Stationary measure for three-state quantum walk
aa r X i v : . [ m a t h - ph ] S e p Stationary measure for three-state quantum walks
TAKAKO ENDO a Department of Applied Mathematics, Faculty of Engineering,Yokohama National University, Hodogaya, Yokohama 240-8501, Japan
TAKASHI KOMATSU b Department of Mathematics and Physics, Faculty of Science,Kanagawa University 2946 Tsuchiya, Hiratsukashi, Kanagawa, 259-1293, Japan
NORIO KONNO c Department of Applied Mathematics, Faculty of Engineering,Yokohama National University, Hodogaya, Yokohama 240-8501, Japan
TOMOYUKI TERADA d Office for Promoting Co-Creation Education Administrative Office, Kanazawa Institute of Technology, Nonoichi, Ishikawa 921-8501, Japan
Abstract
We focus on the three-state quantum walk(QW) in one dimension. In this paper, we give thestationary measure in general condition, originated from the eigenvalue problem. Firstly, we get thetransfer matrices by our new recipe, and solve the eigenvalue problem. Then we obtain the generalform of the stationary measure for concrete initial state and eigenvalue. We also show some specificexamples of the stationary measure for the three-state QW. One of the interesting and crucial futureproblems is to make clear the whole picture of the set of stationary measures.
Owing to their specific properties, quantum walks(QWs) have attracted much attention in various fields,such as quantum algorithms [3, 28], and topological insulators [17]. For the rich application of QWs,it is important to further study both analytically and numerically. Indeed, over the past decade manyresearchers have investigated the asymptotic behaviors of QWs from various viewpoints [5, 14, 18, 22, 23,30]. a [email protected] b [email protected] c [email protected] d [email protected] 1 Stationary measure for three-state quantum walk
So far, localization and the ballistic spreading have been known as the characteristic properties of QWs[2, 12, 13, 14]. They are described mathematically by two kinds of limit theorems, which are composed ofmeasures, i.e., the time-averaged limit measure corresponding to localization, and the weak limit measuresdescribing the ballistic spreading [22]. We should note that the weak limit measure is consisted by theDirac measure part corresponding to localization and absolutely continuous part, expressing the ballisticspreading. The weak convergence theorem for various types of QWs in one dimension, such as Hadamardwalk [18], Grover walk [19], the two-phase QWs [9, 11] were derived.Recently, the stationary measure of the QW has received peculiar interests as another key measure forthe distribution of QW. The stationary measure provides the stationary distribution, for instance. Herewe briefly review the past studies of the stationary measure. Comparing to the study of the stationarydistributions of Markov chains, the corresponding study of QWs has not been done sufficiently. Mostof the present papers deal with stationary measures of the discrete-time case mainly on Z , where Z isthe set of integers. As for the stationary measure of the two-state QW, Konno et al. [22] treated QWwith a single defect at the origin and showed that a stationary measure with exponential decay for theposition is identical to the time-averaged limit measure for the QW. [24] investigated stationary measuresfor various types of QWs. Endo et al. [6] got a stationary measure of the QW with a single defect whosequantum coins are defined by the Hadamard matrix at x = 0 and the rotation matrix at x = 0. Endo andKonno [7] calculated a stationary measure of QW with a single defect which was introduced and studiedby Wojcik et al . [30]. Furthermore, Endo et al. [11] and Endo et al. [8] obtained stationary measuresof the two-phase QWs without defect and with a single defect, respectively, and investigated the relationto localization and the topological insulator. Konno and Takei [25] considered stationary measures ofQWs and provided non-uniform stationary measures of non-diagonal QWs. They also showed that theset of the stationary measures generally contains uniform measure. As for the stationary measure of thethree-state QW, Konno [24] obtained the stationary measures of the three-state Grover walk. Moreover,Wang et al. [29] investigated stationary measures of the three-state Grover walk with a single defect at theorigin. Kawai et al. [15] constructed stationary measures for some types of the three-state QWs by usingthe reduction matrices. As for higher dimensional case, Komatsu and Konno [16] gave the stationarymeasures of the Grover walk on Z d ( d ∈ N ), where N is the set of natural numbers. In this paper, weconsider stationary measures for the three-state QWs.Since there are rich applications of space-inhomogeneous QWs, such as the applications for networks[27] and topological insulator [11], the research on the mathematical aspects of space-inhomogeneous QWto exactly grasp the asymptotic behavior is important topic in the theoretical study of QWs [1, 4, 20, 21].However, comparing to the study on the two-state QW, the three-state QW has not been treated enough.As for the stationary measures, there are few results for the quantum systems in comparison with that ofthe classical systems. In particular, the stationary measures for the classical systems have been known tobe constant or exponential increase: µ ( x ) = c ( ∈ l ∞ − space on Z ) , (cid:18) qp (cid:19) x ( / ∈ l ∞ − space on Z ) , with c, p, q ∈ R ≥ , and 0 = p < q, p + q = 1. Here R ≥ is the set of the real numbers r ≥
0. Also,in the study of the generalized eigenfunctions in the Spectral scattering theory, the eigenfunctions in l ∞ -space have been actively discussed [26]. According to the background, it is very important to studythe stationary measures that are not in l -space on Z . akako Endo, Takashi Komatsu, Norio Konno, Tomoyuki Terada Our main result is the general form of the stationary measure obtained by the transfer matrices of thethree-state QW.The rest of this paper is organized as follows. In Section 2, we give the definition of the three-stateQW which is the main target in this paper, and present our main result. We show specific examples ofour results in Section 3. In Appendix, we give the proof of Theorem 1.
First, we introduce discrete-time space-inhomogeneous QW with three-state on Z , which is a quantumversion of classical random walk with an additional coin state. The particle has a coin state at time n and position x described by a three dimensional vector in C , where C is the set of complex numbers:Ψ n ( x ) = Ψ Ln ( x )Ψ On ( x )Ψ Rn ( x ) ( x ∈ Z ) . The upper and lower elements express the left and right chiralities, respectively, and the middle elementcorresponds to the loop. The time evolution is determined by 3 × U x : U x = a x b x c x d x e x f x g x h x i x , where x ∈ Z , and a x , b x , c x , d x , e x , f x , g x , h x , i x ∈ C .Here the time evolution is determined by the recurrence formulaΨ n +1 ( x ) = P x +1 Ψ n ( x + 1) + R x Ψ n ( x ) + Q x − Ψ n ( x −
1) ( x ∈ Z ) , where P x = a x b x c x , R x = d x e x f x , Q x = g x h x i x , with U x = P x + R x + Q x . Note that P x and Q x correspond to the left and right movements, respectively,and R x represents the loop. Here we remark that the walker steps dependent on position.From now on, we introduce the stationary measure of the QW. LetΨ n = · · · , Ψ Ln ( − On ( − Rn ( − , Ψ Ln (0)Ψ On (0)Ψ Rn (0) , Ψ Ln (1)Ψ On (1)Ψ Rn (1) , · · · T ∈ ( C ) Z , and U ( s ) = . . . ... ... ... ... ... . . . · · · R − P − O O O · · ·· · · Q − R − P O O · · ·· · ·
O Q − R P O · · ·· · · O O Q R P · · ·· · · O O O Q R · · · . . . ... ... ... ... ... . . . with O = , Stationary measure for three-state quantum walk where T means the transposed operation. Then the state of the QW at time n is given by Ψ n = ( U ( s ) ) n Ψ for any n ≥
0. Let R + = [0 , ∞ ). Here we introduce a map φ : ( C ) Z → R Z + such that for Ψ n , we put φ (Ψ)( x ) = | Ψ L ( x ) | + | Ψ O ( x ) | + | Ψ R ( x ) | ( x ∈ Z ) . Then we define the measure µ : Z → R + by µ ( x ) = φ (Ψ)( x ) . We should note that µ ( x ) is a measure of the QW at position x . Now putΣ s = { φ (Ψ ) ∈ R Z + : ∃ Ψ , s . t . φ (( U ( s ) ) n Ψ ) = φ (Ψ ) ∀ n ≥ } , and we call the element of Σ s , the stationary measure of the QW.Here let S = { z ∈ C : | z | = 1 } , and consider the eigenvalue problem U ( s ) Ψ = λ Ψ(Ψ ∈ Map( Z , C ) , λ ∈ S ). Since U ( s ) is unitary, we directly see φ ( U ( s ) Ψ) ∈ Σ s .Now we solve the eigenvalue problem as follows. The proof is devoted to Appendix. Theorem 1
Let { U y } y ∈ Z be the set of y -parameterized unitary matrices of the three-state inhomo-geneous QW, and Ψ( x ) = [Ψ L ( x ) , Ψ O ( x ) , Ψ R ( x )] T be the probability amplitude. Note that there is arestriction for the initial state Ψ(0) [10]. Then the solutions for U ( s ) Ψ = λ Ψ(Ψ ∈ Map( Z , C ) , λ ∈ S ) are Ψ( x ) = Q xy =1 T (+) y Ψ(0) ( x ≥ , Ψ(0) ( x = 0) , Q xy = − T ( − ) y Ψ(0) ( x ≤ − , where T ( ± ) y are the transfer matrices defined by T (+) y = t (+)11 t (+)12 t (+)13 t (+)21 t (+)22 t (+)23 t (+)31 t (+)32 t (+)33 , T ( − ) y = t ( − )11 t ( − )12 t ( − )13 t ( − )21 t ( − )22 t ( − )23 t ( − )31 t (+)32 t (+)33 , with t (+)11 = ( λ − e y )( λ − g y − c y ) − g y − b y f y λ { a y ( λ − e y ) + b y d y } , t (+)12 = − h y − { b y f y + c y ( λ − e y ) } λ { a y ( λ − e y ) + b y d y } t (+)13 = − i y − { b y f y + c y ( λ − e y ) } λ { a y ( λ − e y ) + b y d y } , t (+)21 = λ d y + g y − ( a y f y − c y d y ) λ { a y ( λ − e y ) + b y d y } t (+)22 = h y − ( a y f y − c y d y ) λ { a y ( λ − e y ) + b y d y } , t (+)23 = i y − ( a y f y − c y d y ) λ { a y ( λ − e y ) + b y d y } t (+)31 = g y − λ , t (+)32 = h y − λ , t (+)33 = i y − λ , akako Endo, Takashi Komatsu, Norio Konno, Tomoyuki Terada and t ( − )11 = a y +1 λ , t ( − )12 = b y +1 λ , t ( − )13 = c y +1 λ ,t ( − )21 = − a y +1 ( f y g y − i y d y ) λ { h y f y + i y ( λ − e y ) } , t ( − )22 = − b y +1 ( f y g y − i y d y ) λ { h y f y + i y ( λ − e y ) } ,t ( − )23 = λ f y − c y +1 ( f y g y − i y d y ) λ { h y f y + i y ( λ − e y ) } , t ( − )31 = − a y +1 { h y d y + g y ( λ − e y ) } λ { h y f y + i y ( λ − e y ) } ,t ( − )32 = − b y +1 { h y d y + g y ( λ − e y ) } λ { h y f y + i y ( λ − e y ) } , t ( − )33 = ( λ − e y )( λ − g y c y +1 ) − h y c y +1 d y λ { h y f y + i y ( λ − e y ) } . Furthermore, the stationary measure is given by µ ( x ) = φ (Ψ)( x ) = k Ψ( x ) k ( x ∈ Z ) . We note that the initial state develops by the transfer matrices. We can obtain the stationary measurefor given three-state QW and initial state by calculating the transfer matrices. Also, we should remarkthat we cannot apply Theorem 1 in the case that the elements of the transfer matrices diverge. Our resultis more effective for the three-state QWs in general than the result obtained by Kawai et al. [15], sincethey use the reduced matrix, which restricts the models that can be analyzed.
In this section, we exhibit some concrete typical examples of our result, Theorem 1.1. The Grover walk:First, we consider the Grover walk defined by the unitary matrix U x = 13 − − − . We got the stationary amplitude originated from the eigenvalue problem U ( s ) Ψ = λ Ψ as follows.
Proposition 1
Let { M y } y ∈ Z be the set of y ∈ Z -parameterized unitary matrices of the Grover walk,and Ψ( x ) = [Ψ L ( x ) , Ψ O ( x ) , Ψ R ( x )] T be the probability amplitude. Put α = Ψ L (0) , γ = Ψ O (0) and β = Ψ R (0) . Then we have (1 + 3 λ )Ψ O (0) = 2Ψ L (0) + 2Ψ R (0) , and for every λ ∈ S , wecan choose Ψ(0) = [ α, , − α ] T as an initial state. Now we take λ = − , and the solutions for U ( s ) Ψ = λ Ψ(Ψ ∈ M ap ( Z , C ) , λ ∈ S ) , are Ψ( x ) = Q xy =1 T (+) y Ψ(0) ( x ≥ , Ψ(0) ( x = 0) , Q xy = − T ( − ) y Ψ(0) ( x ≤ − , where T ( ± ) y are the transfer matrices defined by Stationary measure for three-state quantum walk T (+) y = Konno −
13 23 − − −
23 13 , T ( − ) y = − − −
13 23 −
130 0 1 . For Ψ(0) = [ α, , − α ] T , we see Ψ( x ) = [ α, , − α ] T ( x ∈ Z ) , and therefore, we obtain µ ( x ) = 2 | α | ( x ∈ Z ) . We see that the stationary measure is uniform, which is same with the result in Section 4 of [15].2. The Grover walk+1-defect model:Next, we focus on the QW defined by the unitary matrices U x = − − − ( x = 0) ,ρ − − − ( x = 0) , with ρ = e iφ ( φ ∈ [0 , ∞ )).The QW is given by putting the weight ρ at x = 0 to the Grover walk. We derived the stationaryamplitude of the eigenvalue problem U ( s ) Ψ = λ Ψ as follows.
Proposition 2
Let { M y } y ∈ Z be the set of y ∈ Z -parameterized unitary matrices of the Groverwalk+1-defect model, and Ψ( x ) = [Ψ L ( x ) , Ψ O ( x ) , Ψ R ( x )] T be the probability amplitude. Put α =Ψ L (0) , γ = Ψ O (0) and β = Ψ R (0) . Then we see (1 − λ )Ψ O (0) = 2Ψ L (0) + 2Ψ R (0) , and for every λ ∈ S , we can take Ψ(0) = [ α, , − α ] T as an initial state. Now we take λ = − , and the solutionsfor U ( s ) Ψ = λ Ψ(Ψ ∈ M ap ( Z , C ) , λ ∈ S ) , are Ψ( x ) = Q xy =1 T (+) y Ψ(0) ( x ≥ , Ψ(0) ( x = 0) , Q xy = − T ( − ) y Ψ(0) ( x ≤ − , where T ( ± ) y are the transfer matrices defined by akako Endo, Takashi Komatsu, Norio Konno, Tomoyuki Terada T (+)1 = − −
23 1323 23 − , T ( − ) − = −
13 23 2313 − −
530 0 1 ,T (+) y = −
13 23 − − −
23 13 , T ( − ) y = − − −
13 23 −
130 0 1 ( | y | ≥ . For Ψ(0) = [ α, , − α ] T , we obtainΨ( x ) = [ α, , − α ] T ( | x | 6 = 1) , [ α, − α, α ] T ( x = 1) , [ − α, α, − α ] T ( x = − . Hence, we have µ ( x ) = 2 | α | | x | 6 = 1) , | x | = 1) . Thereby, the stationary measure is not uniform, and the result does not coincide with the result inSection 4 of [10]. The defect at the origin seems to influence the value of the stationary measurenot at the origin, but at both sides of the origin.3. The Fourier QW: Here we treat the Fourier QW defined by the unitary matrices U x = 1 √ ω ω ω ω with ω = e iπ .By putting λ = (1+2 ω ) / √
3, we got the stationary amplitude of the eigenvalue problem U ( s ) Ψ = λ Ψas follows.
Proposition 3
Let { M y } y ∈ Z be the set of y ∈ Z -parameterized unitary matrices of the Fourier QW,and Ψ( x ) = T [Ψ L ( x ) , Ψ O ( x ) , Ψ R ( x )] be the probability amplitude. Then we see ( √ λ − ω )Ψ O (0) = Stationary measure for three-state quantum walk Ψ L (0) + ω Ψ R (0) , and for every λ ∈ S , we can choose Ψ(0) = [ α, , − αω − ] T as an initial state.Now we take λ = i , and the solutions for U ( s ) Ψ = λ Ψ(Ψ ∈ M ap ( Z , C ) , λ ∈ S ) , are Ψ( x ) = Q xy =1 T (+) y Ψ(0) ( x ≥ , Ψ(0) ( x = 0) , Q xy = − T ( − ) y Ψ(0) ( x ≤ − , where T ( ± ) y are the transfer matrices defined by T (+) y = ω − ω − ω − ω − ω ω ω (1 − ω ) ω − ω − ω , T ( − ) y = ω (1 − ω ) 1 ω (1 − ω ) 1 ω (1 − ω ) − − ω ω ) − − ω ω ) − − ω ω )0 0 1 ( | y | ≥ . For Ψ(0) = T [ α, , − αω − ], we getΨ( x ) = T (cid:2) α, , − αω − (cid:3) ( x = 3 m ) , T (cid:20) αω, α − − ωω , − αω − (cid:21) ( x = 3 m + 1) , T (cid:20) αω , α − ω − ω , − αω − (cid:21) ( x = 3 m + 2) , T (cid:20) αω − , α − ω − ω , − αω − (cid:21) ( x = − m − , T (cid:20) αω − , α − − ωω , − αω − (cid:21) ( x = − m − , where m ∈ Z ≥ with Z ≥ = { , , , · · · } . Hence, we obtain µ ( x ) = | α | × x = 3 m ) , x = 3 m ) . We notice that the stationary measure has period 3, which is same with the result in Section 4 of[15].
In this paper, we obtained for the three-state QW, the general form of the stationary measure originatedfrom the eigenvalue problem. Our method is mainly based on the transfer matrices, and is more effectivethan the reduction method developed by Kawai et al. [15], since we do not need to reduct to two-state akako Endo, Takashi Komatsu, Norio Konno, Tomoyuki Terada model in our recipe. By using our result, we can derive various types of stationary measures for many typesof the three-state QWs, which contributes to clear the whole picture of the set of stationary measures.One of our future problems is to examine the eigenvalues and eigenspaces, which directly connects toSpectral theory. Also, to investigate the stationary measure which does not come from the eigenvalueproblem is fundamental for the study of the stationary measure of the QW. On the other hand, to explorethe relation between the three-state QWs and physical phenomenon, such as the topological insulator,leads to the application of the three-state QWs to quantum information sciences. Acknowledgments
TE is supported by financial support of Postdoctoral Fellowship from Japan Society for the Promotion ofScience.
References
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In Appendix, we derive the transfer matrices T ( ± ) x of the three-state QW, which leads to Theorem 1.Since λ Ψ = U ( s ) Ψ(Ψ ∈ Map( Z , C ) , λ ∈ S ) holds, we have λ Ψ L ( x ) = a x +1 Ψ L ( x + 1) + b x +1 Ψ O ( x + 1) + c x +1 Ψ R ( x + 1) , (1) λ Ψ O ( x ) = d x Ψ L ( x ) + e x Ψ O ( x ) + f x Ψ R ( x ) , (2) λ Ψ R ( x ) = g x − Ψ L ( x −
1) + h x − Ψ O ( x −
1) + i x − Ψ R ( x − . (3)1. Case of T ( − ) x : By using Eq.(1), we can write down Ψ L ( x )Ψ O ( x )Ψ R ( x ) = a x +1 λ b x +1 λ c x +1 λ ( A )( B ) Ψ L ( x + 1)Ψ O ( x + 1)Ψ R ( x + 1) . From now on, we derive ( A ) and ( B ), which directly leads to T ( − ) x . • For (A): Owing to Eqs.(3) and (2), we have h x Ψ O ( x ) = λ Ψ R ( x + 1) − g x Ψ L ( x ) − i x Ψ R ( x ) , (4)and Ψ R ( x ) = λf x Ψ O ( x ) − d x f x Ψ L ( x ) − e x f x Ψ O ( x ) , (5) akako Endo, Takashi Komatsu, Norio Konno, Tomoyuki Terada respectively. Substituting Eq.(5) into Eq.(4), we get (cid:26) h x + i x f x ( λ − e x ) (cid:27) Ψ O ( x ) = λ Ψ R ( x + 1) − (cid:18) g x − i x d x f x (cid:19) Ψ L ( x ) . Taking into account of Eq.(1), we obtainΨ O ( x ) = − a x +1 ( f x g x − i x d x ) λ { h x f x + i x ( λ − e x ) } Ψ L ( x + 1) − b x +1 ( f x g x − i x d x ) λ { h x f x + i x ( λ − e x ) } Ψ O ( x + 1)+ λ f x − c x +1 ( f x g x − i x d x ) λ { h x f x + i x ( λ − e x ) } Ψ R ( x + 1) , (6)which implies (A). • For (B): From Eq.(3), we get i x Ψ R ( x ) = λ Ψ R ( x + 1) − g x Ψ L ( x ) − h x Ψ O ( x ) . (7)Substituting Eq.(16) and Eq.(1) into Eq.(7), we seeΨ R ( x ) = − a x +1 { h x d x + g x ( λ − e x ) } λ { h x f x + i x ( λ − e x ) } Ψ L ( x + 1) − b x +1 { h x d x + g x ( λ − e x ) } λ { h x f x + i x ( λ − e x ) } Ψ O ( x + 1)+( λ − e x )( λ − g x c x +1 ) − h x c x +1 d x λ { h x f x + i x ( λ − e x ) } Ψ R ( x + 1) , which leads to (B).2. Case of T (+) x : From, Eq.(3), we can write down Ψ L ( x )Ψ O ( x )Ψ R ( x ) = ( C )( D ) g x − λ h x − λ i x − λ Ψ L ( x − O ( x − R ( x − . (8)Hereafter, we calculate ( C ) and ( D ), which contributes to T (+) x . • For (C): By Eqs.(1) and (2), we have a x Ψ L ( x ) = λ Ψ L ( x − − b x Ψ O ( x ) − c x Ψ R ( x ) , (9)and Ψ O ( x ) = d x λ − e x Ψ L ( x ) + f x λ − e x Ψ R ( x ) , (10)respectively. Substituting Eq.(10) into Eq.(9), and taking into account of Eq.(3), we obtainΨ L ( x ) = ( λ − e x )( λ − g x − c x ) − g x − b x f x λ { a x ( λ − e x ) + b x d x } Ψ L ( x − − h x − { b x f x + c x ( λ − e x ) } λ { a x ( λ − e x ) + b x d x } Ψ O ( x − − i x − { b x f x + c x ( λ − e x ) } λ { a x ( λ − e x ) + b x d x } Ψ R ( x − , (11)which implies (C). Stationary measure for three-state quantum walk • For (D): Eq.(1)gives b x Ψ O ( x ) = λ Ψ L ( x − − a x Ψ L ( x ) − c x Ψ R ( x ) . (12)Substituting Eq.(21) and Eq.(3) into Eq.(12), we acquireΨ O ( x ) = λ d x + g x − ( a x f x − c x d x ) λ { a x ( λ − e x ) + b x d x } Ψ L ( x −
1) + h x − ( a x f x − c x d x ) λ { a x ( λ − e x ) + b x d x } Ψ O ( x − i x − ( a x f x − c x d x ) λ { a x ( λ − e x ) + b x d x } Ψ R ( x − ,,