





=





e
φi
cosθ0e
α
x
i
sinθ
000
e
−α
x
i
sinθ0−e
−φi
cosθ





.
DOI:10.12677/pm.2022.1281511387nØêÆ
“+
aq,
D
−
x
=




cosθ
λ
0
e
ω
x
i
sinθ
λ
000
e
−ω
x
i
sinθ
λ
0
λ
2
−e
(ω
x+1
−ω
x
)i
sin
2
θ
−λcosθ




=




cosθ
e
φi
0
e
ω
x
i
sinθ
e
φi
000
e
−ω
x
i
sinθ
e
φi
0
e
2φi
−e
(ω
x+1
−ω
x
)i
sin
2
θ
−e
φi
cosθ




=




cosθ
e
φi
0e
(ω
x
−φ)i
sinθ
000
e
−(ω
x
+φ)i
sinθ0
e
2φi
cos
2
θ
−e
φi
cosθ




=




e
−φi
cosθ0e
α
x
i
sinθ
000
e
−α
x+1
i
sinθ0−e
φi
cosθ




.
N´yD
+
x
,D
−
x
´jÝ, dÚn2.6,Ψ(x) ‰ê† ˜xÃ'. ¤±φ(x) ∈Θ
u
.Ïd, †‚
þTþfiäk˜—²-ÿÝ.
e¡•Ä†‚þüzÝS{U
x
,x∈Z}±Ï5.
½Â3.1.2éuþfi.üzÝS{U
x
,x∈Z},eU
x+n
=U
x
(n∈N),K¡S
{U
x
}äkN±Ï, ÄKvk±Ï.
íØ3.1.3
U
x
=




cosθ0e
ω
x
i
0e
ω
x
i
0
e
−ω
x
i
sinθ0−cosθ




(ω
x
∈[0,2π),θ∈(0,2π)),
Ù¥ω
x
−ω
x−1
=
2π
N
.lkω
x+N
= 2π+ω
x
.¤±
U
x+N
=




cosθ0e
ω
x+N
i
0e
ω
x+N
i
0
e
−ω
x+N
i
sinθ0−cosθ




=




cosθ0e
ω
x
i
0e
ω
x
i
0
e
−ω
x
i
sinθ0−cosθ




= U
x
.
KüzÝS{U
x
}±N•±Ï.
3.2.‚þi˜—²-ÿÝ
‚C
2N
þþfiÏLüzÝU
x
?1üz.3†‚þþfiüzd(2.1)ª¤
û½.‚þžmüzŽfaquU
(s)
•U
(s)
c
.Ùüz•§•Ψ
n+1
=U
(s)
c
Ψ
n
(n≥0),Ψ
n
=
(Ψ
n
(1),···,Ψ
n
(2N)),
DOI:10.12677/pm.2022.1281511388nØêÆ
“+
Ù¥Ψ
n
(1) = (Ψ
L
n
(1),Ψ
O
n
(1),Ψ
Q
n
(1))
T
;Ψ
n
(2N) = (Ψ
L
n
(2N),Ψ
O
n
(2N),Ψ
Q
n
(2N))
T
.
•Ä‚C
2m
þ˜mšàgnþfi,3½n3.1¥^U
(s)
c
“OU
(s)
?1üzke¡
(J,Ù¥2m(m∈N)“Lº:‡ê.éu‚C
2m
=(V,E),∀m∈N,º:8Ú>8©O•
V={x∈Z/mZ}ÚE={(x,x+1),(x+1,x):x∈V}.e¡òþfiüz˜m ˜•›
‚C
2m
þ.Ùüz•§÷veª
Ψ
n+1
(x) = P
x+1
Ψ
n
(x+1)+O
x
Ψ
n
(x)+Q
x−1
Ψ
n
(x−1),(x∈Z/mZ).
·K3.2.1‚C
2m
(m∈N) þ˜mšàgnþfi,3?¿ ˜x?=£ÝXe
D
+
x
=




e
φi
cosθ0e
α
x
i
sinθ
000
e
−α
x
i
sinθ0−e
−φi
cosθ




ω
x
∈[0,2π),θ∈(0,2π),
Ù¥ω
x
−ω
x−1
=
2π
m
,∀x∈{1,2,3,···,2m}.d.3‚C
2m
þk˜—²-ÿÝ.éuЩ
Ψ(0) = (α,β,γ)
T
,˜—²-ÿÝ•ν(Ψ(x)) = |α|
2
+|γ|
2
.Ù¥α,β,γ∈C.
y²:®•ω
x
−ω
x−1
=
2π
m
,Kφ=
2π
m
,¤±
D
+
x
=




e
φi
cosθ0e
α
x
i
sinθ
000
e
−α
x
i
sinθ0−e
−φi
cosθ




=




e
π
m
i
cosθ0e
α
x
i
sinθ
000
e
−α
x
i
sinθ0−e
−
π
m
i
cosθ




qÏ•α
x
=
π
m
+ω
x−1
= ω
x
−
π
m
, ?˜Ú,Œ±α
x+1
−α
x
= (
π
m
+ω
x
)−(
π
m
+ω
x−1
) =
2π
m
. ¤
±,k
D
+
x+1
D
+
x
=




e
φi
cosθ0e
α
x+1
i
sinθ
000
e
−α
x+1
i
sinθ0−e
−φi
cosθ








e
φi
cosθ0e
α
x
i
sinθ
000
e
−α
x
i
sinθ0−e
−φi
cosθ




=




e
2π
m
i
00
000
00e
−2π
m
i




.
dd
2m
Y
x=1
D
+
y
=




e
2π
m
i
00
000
00e
−2π
m
i




m
=




100
000
001




.
éuЩΨ(0) = (α,β,γ)
T
,k
Ψ(x) =




e
2π
m
i
00
000
00e
−2π
m
i




m




α
β
γ




=




α
0
γ




,( 1 ≤x≤2m)
DOI:10.12677/pm.2022.1281511389nØêÆ
“+
ν(Ψ(x)) = |α|
2
+|γ|
2
.-|α|
2
+|γ|
2
= c,kν(Ψ(x)) = c. dd3‚C
2m
þ˜—ÿÝ.
e5`²˜—ÿÝáu²-ÿÝ8Θ
s
,é‚þAŠ¯KU
s
Ψ = λΨ, AŠλ= 1.-
ν
Ψ
0
n
= φ((U
(s)
)
n
)Ψ
0
,k
ν
Ψ
0
n
= (ν
Ψ
0
n
(1),ν
Ψ
0
n
(2),ν
Ψ
0
n
(3),···)
T
.
?˜Ú
ν
Ψ
0
n
= (|α|
2
+|γ|
2
,|α|
2
+|γ|
2
,|α|
2
+|γ|
2
,···)
T
.
Œ±wé∀n≥0, kφ(Ψ
0
(x)) = ν
Ψ
0
n
(x) = c.¤±ν(Ψ) = ν
Ψ
0
n
∈Θ
s
(U).
Ä7‘8
I[g,‰ÆÄ7]Ï‘8(1OÒ:11861057)"
ë•©z
[1]Aharonov,Y.,Davidovich,L.andZagury,N.(1993)QuantumRandomWalks.PhysicalReviewA,48,
1687-1690.https://doi.org/10.1103/PhysRevA.48.1687
[2]Vegenas-Andraca,S.E.(2012)QuantumWalks:AComprehensiveReview.QuantumInformationProcess-
ing,11,1015-1106.https://doi.org/10.1007/s11128-012-0432-5
[3]Shankar,K.H.(2014)QuantumRandomWalksandDecisionMaking.TopicsinCognitiveScience,6,
108-113.https://doi.org/10.1111/tops.12070
[4]‘“,â¬,41.þfxD(©Û[M].ÉÇ:‰ÆEâч,2004.
[5]Machida,T.(2013)RealizationoftheProbabilityLawsintheQuantumCentralLimitTheoremsbya
QuantumWalk.QuantumInformationandComputation,13,430-438.
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maticalPhysics,51,ArticleID:053528.https://doi.org/10.1063/1.3431028
[9]Endo,T.,Kawai,H.andKonno,N.(2017)StationaryMeasuresfortheThree-StateGroverWalkwithOne
DefectinOneDimension.InterdisciplinaryInformationSciences,23,45-55.
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DimensionalHadamardWalk.JournalofStatisticalPhysics,187,ArticleNo.10.
https://doi.org/10.1007/s10955-022-02901-x
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[11]Konno,N.,Luczak,T.andSegawa,E.(2013)LimitMeasuresofInhomogeneousDiscrete-TimeQuantum
WalksinOneDimension.QuantumInformationProcessing,12,33-53.
https://doi.org/10.1007/s11128-011-0353-8
[12]Konno,N.(2014)TheUniformMeasureforDiscrete-TimeQuantumWalksinOneDimension.Quantum
InformationProcessing,13,1103-1125.https://doi.org/10.1007/s11128-013-0714-6
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One-DimensionalLattice.YokohamaMathematicalJournal,63,59-74.
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