Convergence of quantum random walks with decoherence
aa r X i v : . [ m a t h . P R ] A ug Convergence of quantum random walks withdecoherence
November 9, 2018
Shimao Fan ∗ , Zhiyong Feng ∗ , Sheng Xiong ∗∗ and Wei-Shih Yang ∗∗ Department of MathematicsTemple University, Philadelphia, PA 19122 ∗∗ Department of Mathematics and SciencesEdward Waters College, Jacksonville, FL 32209Email: [email protected], [email protected]@ewc.edu, [email protected] WORDS: Quantum walk, Decoherence, Limiting distributionPACS numbers: 05.30.-d, 03.67.Lx, 05.40.-a
Abstract
In this paper, we study the discrete-time quantum random walks on a linesubject to decoherence. The convergence of the rescaled position probabilitydistribution p ( x, t ) depends mainly on the spectrum of the superoperator L kk .We show that if 1 is an eigenvalue of the superoperator with multiplicity oneand there is no other eigenvalue whose modulus equals to 1, then ˆ P ( ν √ t , t ) con-verges to a convex combination of normal distributions. In terms of positionspace, the rescaled probability mass function p t ( x, t ) ≡ p ( √ tx, t ), x ∈ Z/ √ t ,converges in distribution to a continuous convex combination of normal distri-butions. We give an necessary and sufficient condition for a U (2) decoherentquantum walk that satisfies the eigenvalue conditions. We also give a completedescription of the behavior of quantum walks whose eigenvalues do not sat-isfy these assumptions. Specific examples such as the Hadamard walk, walks nder real and complex rotations are illustrated. For the O (2) quantum ran-dom walks, an explicit formula is provided for the scaling limit of p ( x, t ) andtheir moments. We also obtain exact critical exponents for their moments atthe critical point and show universality classes with respect to these criticalexponents. In recent years quantum walks (QWs), as the quantum analog of the classical randomwalks (CRWs), have attracted great attention from mathematicians, computer scien-tists, physicists and engineers. Two forms of QWs, continuous-time QWs (CTQW)[7] and discrete-time QW (DTQW)[1, 10, 3, 11, 4], are widely studied. In this work,we restrict our discussion to DTQW. In this case, an extra “coin” degree of freedomis introduced into the system. Unlike the classical random walk, where the directionof the particle moves is determined by the outcome of a “coin flip”, for quantumrandom walks both the “flip” of the coin and the conditional motion of the particlegiven by unitary transformations are needed.In 2003, Brun, Carteret and Ambainis [4] discussed the decoherent Hadamardwalk on 1-dimensional integer lattice Z , and found the expressions for the first andsecond moments of the position and showed that in the long time limit the variancegrows linearly with time with the diffusive character. However, they did not providean exact expression for the higher order moments nor the limiting distribution, due tothe complicated forms of the eigenvalues of the superoperator L kk and the difficultyto evaluate the position probability analytically. In 2004, Grimmett, Janson andScudo [8] obtained the scaling limit of quantum random walks without decoherence.Their scaling factor is t . Recently in [2], the scaling limit of a quantum random walkwith a Markov controlled coin process converges either with scaling factor t or √ t ,depending on eigenvalue conditions of the walk operator.In this paper, we consider the model with decoherence operators given by [4]. Weovercome the difficulties in there to obtain the scaling limit of decoherent quantumrandom walks, by analyzing ˆ P ( ν, t ), the characteristic function (Fourier transforma-tion) of the position probability distribution p ( x, t ). Here p ( x, t ), x ∈ Z , denotesthe probability that the particle is found at position x at time t . It turns out thatthe convergence of ˆ P ( ν √ t , t ) depends on the spectrum of the superoperator L kk . Weshow that if 1 is an eigenvalue of the superoperator with multiplicity one and there isno other eigenvalue whose modulus equals to 1, then ˆ P ( ν √ t , t ) converges to a convexcombination of normal distributions. In terms of position space, the rescaled prob-2bility mass function p t ( x, t ) ≡ p ( √ tx, t ), x ∈ Z/ √ t , converges in distribution to acontinuous convex combination of normal distributions. We give an necessary andsufficient condition for a U (2) decoherent quantum walk that satisfies the eigenvalueconditions. For the O (2) quantum walks such as the Hadamard walk, and walks un-der real or complex rotations are discussed. An explicit limiting distribution formulais provided for these walks.Our article is organized as follows. In Section 2, we present basic concepts ofquantum random walks. In Section 3, we present our main result about the limitof p t ( x, t ). In Section 4, we give examples that illustrate our results. We obtainthe scaling limit and the exact critical exponents for their moments at the criticalpoint. We show that the decoherent quantum random walks, with coin space unitarytransformation U ∈ O (2), θ = nπ , n = 0 , , ,
3, belong to the same universality classwith respected to the critical exponents of all moments at their critical points.
Consider a general quantum random walk on the 1-dimensional integer lattices Z . Tobe consistent, we adapt analogous definitions and notations as those outlined in [4].We denote the state space by a Hilbert space H p ⊗ H , where H p denotes the positionspace and H denotes the coin space. The basis of the position space are | x > , where x ∈ Z and, the basis of the coin space are | R > and | L > . We will assume that thewalk starts at the origin. The shift operators in H p are defined as follows S + | x > = | x + 1 >, (2.1) S − | x > = | x − >, (2.2)where S − and S + are unitary shift operators on the particle position. Let P R , P L betwo orthogonal projections on the coin space H spanned by | R > or | L > , where P R + P L = I . Let U be a unitary transformation on H , that acts as the ”flipping”of the coin. Then the evolution operator of the quantum random walk is given by E = ( S + ⊗ P R + S − ⊗ P L )( I ⊗ U ) . (2.3)The eigenvectors | k > of S − , S + are | k > = X x e ikx | x >, k ∈ [0 , π ] , (2.4)3ith eigenvalues S + | k > = e − ik | k >,S − | k > = e ik | k > . (2.5)Therefore, in | k > basis, the evolution operator is E ( | k > ⊗| Φ > ) = | k > ⊗ ( e − ik P R + e ik P L ) U | Φ > ≡ | k > ⊗ U k | Φ >, (2.6)where U k = ( e − ik P R + e ik P L ) U is also a unitary operator.The decoherence on the coin space is defined as follows. Suppose before eachunitary transformation acting on the coin, a measurement is performed on the coin.This measurement is given by a set of operators { A n } on H which satisfy X n A ∗ n A n = I. (2.7)Through out this paper, we also assume that the measurement is unital, i.e., itsatisfies X n A n A ∗ n = I. (2.8)After the measurement, a density operator χ on H is transformed by χ → χ ′ = X n A n χA ∗ n . (2.9)The general density operator of quantum random walk is then given by ρ = Z dk π Z dk ′ π | k >< k ′ | ⊗ χ kk ′ , (2.10)where χ kk ′ ∈ L ( H ), and L ( H ) is a vector space of linear operators on H . Thenafter one step of the evolution under coin space decoherence, the density operatorbecomes ρ ′ = Z dk π Z dk ′ π | k >< k ′ | ⊗ X n U k A n χ kk ′ A ∗ n U ∗ k ′ (2.11)Suppose the quantum walk starts at the state | > ⊗| Φ > , then the initial stateis given by the density operator ρ = Z dk π Z dk ′ π | k >< k ′ | ⊗ | Φ >< Φ | . (2.12)4fter t steps, the state evolves to ρ t = Z dk π Z dk ′ π | k >< k ′ | ⊗ X n ,...,n t U k A n t · · · U k A n | Φ >< Φ | A ∗ n U ∗ k ′ · · · A ∗ n t U ∗ k ′ . (2.13)If we define the superoperator L kk ′ to be an operator which maps L ( H ) to L ( H ): L kk ′ B ≡ X n U k A n BA ∗ n U ∗ k ′ , ∀ B ∈ L ( H ) , (2.14)then ρ t = Z dk π Z dk ′ π | k >< k ′ | ⊗ L tkk ′ | Φ >< Φ | . (2.15)The probability of reaching a point x at time t is p ( x, t ) = T r { ( | x >< x | ⊗ I ) ρ t } = 1(2 π ) Z dk Z dk ′ < x | k >< k ′ | x > T r {L tkk ′ | Φ >< Φ |} = 1(2 π ) Z dk Z dk ′ e ix ( k − k ′ ) T r {L tkk ′ | Φ >< Φ |} . (2.16) Let ˆ P ( ν, t ) ≡ < e iνx > t = X x e iνx p ( x, t ) (3.1)be the characteristic function of p ( x, t ). By the property of the δ function12 π X x x m e − ix ( k − k ′ ) = ( − i ) m δ ( m ) ( k − k ′ ) , (3.2)5nd (2.16), we have < e iνx > t = X x e iνx p ( x, t )= X x e iνx Z dk π Z dk ′ π e ix ( k − k ′ ) T r {L tkk ′ | Φ >< Φ |} = Z dk π Z dk ′ π X x e − ix ( k ′ − k − ν ) T r {L tkk ′ | Φ >< Φ |} = Z dk π Z dk ′ π πδ ( k ′ − k − ν ) T r {L tkk ′ | Φ >< Φ |} = 12 π Z dkT r {L tk,k + ν | Φ >< Φ |} . (3.3)For any initial state ˆ O ∈ L ( H ), the generating function of < e iνx > t is given by G ( z, ν ) = ∞ X t =0 z t < e iνx > t (3.4)= 12 π Z dk ∞ X t =0 z t T r {L tk,k + ν ˆ O } (3.5)= 12 π Z dkT r { I − z L k,k + ν ˆ O } , (3.6)where | z | < O ∈ L ( H ). Note that the generating function is well defined sincethe spectrum of L k,k + ν is less than or equal to 1 by the following lemma. Lemma 3.1.
Suppose U ∈ U ( C ) and the set of operators { A n } is unital. Let λ bean eigenvalue of L k,k + ν , then | λ | ≤ . Proof:
Define kL k,k k = sup = ˆ O ∈ L ( H ) kL k,k ˆ O kk ˆ O k with k ˆ O k = T r ( ˆ O ∗ ˆ O ) , then wehave kL k,k k ≤ kL k,k + ν ˆ O k = T r (( X n U k A n ˆ OA ∗ n U ∗ k + ν ) ∗ ( X n U k A n ˆ OA ∗ n U ∗ k + ν ))= T r ( U k + ν ( X n A n ˆ OA ∗ n ) ∗ U ∗ k U k ( X n A n ˆ OA ∗ n ) U ∗ k + ν )= T r ( X n A n ˆ OA ∗ n ) ∗ ( X n A n ˆ OA ∗ n ))= kL k,k ˆ O k . kL k,k + ν k = kL k,k k ≤ | λ | ≤ P ∞ t =0 ( z L k,k + ν ) t ˆ O converges in | z | < I − z L k,k + ν does not have any pole inside the disk | z | < O ∈ L ( H ) can be written as a linear combination ofPauli matrices: ˆ O = r I + r σ + r σ + r σ , (3.7)where σ , , = σ x,y,z are usual Pauli matrices. Hence ˆ O can be represented by acolumn vector ˆ O = r r r r . (3.8)By Lemma 3.1 and 1 − z L k,k + ν ∈ L ( L ( H )), we have < e iνx > t = 12 πi I | z | = r< G ( z, ν ) z t +1 dz, (3.9)for some 0 < r <
1. Let A be the matrix associated with 1 − z L k,k + ν , with respectto the Pauli matrices, then11 − z L k,k + ν ˆ O = A − r r r r = 1det( A ) A A A A A A A A A A A A A A A A r r r r , where A ij is the cofactor of A .Note that T r ( σ i ) = 0 for i = 1 , ,
3, and
T r ( σ ) = 2. So when taking the tracein (3.6) only the first row action h ( z, ν ) = A r + A r + A r + A r remains.Therefore G ( z, ν ) = 12 π Z dk h ( z, ν )det A . (3.10)Let L = ( l ij ( ν )) be the matrix representation of L k,k + ν in terms of Pauli matrices.Then we have the following lemma. Lemma 3.2.
Suppose U ∈ U ( C ) and { A n } is unital. Then L k,k + ν has the followingrepresentation cos ν × × × × × × × × × i sin ν × × × . oreover, if ν = 0 , then L k,k has the following representation × × × × × × × × × . Proof:
Let U ∈ U ( C ). Then | detU | = 1. Let detU = e iγ . We consider thenormalized operator W = e − i γ U. (3.11)Then W ∈ SU ( C ). By (2.14), L kk ′ is the same for U and W . Therefore, withoutloss of generality, we may assume that U ∈ SU ( C ) with the following form U = (cid:18) α − ββ α (cid:19) where α, β ∈ C and | α | + | β | = 1. Then U k = (cid:18) e − ik α − e − ik βe ik β e ik α (cid:19) , (3.12)and L k,k + ν σ i = l i ( ν ) σ + l i ( ν ) σ + l i ( ν ) σ + l i ( ν ) σ = (cid:18) l i ( ν ) + l i ( ν ) − il i ( ν ) + l i ( ν ) il i ( ν ) + l i ( ν ) l i ( ν ) − l i ( ν ) (cid:19) , where i = 0 , , , . On the other hand, L k,k + ν σ = X n U k A n IA ∗ n U ∗ k + ν = U k ( X n A n IA ∗ n ) U ∗ k + ν = U k U ∗ k + ν = (cid:18) e iν e − iν (cid:19) . (3.13)Hence l ( ν ) + l ( ν ) = e iν ,l ( ν ) − l ( ν ) = e − iν , − il ( ν ) + l ( ν ) = 0 ,il ( ν ) + l ( ν ) = 0 , l ( ν ) = cos( ν ), l ( ν ) = l ( ν ) = 0 and l ( ν ) = i sin( ν ). Inparticular, if ν = 0 , then the first column of L k,k is (1 , , , T .Next we finish our proof by showing l i (0) = 0, for i = 2 , ,
4. Suppose X n U k A n σ i − A ∗ n U ∗ k = (cid:18) τ i τ i τ i τ i (cid:19) , then l i (0) + l i (0) = τ i (0) ,l i (0) − l i (0) = τ i (0) , and hence l i (0) = 12 ( τ i (0) + τ i (0)) = 12 T r ( L k,k σ i − ) = 12 T r ( σ i − ) = 0 , (3.14)for i = 2 , ,
4. The third equality in the above holds because L k,k preserves the trace. Theorem 3.1.
Suppose U ∈ U ( C ) , the set of operators { A n } is unital, is aneigenvalue of L k,k with multiplicity one, and | λ | < for any other eigenvalue λ of L k,k . Then lim t →∞ ˆ P ( ν √ t , t ) = 12 π Z π e − z ′′ (0) ν dk, ∀ ν ∈ [0 , π ] , where z ( ν ) is the root of det (1 − z L k,k + ν ) = 0 such that z (0) = 1 . By the Cramer-Levy Theorem (see e.g. Theorem 6.3.2 [6]), the above theoremimplies that under the conditions, the distribution of the scaling limit is a continuousconvex combination of normal distributions with variance z ′′ (0). In other words, ifwe define the rescaled probability mass function on Z √ t by p t ( x, t ) ≡ p ( √ tx, t ) , x ∈ Z √ t , (3.15)then p t converges in distribution to the continuous convex combination of normaldistributions whose density function is given by F ( x ) = 12 π Z π p πz ′′ (0) e − z ′′ x dk, x ∈ R. roof: By (3.11), U can be normalized to an SU operator W . Note that by(2.14)and (2.16), L kk + ν and p ( x, t ) are the same for U and W . Therefore, withoutloss of generality, we may assume that U ∈ SU ( C ). By (3.10) and Cauchy’s integralformula, < e iνx > t = 12 πi I | z | = r< G ( z, ν ) z t +1 dz = 12 π Z dk πi I | z | = r< h ( z, ν ) z t +1 detA dz. Let z ( ν ) , z ( ν ) , z ( ν ) , z ( ν ) be four roots of det A = 0. Then z ( ν ) , z ( ν ) , z ( ν ) , z ( ν ) are four eigenvalues of L k,k + ν . By the assumptions that 1 is an eigenvalue of L k,k with multiplicity one, and | λ | < λ of L k,k , we may makethe following ordering 1 = | z (0) | < | z (0) | ≤ | z (0) | ≤ | z (0) | . Let l ( z, ν ) = h ( z,ν ) z t +1 detA .By continuity of z i ( ν ) , i = 0 , , ,
3, in both variables ν and k , there exist a constant R >
1, and a small neighborhood V of ν = 0 such that | z ( ν ) | < R < | z i ( ν ) | for any ν ∈ ¯ V and k ∈ [0 , π ], for all i = 1 , ,
3. By Cauchy’s Residue Theorem,12 πi I | z | = R h ( z, ν ) z t +1 detA dz = Res ( l, z = 0) + Res ( l, z = z ( ν )) . Note that there exists t such that ν √ t ∈ ¯ V for all ν ∈ (0 , π ) if t ≥ t . Let g ( z, ν ) =det A . By compactness of { ( z, ν, k ); | z | = R, ν ∈ ¯ V , k ∈ [0 , π ] } , there exists aconstant C such that | h ( z, ν √ t ) g ( z, ν √ t ) | ≤ C on { ( z, ν ); | z | = R, ν ∈ [0 , π ] , k ∈ [0 , π ] } , for all t ≥ t . Therefore, | lim t →∞ πi I | z | = R z t +1 h ( z, ν √ t ) g ( z, ν √ t ) dz | ≤ lim t →∞ R t +1 C = 0 . Hence lim t →∞ Res ( l, z = 0) = − lim t →∞ Res ( l, z = z ( ν √ t )) . For any fixed ν , we have Res ( l, z = z ( ν √ t )) = 2 h ( z ( ν √ t ) , ν √ t ) z t +10 ( ν √ t ) ∂g∂z ( z ( ν √ t ) , ν √ t ) . Let t → ∞ , we havelim t →∞ Res ( l, z = z ( ν √ t )) = 2 h (1 , ∂g∂z (1 ,
0) lim t →∞ z (cid:18) ν √ t (cid:19) − t −
10e claim that 2 h (1 , ∂g∂z (1 ,
0) = − z ′ (0) = 0 . (3.16)By the Dominated Convergence Theorem, we havelim t →∞ ˆ P ( ν √ t , t ) = 12 π Z π e − z ′′ (0) ν dk. (3.17)We will finish our proof by proving the claim (3.16). Let M denote the 3 by 3submatrix of L = ( l ij ): M ( ν ) = l l l l l l l l l . Then the matrix A | ν =0 = (cid:18) − z I − zM (0) (cid:19) . The cofactor A = A = A = 0 and A = det( I − zM (0)). By Lemma 3.2, z i (0) are eigenvalues of L k,k for i = 0 , , ,
3. Therefore z (0) , z (0) , z (0) are eigenvalues of M (0). Hence det( I − zM (0)) = (1 − zz (0) )(1 − zz (0) )(1 − zz (0) ) (3.18)= − z (0) z (0) z (0) ( z − z (0))( z − z (0))( z − z (0)) . (3.19)On the other hand, ∂g∂z (1 ,
0) = 1 z (0) z (0) z (0) (1 − z (0))(1 − z (0))(1 − z (0)) . (3.20)Hence2 h (1 , ∂g∂ν (1 ,
0) = 2 r − z (0) z (0) z (0) (1 − z (0))(1 − z (0))(1 − z (0)) z (0) z (0) z (0) (1 − z (0))(1 − z (0))(1 − z (0)) = − r = − T r ( ˆ O ) = − , since ˆ O is a density operator. Next we will show that z ′ (0) = 0. Since g ( z ( ν ) , ν ) = 0,we have 0 = dg ( z ( ν ) , z ) dν = ∂g ( z, ν ) ∂ν | z = z ( ν ) + ∂g ( z, ν ) ∂z | z = z ( ν ) z ′ ( ν ) . ν = 0, z (0) = 1, then the above equation becomes0 = ∂g (1 , ν ) ∂ν | ν =0 + ∂g ( z, ∂z | z =1 z ′ (0) . (3.21)Consider the matrix A at z = 1 , and note that l ( ν ) = l ( ν ) = 0, we have g (1 , ν ) = (1 − l ( ν )) A ( ν ) + l ( ν ) A ( ν ) − l ( ν ) A ( ν ) + l ( ν ) A ( ν )= (1 − cos( ν )) A ( ν ) + i sin νA ( ν ) . By Lemma 3.2, the cofactor A (0) = 0. It follows that ∂g (1 , ν ) ∂ν | ν =0 = 0 . (3.22)By (3.20), ∂g ( z ,ν ) ∂z | z = z = 0 , since z i (0) = 1, for all i = 1 , ,
3, by the assumptionsof the theorem. By (3.21) and (3.22) we have z ′ (0) = 0. Q.E.D
The assumptions that 1 is the largest eigenvalue of L kk with algebraic multiplicity 1and that there is no other eigenvalues whose modulus equals to 1, are crucial in de-termining the convergence of ˆ P ( ν √ t , t ). In this section, we consider the measurementsgiven by A = p − pI,A = √ p | R >< R | ,A = √ p | L >< L | . By analyzing the spectrum of L kk , we obtain a necessary and sufficient conditions inwhich the assumptions of Theorem 3.1 are satisfied. Specific examples such as theHadamard walk, walks under real and complex rotations are illustrated. For certainclass of convergent quantum walks, explicit formulas are obtained for the limits ofthe characteristic functions of properly scaled p ( x, t ). In addition, we will also give acomplete description of the behavior of those walks that do not satisfy the conditions.12 emma 4.1. Let L k,k ′ be a superoperator on the Hilbert space L ( C ) , defined by L k,k ′ ( ˆ O ) = X n =0 U k A n ˆ OA ∗ n U ∗ k ′ , where U k and U k ′ are × unitary matrices and ˆ O ∈ L ( C ) . Then < L k,k ′ ˆ O, L k,k ′ ˆ O > ≤ < ˆ O, ˆ O > .
In particular, < L k,k ′ ˆ O, L k,k ′ ˆ O > = < ˆ O, ˆ O > if and only if the decoherence rate p = 0 or ˆ o = ˆ o = 0 . Part of the above lemma has been obtained by Liu and Petulante in [9], butour lemma extends the equality part of their lemma to a wider scope with moreapplications.
Proof : The inequality follows from Lemma 1 in [9]. By the proof of Lemma 1in [9], the equality holds if and only if (2 p − p )(ˆ o + ˆ o ) = 0. That is, p = 0 orˆ o = ˆ o = 0.If p = 0 and ˆ o = ˆ o = 0, then 1 may not be the unique eigenvalue of L k,k withlargest modulus. In this case, we have the following theorem. Let dim ( λ ) denote thedimensions of the eigenspace associated with the eigenvalue λ . Theorem 4.1.
Let < p < . Let U ∈ U (2) . Suppose λ is an eigenvalue of L kk ,then we have(a) P | λ | =1 dim ( λ ) ≤ . (b) 1 is an eigenvalue of L k,k and dim (1) ≥ . (c) If | λ | = 1 , then λ = 1 or − .(d) dim (1) = 2 if and only if u = u = 0 and | u | = | u | = 1 . In this case itsmultiplicity is 2.(e) There exists eigenvalue λ = − if and only if u = u = 0 and | u | = | u | = 1 .In this case its multiplicity is 1. Proof : By considering the normalized operator W as in (3.11), and noting thatthe statements (a)-(e) do not depend on whether it is U or W , we mat assume withoutloss of generality that U is in SU with the form U = (cid:18) α − ββ α (cid:19) , α, β ∈ C and | α | + | β | = 1.a) Let λ be an eigenvalue of L kk with eigenvector ˆ O = (cid:18) ˆ o ˆ o ˆ o ˆ o (cid:19) . By Lemma3.2, L k,k has the form l l l l l l l l l , (4.23)where l = − k ) Re ( βα ) + 2 sin(2 k ) Im ( βα ) , (4.24) l = − k ) Im ( βα ) − k ) Re ( βα ) , (4.25) l = 2 qRe ( αβ ) , (4.26) l = 2 iqIm ( αβ ) , (4.27) l = | α | − | β | . (4.28)Since | λ | = 1, by Lemma 4.1, we have ˆ o = ˆ o = 0. This implies that the dimensionof the space spanned by the eigenspace for all eigenvalues with modulus 1 is at most2. Moreover, the intersections of eigenspace corresponding to different eigenvalues is { } . Therefore a) holds.b) By (4.23), 1 is an eigenvalue of L k,k . Furthermore, (1 , , , T is one of itseigenvectors. Therefore dim (1) ≥ O = 12 (ˆ o + ˆ o ) σ + i o − ˆ o ) σ + 12 (ˆ o + ˆ o ) σ + 12 (ˆ o − ˆ o ) σ , (4.29)so if L k,k ˆ O = λ ˆ O , with | λ | = 1, then by Lemma 4.1,12 (ˆ o + ˆ o ) = λ o + ˆ o ) , (4.30)(2 cos(2 k ) Re ( βα ) − k ) Im ( βα )) 12 (ˆ o − ˆ o ) = 0 , (4.31)(2 cos(2 k ) Im ( βα ) + 2 sin(2 k ) Re ( βα )) 12 (ˆ o − ˆ o ) = 0 , (4.32)( | α | − | β | ) 12 (ˆ o − ˆ o ) = λ o − ˆ o ) . (4.33)144.31) and (4.32) can be written as the following matrix form (cid:18) cos 2 k − sin 2 k sin 2 k cos 2 k (cid:19) ( Reβα, Imβα ) T (ˆ o − ˆ o ) = 0 . (4.34)Note that the matrix on the left hand side of (4.34) has determinant 1, for all k ,hence invertible. Therefore, if ˆ o − ˆ o = 0 , then | βα | = 0, or equivalently, αβ = 0.Consequently, if β = 0, then λ = 1. If α = 0, then λ = −
1. On the other hand, ifˆ o − ˆ o = 0 , then ˆ o + ˆ o = 0 (otherwise ˆ O = 0 ). In this case, λ = 1 by (4.30).So λ = 1 or − dim (1) = 2. This implies that there exists an eigenvectorof the form ( a, , , b ) T , with b = 0. Since (1 , , , T is an eigenvector, (0 , , , T is also an eigenvector. By (4.29), (1 , , , T and (0 , , , T are two eigenvectorscorresponding to ˆ o − ˆ o = 0 and ˆ o + ˆ o = 0, respectively. Hence | α | − | β | = 1by (4.33). Therefore β = 0 and | α | = 1.Conversely, if β = 0 and | α | = 1, then L k,k has the form l l l l l l . (4.35)It follows from (4.26) and (4.27) that l = l = 0. Therefore, dim (1) ≥
2. So dim (1) = 2 by part a). Therefore, by (4.35) again, the multiplicity of 1 is also two,otherwise dim (1) > − a, , , T is the associated eigenvector, then a = 0 by (4.30). Therefore | α | −| β | = − α = 0 and | β | = 1.Conversely, if α = 0 and | β | = 1, then L k,k has the form l l l l
00 0 0 − . (4.36)So dim(-1)=1 by part a) and b). Therefore, by (4.36) again, the multiplicity of − dim ( − ≥ Corollary 4.1. If U ∈ O (2) , i.e. U = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) or (cid:18) cos θ sin θ sin θ − cos θ (cid:19) forsome θ ∈ [0 , π ] , then ) is an eigenvalue of L k,k with multiplicity one, and | λ | < for any othereigenvalue holds if and only if θ = nπ where n = 0 , , , .b) If θ = 0 , π , 1 is an eigenvalue of L k,k with multiplicity 2.c) If θ = π , π , L k,k has eigenvalues 1 and -1, each has multiplicity 1. Proof : Note that θ = 0 , π if and only if u = u = 0 and | u | = | u | = 1.,and θ = π , π if and only if u = u = 0 and | u | = | u | = 1. Therefore corollaryfollows.We are now ready to discuss examples according to different values of θ .In the Hadamard walk, the evolution operator is given by U k = 1 √ (cid:18) e − ik e − ik e ik − e − ik (cid:19) , (4.37)where U = (cid:18) cos π sin π sin π − cos π (cid:19) . By Theorem 4.1, the associated superoperator L k,k + ν satisfies the assumptions in Theorem 3.1, and we have z ′ (0) = 0; z ′′ (0) = 1 + q + 2 q cos 2 k − q , where q = 1 − p . By Theorem 3.1, we havelim t →∞ ˆ P ( ν √ t , t ) = 12 π Z π e −
12 1+ q q cos 2 k − q ν dk (4.38)= 12 π Z π e −
12 1+ q q cos k − q ν dk, (4.39)by change of variables and periodicity of cosine.In general, if U ∈ O (2) with det U = − θ = nπ where n = 0 , , ,
3. Then U k = (cid:18) e − ik cos( θ ) e − ik sin( θ ) e ik sin( θ ) − e ik cos( θ ) (cid:19) , (4.40)and L k,k + ν = cos( ν ) (1 − p ) i sin( ν ) sin(2 θ ) 0 i sin( ν ) cos(2 θ )0 − (1 − p ) cos(2 k + ν ) cos(2 θ ) (1 − p ) sin(2 k + ν ) cos(2 k + ν ) sin(2 θ )0 − (1 − p ) sin(2 k + ν ) cos(2 θ ) − (1 − p ) cos(2 k + ν ) sin(2 k + ν ) sin(2 θ ) i sin( ν ) (1 − p ) cos( ν ) sin(2 θ ) 0 cos( ν ) cos(2 θ ) z ′ (0) = 0; z ′′ (0) = 1 + 2 q cos(2 k ) + q − q cot ( θ ) . where q = 1 − p . Hence by Theorem 3.1,lim t →∞ ˆ P ( ν √ t , t ) = 12 π Z π e −
12 1+2 q cos(2 k )+ q − q cot ( θ ) ν dk (4.41)= 12 π Z π e −
12 1+2 q cos( k )+ q − q cot ( θ ) ν dk, (4.42)by change of variables and periodicity of cosine.Similar calculations also show that (4.42) holds for U ∈ O (2) with det U = 1 and θ = nπ , n = 0 , , , M n of the limiting distribution can be calculated from (4.42)by using moment generation functions. Let ϕ ( ν ) = 12 π Z π e −
12 1+2 q cos( k )+ q − q cot ( θ ) ν dk. Then ∞ X n =0 n ! i n M n ν n = ϕ ( ν ) (4.43)= ∞ X n =0 n ! ( −
12 ) n cot n ( θ ) ν n − q ) n π Z π (1 + 2 q cos( k ) + q ) n dk (4.44)It follows that for U ∈ O (2) with θ = nπ , n = 0 , , ,
3, we have M n +1 = 0 , n = 0 , , , , ..., (4.45)and for even moments, M n = (2 n )! n ! cot n ( θ ) 12 n (1 − q ) n π Z π (1 + 2 q cos(2 k ) + q ) n dk (4.46)= (2 n )! n ! cot n ( θ ) 12 n (1 − q ) n π Z π ( e i k + q ) n ( e − i k + q ) n dk (4.47)= (2 n )! n ! cot n ( θ ) 12 n (1 − q ) n π Z π n X l =0 n X l ′ =0 (cid:18) nl (cid:19) e il k q n − l (cid:18) nl ′ (cid:19) e − il ′ k q n − l ′ dk (4.48)= (2 n )! n ! cot n ( θ ) 12 n (1 − q ) n n X l =0 (cid:18) nl (cid:19) q n − l ) , n = 0 , , , ... (4.49)17herefore we have M n = (2 n )! n !2 n ( cot θ − q ) n T n ( q ) , n = 0 , , , ... (4.50)where T n ( q ) is a polynomial of q of order 2 n given by T n ( q ) = n X l =0 (cid:18) nl (cid:19) q l . (4.51)In particular, for Hadamard walk, the second moment of the limiting distributionis given in terms of T ( q ) = 1 + q . (4.52)This result agrees with the results given in [4].Comparing to the well known 2 n -th moment N n for the normal distribution withmean 0 and variance σ = cot θ − q , N n = (2 n )! n !2 n ( cot θ − q ) n , n = 0 , , , ..., (4.53)we see that the scaling limits of the decoherent quantum random walks are notnormally distributed if q = 0. The deviation from the normal distribution gets largeras the even moments gets larger. However the deviations of the 2 n -th moment areby the same factor T n ( q ) for all θ = jπ , where j = 0 , , , M n at p = 0: γ n ≡ lim p → − ln M n ln p = n. (4.54)This result shows universality in which the critical exponents do not depend on θ as long as it converges. In other words, the coin-space decoherent quantum randomwalks, with coin space unitary transformation U ∈ O (2), θ = nπ , n = 0 , , ,
3, belongto the same universality class with respected to the critical exponents of all momentsas p → θ = 0 , π . If the initial state is | > ⊗| R > , then E ( | > ⊗| R > ) = | > ⊗| R > .If the initial state is | > ⊗| L > , then E ( | > ⊗| R > ) = −| − > ⊗| L > . That is,the walk goes either left or right forever. Henceˆ P ( ν, t ) = e ± itν φ = c R | R > + c L | L > , with | c R | + | c L | = 1 , thenˆ P ( νt , t ) = | c R | e iν + | c L | e − iν . b) θ = π , π . If the initial state is | > ⊗| R > , then E ( | > ⊗| R > ) = −| > ⊗| L > . If the initial state is | > ⊗| L > , then E ( | > ⊗| R > ) = −| − > ⊗| R > .That is, the walk switches back and forth between two positions, which is trivial andlim t →∞ ˆ P ( νt κ , t ) = 1 , for any κ > In this paper we consider coin space decoherent quantum random walks with coinspace unitary transformation U . We prove that under the eigenvalue conditions, thescaling limit of the probability distribution converges in distribution to a continuousconvex combination of normal distributions. An necessary and sufficient conditionis obtained for U to satisfy the eigenvalue conditions. For U in O (2), an exact formof the limiting distribution is given and the moments of all orders are obtained. Forthis case, the critical exponents are obtained and we show that all U with rotationangles θ = nπ , n = 0 , , ,
3, belong to the same universality class with respectedto the critical exponents of all moments as p →
0. Our analysis is based on thecharacteristic functions of the position distribution and the analysis of eigenvalues,Theorem 4.1 and its corollary, which plays an important role in the applications of ourmain convergence theorem, Theorem 4.1. We believe that a wider class of universalityshould hold for general quantum random walks in general d-dimensional lattices withgeneral rotation U ∈ U ( n ). For the future research, it would be very interesting toexplore and classify their universality classes with respect to their critical points. Onthe other hand, we have fixed a measurement in our applications, while our generalconvergence theorem does not depend on the special form as that in our applications.An interesting problem would be to understand how the general measurements affectthe limiting distributions and, especially their universality classes. References [1] Y. Aharonov, L. Davidovich and N. Zagury, Phys. Rev. A 48, 1687, (1993).192] A. Ahlbrecht, H. Vogts, A. H. Werner, and R. F. Werner, Asymptotic evolutionof quantum walks with random coin, J. Math. Phys. 52, 042201 (2011).[3] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath and J. Watrous, Proceeding ofthe 33rd ACM Symposium on Theory of Computing (ACM Press, New York,2001), p.60.[4] T. A. Brun, H. A. Carteret, and A. Ambainis, Phys. Rev. A 67, 032304 (2003).[5] W. Bruzda, V. Cappellini, H. J. Sommers and K. Zyczkowski, Phys. Lett. A 375,320-324 (2006).[6] K. L. Chung,