Limit Theorems for the Discrete-Time Quantum Walk on a Graph with Joined Half Lines
aa r X i v : . [ qu a n t - ph ] J a n Limit theorems for the discrete-time quantum walkon a graph with joined half lines
Kota Chisaki, ∗ Norio Konno, † Etsuo Segawa, ‡ , Department of Applied Mathematics, Faculty of Engineering, Yokohama National UniversityHodogaya, Yokohama 240-8501, Japan Department of Mathematical Informatics, The University of Tokyo,Bunkyo, Tokyo, 113-8656, Japan
Abstract . We consider a discrete-time quantum walk W t,κ at time t on a graph with joined half lines J κ , whichis composed of κ half lines with the same origin. Our analysis is based on a reduction of the walk on a half line.The idea plays an important role to analyze the walks on some class of graphs with symmetric initial states. In thispaper, we introduce a quantum walk with an enlarged basis and show that W t,κ can be reduced to the walk on ahalf line even if the initial state is asymmetric . For W t,κ , we obtain two types of limit theorems. The first one isan asymptotic behavior of W t,κ which corresponds to localization. For some conditions, we find that the asymptoticbehavior oscillates. The second one is the weak convergence theorem for W t,κ . On each half line, W t,κ converges toa density function like the case of the one-dimensional lattice with a scaling order of t . The results contain the casesof quantum walks starting from the general initial state on a half line with the general coin and homogeneous treeswith the Grover coin. Random walks have a very important role in various fields, such as physical systems, mathematical modelingand computer algorithms. In 1990s, quantum walks arise as a quantum counterpart of random walks [1–3].They are defined by unitary evolutions of probability amplitudes, whereas random walks are obtained byevolutions of probabilities by transition matrices. Discrete-time quantum walks are introduced by Refs. [1,2].In recent years, quantum walks have been well developed in fields of quantum algorithms, for example [4–6].On the other hand, studies of the walks from the mathematical point of view also arise. Especially, asa limiting behaver, localization appears in quantum cases [7–12]. Furthermore the quantum walk has aquadratically faster scaling order than the random walk in the weak convergence [13–17]. Cantero et al.introduced an analysis using the CMV matrix [11, 12]. This method is very useful to consider localization.To analyze the quantum walk, we use the generating function. By using the generating function, we cancompute not only localization but also the weak convergence of the walk. A reduction technique [20–22],which reduces the walk to a one-dimensional quantum walk, is very important to apply a path countingmethod [13,14,23] which gives an explicit expression for the generating function. To treat the quantum walkwith asymmetric initial states, we introduce a quantum walk with enlarged bases.Our main results are two limit theorems for the quantum walk W t,κ on a graph with joined half lineswith arbitrary initial state starting from the origin. In case of κ = 1, W t, corresponds to a quantum walkon a half line with the general coin. Furthermore, by considering the reduction of the walks, the two limittheorems can be adopted to quantum walks on homogeneous trees and semi-homogeneous trees with theGrover coin operator. One of two our main results is the explicit expression for the limit probability of W t,κ . It is corresponding to localization which is defined that there exists a vertex of the graph x such thatlim sup t →∞ P ( W t,κ = x ) >
0. We find that, for some conditions, the asymptotic behavior oscillates. Sameas other results on quantum walks [7–9], localization has an exponential decay for position x on each half ∗ [email protected] † [email protected] ‡ To whom correspondence should be addressed. [email protected]
Key words. quantum walk, localization, weak convergence, homogeneous tree W t,κ . On each half line, W t,κ has a scaling order t .Moreover the limit measure has a typical density function which appears on other quantum walks [8, 13–17].For related works, Chisaki et al. [8] obtained the same type of limit theorems for a quantum walk onhomogeneous trees with two special initial states. This result induces limit theorems for a quantum walk ona half line with a special coin operator. Konno and Segawa [18] showed localization of quantum walks on ahalf line by using the spectral analysis of the corresponding CMV matrices.The remainder of the present paper is organized as follows. In Section 2, we give definitions of discrete-time quantum walks treated in this paper. Section 3 presents our results. Section 4 gives proofs of ourmain theorems. In Subsection 4.1, we introduce a quantum walk with an enlarged basis and reduce W t,κ to the walk on a half line. Subsection 4.2 presents a proof of Theorem 1 based on the generating function.Subsection 4.3 is devoted to a proof of Theorem 2 using the Fourier transform of the generating function.In Appendix, we compute the generating function. This section gives the definition of the quantum walk on undirected connected graph G . Let V ( G ) be aset of all vertices in G and E ( G ) be a set of all edges in G . Here we define E x ( G ) ⊂ E ( G ) as a set ofall edges which connect the vertex x ∈ V ( G ). Now we take a Hilbert space spanned by an orthonormalbasis {| x i ; x ∈ V ( G ) } as a position space H p and a Hilbert space generated by an orthonormal basis {| l i ; l ∈ E x ( G ) } for x ∈ V ( G ) as a local coin space H c x . A discrete-time quantum walk on G is definedon a Hilbert space H spanned by an orthonormal basis {| x, l i ; x ∈ V ( G ) , l ∈ E x ( G ) } . Note that if wetake G as a regular graph, H can be written as H = H p ⊗ H c x for any x ∈ V ( G ). On the space H , theevolution operator U is given by U = SF , where S : H → H is a shift operator and F : H → H is a coinoperator. Here we define F = P x ∈ V ( G ) | x ih x | ⊗ C x as a coin operator and C x : H c x → H c x for x ∈ V ( G ) asa local coin operator. If the graph is regular and the local coin operator is all the same, we can rewrite thecoin operator as F = I p ⊗ C , where I p is the identity operator on H p . As typical local coin operators, theHadamard operator H and the Grover operator G d are often used, where H and G d ( d ≥
2) are defined by H = 1 √ (cid:20) − (cid:21) ,G d = a d b d · · · b d b d a d · · · b d ... ... . . . ... b d b d · · · a d = d − d · · · d d d − · · · d ... ... . . . ... d d · · · d − . In this paper we define a = 1, b = 2 and G = 1. From the construction, the state at time t and position x is described as Ψ t ( x ) = X l ∈ E x ( G ) α t ( x, l ) | x, l i , (2.1)where α t ( x, l ) ∈ C is the amplitude of the base | x, l i at time t and C is the set of all complex numbers. Theprobability of the state is given by a square norm of Ψ t ( x ), i.e., k Ψ t ( x ) k = P l ∈ E x ( G ) | α t ( x, l ) | . We onlyconsider the initial state starting from the origin “ o ” with the state Ψ ( o ) such that k Ψ ( o ) k = 1. This subsection gives the definition of a graph with joined half lines J κ and the quantum walk W t,κ on J κ .Let K κ = { , , . . . , κ − } and Z r = { h r (1) , h r (2) , . . . } for r ∈ K κ , we define V ( J κ ) = { } ∪ {∪ j ∈ K κ Z j } . Avertex h i ( x ) connects h j ( y ) if and only if | x − y | = 1 with i = j , and the origin 0 connects h r (1) for any r (see Fig. 1 (a) for example).The quantum walk on J κ is defined on H ( J κ ) which is a Hilbert space spanned by an orthonormal basis {| , l i ; l ∈ { ǫ , ǫ , . . . , ǫ κ − }} ∪ {| x, l i ; x ∈ V ( J κ ) \ { } , l ∈ { U p, Down }} . Throughout this paper, we put2he base | ǫ r i as T [ r z }| { · · · κ − r − z }| { · · · , where T is the transposed operator. We define a local coin operator C as C = (cid:20) a bc d (cid:21) ∈ U (2) with abcd = 0 , (2.2)where U ( d ) is the set of d × d unitary matrices. The coin operator F J is given by F J = | ih | ⊗ G κ + X x ∈ V ( J κ ) \{ } | x ih x | ⊗ C, (2.3)where G κ is the Grover operator. The shift operator S J is given by S J | , l i = | h r (1) , Down i , l = ǫ r ,S J | h r (1) , l i = (cid:26) | , ǫ r i , l = U p, | h r (2) , Down i , l = Down,S J | h r ( x ) , l i = (cid:26) | h r ( x − , U p i , l = U p, | h r ( x + 1) , Down i , l = Down, x ≥ . Then the evolution operator of the walk U J is obtained by U J = S J F J . An expression of W t, using weightsis shown in Fig. 1 (a), where P ǫ = a b , Q ǫ = (cid:20) −
13 23 23 (cid:21) ,P ǫ = a b , Q ǫ = (cid:20) −
13 23 (cid:21) ,P ǫ = a b , Q ǫ = (cid:20)
23 23 − (cid:21) ,P = (cid:20) a b (cid:21) , Q = (cid:20) c d (cid:21) . Note that W t, is a quantum walk on a half line with a reflecting wall.(a) (b)Figure 1: (a) Quantum walk on J , (b) Quantum walk on T .2 Quantum walk on homogeneous trees We define a homogeneous tree T κ and a quantum walk V t,κ on T κ . Fix κ ≥
2, let Σ = { σ , σ , . . . , σ κ − } be the set of generators subjected to the relation σ j = e for j ∈ K κ , where the empty word e is the unit ofthis group. Then we put V ( T κ ) = { e } ∪ { σ i n . . . σ i σ i : n ≥ , σ i j ∈ Σ , i j +1 = i j for j = 1 , , . . . , n − } .Here vertices g and h are connected if and only if gh − ∈ Σ. On this graph, H ( T κ ) p is generated by anorthonormal basis {| g i ; g ∈ V ( T κ ) } and H ( T κ ) c is associated with an orthonormal basis {| σ j i ; σ j ∈ Σ } . Wechoose G κ as the local coin operator, then the coin operator F T and the shift operator S T are defined asfollows: for σ ∈ Σ F T = | e ih e | ⊗ ˜ cG κ + X g ∈ V ( T κ ) \{ e } | g ih g | ⊗ G κ ,S T | g, σ i = | σg, σ i , where we put | σ r i as T [ r z }| { · · · κ − r − z }| { · · ·
0] and ˜ c ∈ C with | ˜ c | = 1. The phase ˜ c works as a defect on the origin,which is an extension of our model in [8]. An expression of V t, using weights is shown in Fig. 1 (b), where P = −
13 23 23 , Q = −
13 23 , R =
23 23 − , and ˜ P = ˜ cP , ˜ Q = ˜ cQ , ˜ R = ˜ cR .In the case of the one point initial state on the origin, V t,κ can be reduced to the equivalent walk on J κ even if the initial state is not symmetric. To explain it, we define subgraph T ( r ) κ ∈ T κ as V ( T ( r ) κ ) = { σ i n · · · σ i σ i : n ≥ , σ i j ∈ Σ , σ i = σ r , i j +1 = i j for j = 1 , , . . . , n − } . They are subtrees whose rootsare the children of the root of T κ . Now we consider the following new basis, for x ≥ | x, U p i σ r = 1 p ( κ − x − X g ∈ V ( T ( r ) κ ) | g | = x X σ j : | σ j g | = x − | g, σ j i , | x, Down i σ r = 1 p ( κ − x X g ∈ V ( T ( r ) κ ) | g | = x X σ j : | σ j g | = x +1 | g, σ j i . The new space H ( T κ ) ′ spanned by a basis {| e, l i : l ∈ Σ } ∪ {| x, l i σ r : r ∈ K κ , x ∈ Z + , l ∈ { U p, Down }} isisomorphic to H ( J κ ) under the following one-to-one correspondence | x, l i σ r ↔ | h r ( x ) , l i for l ∈ { U p, Down } , x ≥ , | e, σ r i ↔ | , ǫ r i , (2.4)where Z + = { , , . . . } . Then the direct computation gives the following lemma. Lemma 1 (Homogeneous tree)
The subspace of H ( T κ ) ′ is invariant under the action of the time evolu-tion of V t,κ . In particular, when we take the bijection from H ( T κ ) ′ to H ( J κ ) given by Eq. (2.4), the walk isequivalent to W t,κ with the following coin operator F ( T κ ) J = | ih | ⊗ ˜ cG κ + X x ∈ V ( J κ ) \{ } | x ih x | ⊗ (cid:20) a κ √ κ − b κ √ κ − b κ − a κ (cid:21) . When we consider ˜ c − F ( T κ ) J , the above equation becomes a special case of Eq. (2.3), since | ˜ c | = 1.Similar to V t,κ , we can define a quantum walk V t,κ ′ ,κ on a semi-homogeneous tree T κ ′ ,κ , which is a κ -regular tree except the origin whose degree is κ ′ ≥
2, with the local coin operator ˜ cG κ ′ at the origin and G κ W t,κ ′ with the coin operator F ( T κ ′ ,κ ) J given by F ( T κ ′ ,κ ) J = | ih | ⊗ ˜ cG κ ′ + X x ∈ V ( J κ ′ ) \{ } | x ih x | ⊗ (cid:20) a κ √ κ − b κ √ κ − b κ − a κ (cid:21) . The infinite binary tree is a special case for this graph ( κ = 3 , κ ′ = 2). In our main theorems, we give explicit formulae with respect to each half line in J κ . Let Ψ t ( x ) be the stateof the quantum walk W t,κ at time t and position x . For x ∈ Z + ∪ { } , we introduce random variables X t,r as P ( X t,r = 0) = | α t (0 , ǫ r ) | and P ( X t,r = 0) = || Ψ t ( h r ( x )) || . Remark that P ( W t,κ = h r ( x )) = P ( X t,r = x )for x ≥ P ( W t,κ = 0) = P j ∈ K κ P ( X t,j = 0) and P j ∈ K κ P x ∈ Z + ∪{ } P ( X t,j = x ) = 1.In order to describe the limit theorems for W t,κ , we first introduce several parameters. φ = arg ( c ) ,K ± = | ± c | ,K × = (1 − c )(1 + ¯ c ) , where ¯ a is the complex conjugation of a ∈ C . Next, we denote the following notations to state Theorem 1.Localization is described by three terms L m ( x ), L rp ( x ) and L rc ( x ).Ψ (0) = X j ∈ K κ ψ j | , ǫ j i ,L m ( x ) = Γ − ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ∈ K κ ψ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , L rp ( x ) = Γ + ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ∈ K κ ( ψ j − ψ r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,L rc ( x, t ) = 2 b κ Re Γ × ( x, t ) X j ∈ K κ ψ j X j ∈ K κ ( ψ j − ψ r ) , Γ ± ( x ) = b κ | c | (cos φ ± | c | ) K ± ( δ ( x ) + (1 − δ ( x )) (cid:18) | a | K ± (cid:19) x − (cid:18) | a | K ± (cid:19)) , Γ × ( x, t ) = s K × K × ! t +1 | c | (cos φ − | c | ) K × × − δ ( x ) s K × K × + (1 − δ ( x )) | a | p K + K − ! x − (cid:18) − | a | K × (cid:19) . Then we have the following theorem.
Theorem 1 (Localization)
For κ ≥ , x ∈ Z + ∪ { } , r ∈ K κ , P ( X t,r = x ) ∼ − t + x (cid:8) I [ − , | c | ) (cos φ ) L m ( x ) + I ( −| c | , (cos φ ) L rp ( x ) + I ( −| c | , | c | ) (cos φ ) L rc ( x, t ) (cid:9) , where f ( t ) ∼ g ( t ) means f ( t ) /g ( t ) → t → ∞ ) . We see that in many cases the quantum walk on J κ exhibits localization. Localization does not occur onlyin the following two cases, “ P j ( ψ j − ψ r ) = 0 for any r and cos φ ≥ | c | ” and “ P j ψ j = 0 and cos φ ≤ −| c | ”.Only the symmetric initial state (i.e., ψ i = 1 / √ κ for any i ) satisfies the first condition. Moreover L rc ( x, t ) is5igure 2: Comparison between theoretical values with circles and numerical estimations with crosses of W t, with C = e iϕ H . The probability P ( X t, = 1) is plotted. The initial state is ψ = e i π/ / √ ψ = e i π/ / √ ψ = e i π/ / √ ϕ = 0, (b) ϕ = 40 π/ ϕ = 50 π/ ϕ = 80 π/ | c | = 1 / √ ϕ = 45 π/
180 is a critical point for the oscillatory behavior. In the large figure of (c),theoretical value are omitted.an oscillatory term, so the probability oscillates if L rc ( x, t ) exists. The probability P ( X t, = 1) is shown inFig. 2, where we choose the local coin operator as e iϕ H . From Theorem 1, the condition for the existence L rc ( x, t ) is −| c | < cos φ < | c | . Therefore, in this case, the oscillation emerges when π/ < ϕ < π/
4. Remarkthat from Theorem 1 we can see the following relation, X r ∈ K κ L rc ( x, t ) = 2 b κ Re Γ × ( x, t ) X j ∈ K κ ψ j X r ∈ K κ X j ∈ K κ ( ψ j − ψ r ) = 0 , This means that the oscillation disappears when we take the probability summed over all vertices with asame distance from the origin. In addition, since P ( W t,κ = 0) = P j P ( X t,j = 0), the probability of theorigin does not oscillate for any condition. We also find that the distribution has an exponentially decaywith x from Theorem 1. The probability P ( X , = x ) is shown in Fig. 3.In order to state the weak convergence theorem, at first we define some parameters depending on theinitial state. For r ∈ K κ , we put θ r ( ψ ) = | ψ r | , (3.5) θ r ( ψ ) = X j ∈ K κ \{ r } ψ r ψ j , (3.6) θ r ( ψ ) = X j ∈ K κ \{ r } | ψ j | + X j,k ∈ K κ \{ r } j = k (cid:0) ψ j ψ k + ψ j ψ k (cid:1) . (3.7)Next, we introduce the following notations. Terms C m and C rd are delta measures which are caused by local-ization and C rd ( x ) is a weight on density function f K ( x ) which is formed a typical shape of one dimensional6igure 3: Comparison between theoretical values with circles and numerical estimations with crosses of W t, with C = e iϕ H . The probability P ( X , = x ) is plotted. The initial state is ψ = e i π/ / √ ψ = e i π/ / √ ψ = e i π/ / √ ϕ = 0, (b) ϕ = 40 π/ ϕ = 50 π/ ϕ = 80 π/ C m = b κ | c | ( | c | − cos φ )2 K − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ∈ K κ ψ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , C rp = b κ | c | ( | c | + cos φ )2 K + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ∈ K κ ( ψ j − ψ r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ,C rd ( x ) = Γ ( x ) θ r ( ψ ) + 2Re(Γ ( x ) θ r ( ψ )) + Γ ( x ) θ r ( ψ )( K + − (1 − x ) sin φ )( K − − (1 − x ) sin φ ) x , Γ ( x ) = 4 a κ | c | ( | a | − x ) cos φ sin φ + (cid:0) a κ + 2 a κ | c | cos φ + 1 (cid:1) (cid:0) | c | − | c | cos φ − (1 − x ) sin φ (cid:1) , Γ ( x ) = − b κ | c | ( | a | − x ) ie iφ cos φ sin φ + b κ ( a κ + | c | e iφ )(1 + | c | − | c | cos φ − (1 − x ) sin φ ) , Γ ( x ) = b κ (1 + | c | − | c | cos φ − (1 − x ) sin φ ) . The weak convergence theorem is derived as follows.
Theorem 2 (Weak convergence)
For κ ≥ , r ∈ K κ , as t → ∞ P (cid:18) u ≤ X t,r t ≤ v (cid:19) → Z vu ρ rW ( x ) dx. The limit measure is defined by ρ rW ( x ) = (cid:8) I [ − , | c | ) (cos φ ) C m + I ( −| c | , (cos φ ) C rp (cid:9) δ ( x ) + C rd ( x ) f K ( x ) , where f K ( x ) = I [0 ,a ) ( x ) p − | a | π (1 − x ) p | a | − x . W t, with C = e iϕ H . Density function C d ( x ) f K ( x ) and scaled numerical values at time 2000 are plotted. Intheoretical values, the delta measure is omitted. Since localization has an exponentially decay, the scalednumerical values corresponding to localization converge to the delta measure at the infinite time. The initialstate is ψ = e i π/ / √ ψ = e i π/ / √ ψ = e i π/ / √ ϕ = 0, (b) ϕ = 40 π/ ϕ = 50 π/ ϕ = 80 π/ C d ( x ) f K ( x ) and the scaled numerical values are shown in Fig. 4. Same as other one-dimensionalcases [13–17], this distribution has scaling order t and the typical density function of quantum walks f K ( x ).The delta measures C m and C rp are caused by localization, i.e., C m = P x L m ( x ) and C rp = P x L rp ( x ).The above expressions of both theorems seem to be complicated, however for the following cases, theyare written in simpler forms. Corollary 3 ( φ = 0 ) For x ∈ Z + ∪ { } , r ∈ K κ and φ = 0 , P ( X t,r = x ) ∼ (cid:18) − t + x (cid:19) b κ | c | | c | ( δ ( x ) + (1 − δ ( x )) (cid:18)
21 + | c | (cid:19) (cid:18) − | c | | c | (cid:19) x − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ∈ K κ ( ψ j − ψ r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , and X t,r /t converges weakly to a limit measure ρ rW (0) as t → ∞ , where ρ rW (0) ( x ) = b κ | c | | c | ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X j ∈ K κ ( ψ j − ψ r ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ ( x ) + 1 | a | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( | c | − ψ r + b κ X j ∈ K κ ψ j (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + | a | | ψ r | x f K ( x ) . Proof : From Theorem 1, when φ = 0, we have P ( X t,r = x ) ∼ (1 + ( − t + x ) L rp ( x ) /
2. Also Theorem 2implies Γ ( x ) = | a | ( a κ + 2 a κ | c | + 1), Γ ( x ) = | a | b κ ( a κ + | c | ), Γ ( x ) = | a | b κ , and K + K − = | a | . After somecalculations of θ r ( ψ ) , θ r ( ψ ) , θ r ( ψ ), we have the desired conclusion. (cid:3) W t,κ , W t, is simply a quantum walk on a half line with a reflecting wall on theorigin. In the next corollary, we denote X t, ≡ X t to assert that the walk is defined on a half line. Corollary 4 (Half line)
For x ∈ Z + ∪ { } , P ( X t = x ) ∼ I [ − , | c | ) (cos φ ) (cid:18) − t + x (cid:19) × | c | (cos φ − | c | ) K − ( δ ( x ) + (1 − δ ( x )) (cid:18) | a | K − (cid:19) x − (cid:18) | a | K − (cid:19)) , and X t /t converges weakly to a limit measure ρ X as t → ∞ , where ρ X ( x ) = I [ − , | c | ) (cos φ ) 2 | c | ( | c | − cos φ ) K − δ ( x ) + 2(1 − | c | cos φ ) K − − (1 − x ) sin φ x f K ( x ) . Proof : For κ = 1, we have b = 2 /κ = 2, a = b − θ ( ψ ) = 1 and θ ( ψ ) = θ ( ψ ) = 0. From Theorem1, we get L p ( x ) = 0 and L c ( x, t ) = 0, thus we should consider only L m ( x ) as localization factor of X t . FromTheorem 2, we have C p = 0 andΓ ( x ) = 2(1 − | c | cos φ ) (cid:8) K + − (1 − x ) sin φ (cid:9) . (3.8)Combining C m and C rd ( x ) with Eq. (3.8) implies ρ X ( x ). (cid:3) Remark that we get another proof of Corollary 4 by considering W t,κ with the symmetric initial state.We can adopt Corollary 3 for V t,κ with no perturbation, i.e., ˜ c = 1. In φ = 0 case, the formula for the r th half line is directly expressed by ψ , ψ , . . . , ψ κ − instead of θ r ( ψ ) , θ r ( ψ ) , θ r ( ψ ). Both cases of φ = 0and half line, the oscillatory term L rc ( x, t ) appearing in Theorem 1 vanishes. In order to prove Theorems 1 and 2, we consider a reduction of W t,κ on a half line. For W t,κ with arbitraryinitial states, we can not construct the reduction of the walk directly, since the states with the same distancefrom the origin have different amplitudes. To solve this problem, we introduce W ′ t,κ which is a quantumwalk with an enlarged basis of W t,κ . After that, we construct X ∗ t as a reduction of W ′ t,κ on a half line. Toanalyze X ∗ t , we give the generating function of the states. By using it, we obtain the limit states and thecharacteristic function of W t,κ . Let Ψ t ( x ) be the state of the quantum walk W t,κ at time t and position x . We denote the initial state Ψ (0)as ψ = P j ∈ K κ ψ j | , ǫ j i . Now we rewrite ψ using a new orthogonal basis {| ǫ ′ j i ; j ∈ K κ } as ψ = X j ∈ K κ X r ∈ K κ ψ r h ǫ ′ r | ⊗ I W ! | ǫ ′ j i| , ǫ j i = X j ∈ K κ Λ( ψ ) | ǫ ′ j i| , ǫ j i , where I W is the identity operator on H ( J κ ) and we defined asΛ( ψ ) = X r ∈ K κ ψ r h ǫ ′ r | ⊗ I W . Now let H ′ be a Hilbert space spanned by an orthonormal basis {| ǫ ′ j i ; j ∈ K κ } . Then we define W ′ t,κ as a quantum walk on H ′ ⊗ H ( J κ ) with the evolution operator U ′ J = ( I κ ⊗ S J )( I κ ⊗ F J ) = I κ ⊗ U J andthe initial state P j ∈ K κ | ǫ ′ j i| , ǫ j i , where I κ is the identity operator on H ′ . Let l = { ǫ , ǫ , . . . , ǫ κ − } and l x = { U p, Down } for x ∈ V ( J κ ) \ { } . Then the state of quantum walk W ′ t,κ at time t and position x iswritten as Ψ ′ t ( x ) = P j ∈ K κ ,u ∈ l x α ′ t ( ǫ ′ j , x, u ) | ǫ ′ j i| x, u i , where α ′ t ( a, b, c ) is the amplitude of the base | a, b, c i attime t . From the construction we obtain the following lemma.9 emma 2 (Enlarging basis) For any t ≥ and x ∈ V ( J κ ) , Ψ t ( x ) = Λ( ψ )Ψ ′ t ( x ) . Proof : We show the equation by induction with respect to t . At t = 0, by definition of Λ( ψ ), it is trivial.For fixed t ≥
1, we assume Ψ t ( x ) = Λ( ψ )Ψ ′ t ( x ), then for x ∈ V ( J κ ), U ′ J Ψ ′ t ( x ) = ( I κ ⊗ U J ) X j ∈ K κ ,u ∈ l x α ′ t ( ǫ ′ j , x, u ) | ǫ ′ j i| x, u i = X j ∈ K κ ,u ∈ l x α ′ t ( ǫ ′ j , x, u ) | ǫ ′ j i ( U J | x, u i ) ,U J Ψ t ( x ) = U J Λ( ψ )Ψ ′ t ( x )= U J Λ( ψ ) X j ∈ K κ ,u ∈ l x α ′ t ( ǫ ′ j , x, u ) | ǫ ′ j i| x, u i = U J X j ∈ K κ ,u ∈ l x ψ ′ j α ′ t ( ǫ ′ j , x, u ) | x, u i = X j ∈ K κ ,u ∈ l x ψ ′ j α ′ t ( ǫ ′ j , x, u ) U J | x, u i = Λ( ψ ) X j ∈ K κ ,u ∈ l x α ′ t ( ǫ ′ j , x, u ) | ǫ ′ j i ( U J | x, u i )= Λ( ψ ) U ′ J Ψ ′ t ( x ) . This relation holds for any x , so we conclude Ψ t +1 ( x ) = Λ( ψ )Ψ ′ t +1 ( x ). (cid:3) For x ∈ V ( J κ ), we define the probability of “ W ′ t,κ = x ” by P ( W ′ t,κ = x ) = k Λ( ψ )Ψ ′ t ( x ) k . Then it followsform Lemma 2 that P ( W ′ t,κ = x ) = P ( W t,κ = x ) for any x ∈ V ( J κ ). For W ′ t,κ , the information of the initialstate is covered by Λ( ψ ). In other words, for any initial state of W t,κ , it is enough to consider the initialstate P j ∈ K κ | ǫ ′ j i| , ǫ j i on W ′ t,κ . Consequently, the states of the quantum walk W ′ t,κ have a good symmetry,so we can treat the reduction of the walk.Now we introduce X ∗ t as a reduction of W ′ t,κ on a half line. Here X ∗ t is defined on a Hilbert spacegenerated by the following new basis. For all l ∈ { U p, Down } and x ∈ Z + , | Own, , ǫ i = X j ∈ K κ | ǫ ′ j , , ǫ j i , | Other, , ǫ i = 1 √ κ − X j ∈ K κ X k ∈ K κ \{ j } | ǫ ′ k , , ǫ j i , | Own, x, l i = X j ∈ K κ | ǫ ′ j , h j ( x ) , l i , | Other, x, l i = 1 √ κ − X j ∈ K κ X k ∈ K κ \{ j } | ǫ ′ k , h j ( x ) , l i . On this basis, we obtain the one-step time evolution. For x ∈ Z + , U ′ J : | Own, , ǫ i → a κ | Own, , Down i + √ κ − b κ | Other, , Down i , | Other, , ǫ i → √ κ − b κ | Own, , Down i − a κ | Other, , Down i , | Own, x, U p i → a | Own, x − , U p i + c | Own, x + 1 , Down i , | Other, x, U p i → a | Other, x − , U p i + c | Other, x + 1 , Down i , | Own, x, Down i → b | Own, x − , U p i + d | Own, x + 1 , Down i , | Other, x, Down i → b | Other, x − , U p i + d | Other, x + 1 , Down i . U ′ J . Moreover the initial state of W ′ t,κ can be written as | Own, , ǫ i . Therefore we can write the evolution operator of X ∗ t as U ∗ H = F ∗ H S ∗ H . Thecoin operator F ∗ H is defined by F ∗ H = (cid:20) a κ √ κ − b κ √ κ − b κ − a κ (cid:21) ⊗ | ih | ⊗ X x ∈ Z + I ⊗ | x ih x | ⊗ C. For m ∈ { Own, Other } , l ∈ { U p, Down } , the shift operator S ∗ is defined by S ∗ H | m, , ǫ i = | m, , Down i ,S ∗ H | m, , l i = (cid:26) | m, , ǫ i , l = U p, | m, , Down i , l = Down,S ∗ H | m, x, l i = (cid:26) | m, x − , U p i , l = U p, | m, x + 1 , Down i , l = Down, x ≥ . Throughout this paper, we put | Own i = T [1 ,
0] and | Other i = T [0 , X ∗ t using weights isshown by Fig. 5 and Eqs. (.18)-(.20) in Appendix.Let Ψ ∗ t ( x ) be the state of the quantum walk X ∗ t . Now we define for r ∈ K κ ,Λ r ( ψ ) = ψ r h Own | + √ κ − X j ∈ K κ \{ r } ψ j h Other | ⊗ I W . Then we introduce X t,r whose probability of “ X t,r = x ” is defined by P ( X t,r = x ) = k Λ r ( ψ )Ψ ∗ t ( x ) k . (4.9)This probability is described by the state of W t,κ in the following, P ( X t,r = x ) = (cid:26) | α t (0 , ǫ r ) | , x = 0 , k Ψ t ( h r ( x )) k , otherwise . Hence the relation between the probabilities of “ W t,κ = h r ( x )” and “ X t,r = x ” is obtained as P ( W t,κ = h r ( x )) = ( P j ∈ K κ P ( X t,j = 0) , x = 0 ,P ( X t,r = x ) , otherwise . Note that P j ∈ K κ P x ∈{ }∪ Z + P ( X t,j = x ) = 1.In Subsections 4.2 and 4.3, we analyze Ψ ∗ t ( x ) by the generating function. We compute the limit state of X ∗ t from the generating function which is defined by˜Ψ ∗ ( x ; z ) = ∞ X t =0 Ψ ∗ t ( x ) z t = X l ∈{ Own,Other } m ∈{ Up,Down } ˜ α ∗ ( l, x, m ; z ) | l, x, m i . < r < z with | z | < r ,˜ α ∗ ( Own, x, U p ; z ) = (cid:26) − dac ( λ ( z ) − az )( µ ( z ) + a κ )Φ( x ; z ) , x > , − ( µ ( z ) + a κ ) µ ( z )Φ( x ; z ) , x = 0 , (4.10)˜ α ∗ ( Other, x, U p ; z ) = (cid:26) − d √ κ − ac ( λ ( z ) − az ) b κ Φ( x ; z ) , x > , −√ κ − b κ µ ( z )Φ( x ; z ) , x = 0 , (4.11)˜ α ∗ ( Own, x, Down ; z ) = (cid:26) − z ( µ ( z ) + a κ )Φ( x ; z ) , x > , , x = 0 , (4.12)˜ α ∗ ( Other, x, Down ; z ) = (cid:26) − z √ κ − b κ Φ( x ; z ) , x > , , x = 0 , (4.13)Φ( x ; z ) = (cid:26) dλ ( z ) a (cid:27) x − w w − (cid:16) η + ( z ) + p ν ( z ) (cid:17) (cid:16) η − ( z ) − p ν ( z ) (cid:17) − c )( z − w )( z − w − ) , (4.14)where λ ( z ) = ∆ z + 1 − p ∆ z + 2∆(1 − | a | ) z + 12 dz ,µ ( z ) = dλ ( z ) − ∆ zc z,ν ( z ) = (1 + ∆ z ) − | a | z ,η ± ( z ) = 2 c ± ∓ ∆ z ,w ± = ∓ c (1 ± c )∆( | a | − ∓ c ) , ∆ = ad − bc. Note that | w ± | = (cid:12)(cid:12) (1 ± c ) / (1 ± c ) (cid:12)(cid:12) = 1. From Cauchy’s theorem, we have for 0 < r < r < ∗ t ( x ) = 12 πi I | z | = r ˜Ψ ∗ ( x ; z ) dzz t +1 . Therefore as t → ∞− Ψ ∗ t ( x ) ∼ Res( ˜Ψ ∗ ( x ; z ) , w + ) w − ( t +1)+ + Res( ˜Ψ ∗ ( x ; z ) , − w + )( − w + ) − ( t +1) + Res( ˜Ψ ∗ ( x ; z ) , w − ) w − ( t +1) − + Res( ˜Ψ ∗ ( x ; z ) , − w − )( − w − ) − ( t +1) , where Res( f ( z ) , w ) is the residue of f ( z ) for z = w . Taking the residues of the generating function, we cancompute Ψ ∗ t ( x ). After some calculations with Eq.(9) and Ψ ∗ t ( x ), the proof of Theorem 1 is complete. In order to prove Theorem 2, we calculate the Fourier transform of the generating function as ˆ˜Ψ ∗ ( s ; z ) = P x ˜Ψ ∗ ( x ; z ) e isx by Eqs. (4.10)-(4.13). Then we obtain the characteristic function from the following relation E (cid:2) e iξX t,r (cid:3) = X x ∈ Z h Λ r ( ψ )Ψ ∗ t ( x ) , Λ r ( ψ )Ψ ∗ t ( x ) i e iξx = X x,y ∈ Z h Λ r ( ψ )Ψ ∗ t ( x ) , Λ r ( ψ )Ψ ∗ t ( y ) i e iξx Z π e ik ( x − y ) dk π = Z π X x,y ∈ Z h Λ r ( ψ )Ψ ∗ t ( x ) , Λ r ( ψ )Ψ ∗ t ( y ) i e ik ( x − y ) e iξx dk π = Z π h Λ r ( ψ ) ˆΨ ∗ t ( s ) , Λ r ( ψ ) ˆΨ ∗ t ( s + ξ ) i ds π , (4.15)12here h u , v i is the inner product of vectors u and v .Now we write the Fourier transform of the generating function asˆ˜Ψ ∗ ( s ; z ) = X l ∈ { Own, Other } m ∈ { Up, Down } ˆ˜ α ∗ ( l, m ; s ; z ) | l, m i . From Eqs. (4.10)-(4.13), we have ˆ˜Ψ ∗ ( s ; z ) asˆ˜ α ∗ ( Own, U p ; s ; z ) = (cid:18) − µ ( z ) + dac ( λ ( z ) − az )Φ ( s ; z ) (cid:19) ( a κ + µ ( z ))Φ ( s ; z ) , ˆ˜ α ∗ ( Own, Down ; s ; z ) = (cid:18) − µ ( z ) + dac ( λ ( z ) − az )Φ ( s ; z ) (cid:19) √ κ − b κ Φ ( s ; z ) , ˆ˜ α ∗ ( Other, U p ; s ; z ) = z ( a κ + µ ( z ))Φ ( s ; z )Φ ( s ; z ) , ˆ˜ α ∗ ( Other, Down ; s ; z ) = z √ κ − b κ Φ ( s ; z )Φ ( s ; z ) , where Φ ( s ; z ) = w w − − c ) (cid:16) η + ( z ) + p ν ( z ) (cid:17) (cid:16) η − ( z ) − p ν ( z ) (cid:17) ( z − w )( z − w − ) , Φ ( s ; z ) = e ik (cid:16) ζ ( s ; z ) − p ν ( z ) (cid:17) z − v + ( s ))( z − v − ( s )) ,ζ ( s ; z ) = 2 ae − is z − − ∆ z ,v ± ( s ) = ae − is + a ∆ e is ± p ( ae − is + a ∆ e is ) − . Here we can rewrite v ± ( s ) as v ± ( s ) = e − iρ ( | a | cos γ ( s ) ± p | a | cos γ ( s ) −
1) = e − iρ e ± iθ ( s ) , where we take ∆ = e iρ , a = | a | e iσ , γ ( s ) = s − σ + ρ , cos θ ( s ) = | a | cos γ ( s ). Note that | v ± ( s ) | = 1. Now || ˆ˜Ψ ∗ ( s ; z ) || < ∞ for 0 < | z | < r , we can rewrite ˆ˜Ψ ∗ ( s ; z ) = P t ≥ ˆΨ ∗ t ( s ) z t . So we have for 0 < r < r ˆΨ ∗ t ( s ) = 12 πi I | z | = r ˆ˜Ψ( s ; z ) dzz t +1 . Therefore we get the Fourier transform of the state ˆΨ ∗ t ( s ) as follows: − ˆΨ ∗ t ( s ) ∼ ψ w + ( s )( w + ) − ( t +1) + ψ − w + ( s )( − w + ) − ( t +1) + ψ w − ( s )( w − ) − ( t +1) + ψ − w − ( s )( − w − ) − ( t +1) + ψ v + ( s )( v + ( s )) − ( t +1) + ψ v − ( s )( v − ( s )) − ( t +1) , (4.16)where ψ ± w ± ( s ) = Res( ˆ˜Ψ( s ; z ); ± w ± ) and ψ v ± ( s ) = Res( ˆ˜Ψ( s ; z ); v ± ( s )).Finally we compute the characteristic function by Eqs.(4.15) and (4.16). Now we have Z π (cid:0) k Λ r ( ψ ) ψ w + ( s ) k + k Λ r ( ψ ) ψ − w + ( s ) k (cid:1) ds π = C rp , Z π (cid:0) k Λ r ( ψ ) ψ w − ( s ) k + k Λ r ( ψ ) ψ − w − ( s ) k (cid:1) ds π = C m , Z π (cid:0) h Λ r ( ψ ) ψ w + ( s ) , Λ r ( ψ ) ψ − w + ( s ) i + h Λ r ( ψ ) ψ − w + ( s ) , Λ r ( ψ ) ψ w + ( s ) i (cid:1) ds π = 0 , Z π (cid:0) h Λ r ( ψ ) ψ w − ( s ) , Λ r ( ψ ) ψ − w − ( s ) i + h Λ r ( ψ ) ψ − w − ( s ) , Λ r ( ψ ) ψ w − ( s ) i (cid:1) ds π = 0 . e i ( t +1) θ ( s + ξ/t ) = e i ( t +1) θ ( s )+ iξh ( s )+ o ( t − ) , where h ( s ) = dθ ( s ) /ds , the above equation and Eq.(4.15)with the Riemann-Lebesgue lemma implylim t →∞ E h e iξX t,r /t i = C rp + C m + Z π e − iξh ( s ) p ( s ) ds π + Z π e iξh ( s ) q ( s ) ds π , (4.17)where p ( s ) = k Λ r ( ψ ) ψ v + ( s ) k and q ( s ) = k Λ r ( ψ ) ψ v − ( s ) k . Moreover, from a change of variable for last twoterms in Eq. (4.17), we have Z π (cid:16) e − iξh ( s ) p ( s ) + e iξh ( s ) q ( s ) (cid:17) ds π = Z ∞ e iξx w ( x ) f K ( x ) dx. After some calculations for w ( x ) with p ( s ) and q ( s ), we have the desired conclusion. We introduced a quantum walk with an enlarged basis to consider a reduction of quantum walks witharbitrary initial state. This method is based on an idea canceling the asymmetry caused from initial stateby a new tensor product. From our results in this paper, we discuss two interesting points. First, we foundthe oscillating probability as localization. From Theorem 1, the oscillatory term is expressed by L rc ( x, t ).We can see that this term vanishes with some initial states or local coins. For example, we consider V t,κ ,which is a quantum walk on T κ with local coin operators G κ with additional complex phase ˜ c at the origin.If ˜ c = 1 Corollary 3 implies that the oscillation does not occur with arbitrary initial state. Also if the initialstate is symmetric, the walk is reduced on a half line. Then it follows from Corollary 4 that no oscillatorybehavior arise with any complex phase ˜ c . Thus the initial state and differences on complex phase of localcoins are important factors for the oscillatory behavior on localization. Especially in quantum walks onthe one-dimensional lattice with homogeneous local coins, localization does not occur [13–15]. If localizationoccurs with perturbations of local coin operators on the one-dimensional lattice, there seems to be a conditionthat an oscillatory behavior arises in localization. Second, W t,κ has the scaling order t and the limit measurehas the density function f K ( x ) which is a half-line version of one appearing in the quantum walk on aline [13–15]. This is a typical property of quantum walks [8, 16, 17]. To show the universality of the limittheorems for quantum walks is one of the interesting future’s problems. Acknowledgments.
We thank Noriko Saitoh and Jun Kodama for useful discussions. N.K. is supportedby the Grant-in-Aid for Scientific Research (C) (No. 21540118).
References [1] Y. Aharonov, L. Davidovich, N. Zagury (1993),
Quantum random walks , Phys. Rev. A, , 1687[2] D. Meyer (1996), From quantum cellular automata to quantum lattice gases , J. Stat. Phys., , 551-574[3] E. Farhi, S. Gutmann (1998), Quantum computation and decision trees , Phys. Rev., A (2), 915-928[4] A.M. Childs, R. Cleve, E. Deotto, E. Farhi, S. Gutmann, D.A. Spielman (2003), Exponential algorithmic speedupby quantum walk , Proc. 35th ACM Symposium on Theory of Computing, 59-68[5] N. Shenvi, J. Kempe, K.B. Whaley (2003),
A quantum random walk search algorithms , Phys Rev. A, (6),062311[6] A. Ambainis, A.M. Childs, B.W. Reichardt, R. ˘Spalek, S. Zhang (2007), Any AND-OR formula of size N can beevaluated in time N / o (1) on a quantum computer , Proc. 48th IEEE Symposium on Foundations of ComputerScience, 363-372[7] N. Inui, N.Konno, E. Segawa (2005), One-dimensional three-state quantum walk , Phys. Rev. E, , 056112[8] K. Chisaki, M. Hamada, N. Konno, E. Segawa (2009), Limit theorems for discrete-time quantum walks on trees ,Interdiscip. Inform. Sci. , 423-429
9] N. Konno (2010),
Localization of an inhomogeneous discrete-time quantum walk on the line , Quantum Inf. Proc., , 405-418[10] Y. Shikano, H. Katsura (2010), Localization and fractality in inhomogeneous quantum walks with self-duality ,Phys. Rev. E, , 031122[11] M.J. Cantero, F.A. Gr¨unbaum, L. Moral, L. Ver´azquez (2010), Matrix valued szeg¨o polynomials and quantumrandom walks , Commun. Pure and Appl. Math., , 464-507[12] M.J. Cantero, F.A. Gr¨unbaum, L. Moral, L. Ver´azquez (2010), One dimensional quantum walks with one defect ,arXiv:1010.5762[13] N. Konno (2002),
Quantum random walks in one dimension , Quantum Inf. Proc., , 345-354[14] N. Konno (2005), A new type of limit theorems for the one-dimensional quantum random walk . J. Math. Soc.Jpn., , 1179-1195[15] G. Grimmett, S. Janson, P.F. Scudo (2004), Weak limits for quantum random walks , Phys. Rev. E, , 026119[16] T. Miyazaki, M. Katori, N. Konno (2007), Wigner formula of rotation matrices and quantum walks . Phys. Rev.A , 012332[17] E. Segawa, N. Konno (2008), Limit theorems for quantum walks driven by many coins . Int. J. Quantum Inf., ,1231-1243[18] N. Konno, E. Segawa (2011), Localization of discrete-time quantum walks on a half line via the CGMV method ,Quantum Inf. Comput., (5,6), 0485–0495.[19] N. Konno (2006), Continuous-time quantum walks on trees in quantum probability theory , Inf. Dim. Anal. Quan-tum Probab. Rel. Topics, , 287-297[20] H. Krovi, T.A. Brun (2007), Quantum walks on quotient graphs , Phys. Rev. A, , 062332[21] B. Tregenna, W. Flanagan, R. Maile, V. Kendon (2003), Controlling discrete quantum walks: coins and initialstates , New J. Phys., , 83[22] I. Carneiro, M. Loo, X. Xu, M. Girerd, V. Kendon, P.L. Knight (2005), Entanglement in coined quantum walkson regular graphs , New J. Phys., , 156[23] T. Oka, N. Konno, R. Arita, H. Aoki (2005), Breakdown of an electric-field driven system: a mapping to aquantum walk , Phys. Rev. Lett., , 100602 Appendix
Figure 5: Quantum walk with enlarged bases X ∗ t on a half lineWe calculate the generating function of Ψ ∗ t ( x ) by using the method in [23]. To simplify notations, for l ∈ { Own, Other } , we denote | l, , ǫ i = | l, , U p i and construct | l, , Down i as a dummy base, which alwayshas value 0 as its amplitude, so that the local coin operator on the origin has 2 × → x ; τ + τ + 1)evolution operator of the walk, we use an expression using weights (see Fig.5), where˜ Q = (cid:20) a κ √ κ − b κ √ κ − b κ − a κ (cid:21) ⊗ (cid:20) (cid:21) , (.18) P = I ⊗ (cid:20) a b (cid:21) , Q = I ⊗ (cid:20) c d (cid:21) , (.19)Ψ ∗ (0) = (cid:20) (cid:21) ⊗ (cid:20) (cid:21) . (.20)We define the generating function for the state by˜Ψ ∗ ( x ; z ) = ∞ X t =0 Ψ ∗ t ( x ) z t . In order to compute ˜Ψ ∗ ( x ; z ), we first define the transition amplitude ˜Ξ(0 → x ; τ ) as the weight of all pathsstarting from 0 ending at x after τ steps, and Ξ(0 → x ; τ ) as the weight of all paths on another walk defined by Q ′ = Q . For example, ˜Ξ(0 →
2; 4) =
QP Q ˜ Q + P QQ ˜ Q + Q ˜ QP ˜ Q and Ξ(0 →
2; 4) =
QP QQ + P QQQ + QQP Q .From ˜Ξ(0 → τ ) and Ξ(0 → x − τ ), we can obtain ˜Ξ(0 → x ; τ + τ + 1) as Fig.6. Then we get ˜Ψ ∗ ( x ; z )from the generating function for ˜Ξ(0 → x ; τ ).We now calculate the generating function for Ξ(0 → x ; τ ). Since the first operator should be Q on thehalf line, the weights of paths form Q · · · Q or P · · · Q . So we express Ξ(0 → x ; τ ) as a linear combination of Q and R : Ξ(0 → x ; τ ) = b q (0 → x ; τ ) Q + b r (0 → x ; τ ) R + δ ( x ) δ ( τ ) I ⊗ I where R = I ⊗ (cid:20) c d (cid:21) and we define b q (0 → x ; 0) = b r (0 → x ; 0) = 0. The generating function forΞ(0 → x ; τ ) is defined by ∞ X τ =0 Ξ(0 → x ; τ ) z τ = B q (0 → x ; z ) Q + B r (0 → x ; z ) R + δ ( x ) I ⊗ I , with B q (0 → x ; z ) = P ∞ τ =0 b q (0 → x ; τ ) z τ and B r (0 → x ; z ) = P ∞ τ =0 b r (0 → x ; τ ) z τ . Since the left-handtensor product of P and Q is I , the generating function for Ξ(0 → x ; τ ) corresponds to the result in [23],i.e., for sufficiently small z , B q (0 → x ; z ) = (cid:26) da λ ( z ) (cid:27) x d , x ≥ ,B q (0 → z ) = 0 ,B r (0 → x ; z ) = (cid:26) da λ ( z ) (cid:27) x λ ( z ) − azacz , x ≥ ,λ ( z ) = ∆ z + 1 − p ∆ z + 2∆(1 − | a | ) z + 12 dz . (.21)16igure 7: ˜Ξ(0 → τ ; 0)Here we take λ ( z ) for the smaller solution of the absolute value of λ ( z ) − d (cid:18) ∆ z + 1 z (cid:19) λ ( z ) + ad = 0 . (.22)Note that for sufficiently small z we can write λ ( z ) by Eq. (.21). Moreover since | a/d | = 1, we can take r < | λ ( z ) | < | z | < r . Next we calculate the generating function for ˜Ξ(0 → τ ). To do so,we introduce a new notation ˜Ξ(0 → τ ; n ) as the weight of all paths starting from the origin reaching theorigin n times before ending at the origin at time τ . Now we consider ˜Ξ(0 → τ ; 0). For τ ≥
2, we obtain˜Ξ(0 → τ ; 0) as (see Fig.7)˜Ξ(0 → τ ; 0) = (1 − δ ( τ )) P { b r (0 → τ − R } ˜ Q + δ ( τ ) P ˜ Q = { (1 − δ ( τ )) abb r (0 → τ −
2) + δ ( τ ) b } ˜ R, where ˜ R = (cid:20) a κ √ κ − b κ √ κ − b κ − a κ (cid:21) ⊗ (cid:20) (cid:21) and for τ < → τ ; 0) = 0. Therefore weget the generating function for ˜Ξ(0 → τ ; 0) as ∞ X τ =0 ˜Ξ(0 → τ ; 0) z τ = ( adB r (0 → z ) + b ) z ˜ R = B ˜ r (0 → z ; 0) ˜ R. Similarly, for τ ≥ → τ ; 1) as˜Ξ(0 → τ ; 1) = X τ + τ +4= τ { (1 − δ ( τ )) abb r (0 → τ ) + δ ( τ ) b } ˜ R × { (1 − δ ( τ )) abb r (0 → τ ) + δ ( τ ) b } ˜ R, and for τ < → τ ; 1) = 0. Thus the generating function for ˜Ξ(0 → τ ; 1) is obtained by ∞ X τ =0 ˜Ξ(0 → τ ; 1) z τ = { ( adB r (0 → z ) + b ) z } ˜ R I = B ˜ r I (0 → z ; 1) ˜ R I , where ˜ R I = I ⊗ (cid:20) (cid:21) . Recursively we have the following formulae: for n ≥ B ˜ r (0 → z ; n ) = (cid:18) − n (cid:19) { ( adB r (0 → z ) + b ) z } n +1 , (.23) B ˜ r I (0 → z ; n ) = (cid:18) − n +1 (cid:19) { ( adB r (0 → z ) + b ) z } n +1 . (.24)From Eqs. (.23) and (.24), we get the generation function for ˜Ξ(0 → τ ) by summing over n . Here( dB r (0 → z ) − ∆ z ) z = ( dλ ( z ) − ∆ z ) z/c , so we see that for z with | z | < r ≡ min( | c | , r ), | ( dλ ( z ) − ∆ z ) z/c | ≤ ( | dλ ( z ) | + | z | ) | z/c | < | d | + | c | = 1 . z such that | z | < r , ∞ X τ =0 ˜Ξ(0 → τ ) z τ = B ˜ r (0 → z ) ˜ R + B ˜ r I (0 → z ) ˜ R I + I ⊗ I ,B ˜ r (0 → z ) = ∞ X n =0 B ˜ r (0 → z ; n ) = ( dλ ( z ) − ∆ z ) z/c − { ˜ c ( dλ ( z ) − ∆ z ) z/c } ,B ˜ r I (0 → z ) = ∞ X n =0 B ˜ r I (0 → z ; n ) = { ( dλ ( z ) − ∆ z ) z/c } − { ( dλ ( z ) − ∆ z ) z/c } . For x ≥
1, ˜Ξ(0 → x ; τ ) is written by ˜Ξ(0 → τ ) and Ξ(0 → x ; τ ) (see Fig.6) as˜Ξ(0 → x ; τ ) = X τ + τ +1= τ Ξ(0 → x − τ ) ˜ Q ˜Ξ(0 → τ ) + δ ( τ ) δ ( x ) I ⊗ I . From the generating function for ˜Ξ(0 → τ ) and Ξ(0 → x ; τ ), we can compute the generating function for˜Ξ(0 → x ; τ ) as follows: for x ≥ ∞ X τ =0 ˜Ξ(0 → x ; τ ) z τ = { B q (0 → x − z ) Q + B r (0 → x − z ) R + δ ( x ) I ⊗ I } ˜ Qz ×{ B ˜ r (0 → z ) ˜ R + B ˜ r I (0 → z ) ˜ R I + I ⊗ I } + δ ( x ) I ⊗ I . Then we obtain the generating function ˜Ψ ∗ ( x ; z ) as follows:˜Ψ ∗ ( x ; z ) = ∞ X τ =0 ˜Ξ(0 → x ; τ ) z τ Ψ ∗ ( x ) ..