There is an abundance of research on discrete-time quantum walks since 1993 [1] [2] [3] . Then quantum walks as a quantum mechanical attract large number of scholars [4] [5] [6] . For instance, the two-phase quantum walks are related to the research of topological insulator [7] [8] , and one-defect quantum walks are applied to quantum search algorithms [9] [10] .

Recently, the asymptotic behaviors of the quantum walks have received much attention [11] [12] . Konno gave the uniform measure as a stationary measure of the one-dimensional discrete-time quantum walks [13] . Endo et al., solve the eigenvalue problem and present a stationary measure by using SGF method [14] . Then Wang et al., obtain the stationary measures of three-state Wojcik walk by adopting SGF method [15] . Shortly afterwards, Kawai et al. raised Reduced matrix method [16] . Lately, Endo et al. got the transfer matrices and solve the eigenvalue [17] . In this paper, we will use this method to further derive one-phase and two-phase model of space-inhomogeneous three-state quantum walks.

In this section, we give the definition of three-state quantum walk on ℤ , where ℤ is the set of integers. The discrete-time quantum walk on ℤ defined by a unitary matrix;

U x = [ a x b x c x d x e x f x g x h x i x ] (2.1)

We let ℕ be the set of nonnegative integers, and Ψ n ( x ) = ( Ψ n L ( x ) , Ψ n O ( x ) , Ψ n R ( x ) ) T be the amplitude of the wave function corresponding to the chiralities “L”, “O”, and “R” at position x ∈ ℤ and time n ∈ ℕ . Obviously, for each position x ∈ ℤ , the matrix U x can be divided into three parts.

U x = U x L + U x O + U x R (2.2)

Through these matrix, we can define time evolution of a quantum walk in the following way:

Ψ n + 1 ( x ) ≡ U x + 1 L Ψ n ( x + 1 ) + U x O Ψ n ( x ) + U x − 1 R Ψ n ( x − 1 ) (2.3)

Then let

Ψ n = ( ⋯ , Ψ n ( − 1 ) , Ψ n ( 0 ) , Ψ n ( 1 ) ) T , U ( s ) = [ ⋱ ⋮ ⋮ ⋮ ⋮ ⋮ ⋯ ⋯ U − 2 O U − 1 L O O O ⋯ ⋯ U − 2 R U − 1 O U 0 L O O ⋯ ⋯ O U − 1 R U 0 O U 1 L O ⋯ ⋯ O O U 0 R U 1 R U 2 L ⋯ ⋯ O O O U 1 R U 2 O ⋯ ⋯ ⋮ ⋮ ⋮ ⋮ ⋮ ⋱ ] , O = [ 0 0 0 0 0 0 0 0 0 ] (2.4)

Then the sate at time n can be expressed as

Ψ n = ( U ( s ) ) n Ψ 0 , n ≥ 0 (2.5)

where Ψ 0 is the initial state.

Definition 2.1. The one-phase model of space-inhomogeneous three-state quantum walk is defined on the set ℤ of integers. which is characterized by a chirality-state space { | L 〉 , | O 〉 , | R 〉 } and a position space { | x 〉 : x ∈ ℤ } , and the chiralities “L”, “R” and “O” express the left, right and neutral state for the motion of the walker. Its time evolution is determined by the following 3 × 3 unitary matrices

U x = e i θ x 3 [ − 1 2 2 2 − 1 2 2 2 − 1 ] , x ∈ ℤ (2.6)

where

θ x = { 0 , x = ± 1 , ± 2 , ⋯ , 2 π τ , x = 0 ,

with τ ∈ ( 0,1 ) , which θ x shows the phase 2 π τ of the walk.

Then

U x L = e i θ x 3 [ − 1 2 2 0 0 0 0 0 0 ] , U x O = e i θ x 3 [ 0 0 0 2 − 1 2 0 0 0 ] , U x R = e i θ x 3 [ 0 0 0 0 0 0 2 2 − 1 ] .

Definition 2.2. The two-phase model of space-inhomogeneous three-state quantum walk is defined on the set ℤ of integers, which is characterized by a chirality-state space { | L 〉 , | O 〉 , | R 〉 } and a position space { | x 〉 : x ∈ ℤ } . Its time evolution is determined by the following unitary matrices

U x = { U + , x ≥ 1 , U 0 , x = 0 , U − , x ≤ − 1 , (2.7)

where

U ± = [ − 1 + g ± 2 ℏ ± 2 1 − g ± 2 ℏ ± 2 g ± ℏ ± 2 1 − g ± 2 ℏ ± 2 − 1 + g ± 2 ] , U 0 = ξ 3 [ − 1 2 2 2 − 1 2 2 2 − 1 ] . (2.8)

where g ± = cos γ ± , ℏ ± = sin γ ± , γ ± ∈ [ 0 , 2 π ) , ξ = e 2 π i τ , τ ∈ ( 0 , 1 ) . Then

U x L = { U x + L , x ≥ 1 , U x L , x = 0 , U x − L , x ≤ − 1 ,     U x O = { U x + O , x ≥ 1 , U 0 O , x = 0 , U x − O , x ≤ − 1 ,     U x R = { U x + R , x ≥ 1 , U 0 R , x = 0 , U x − R , x ≤ − 1 , (2.9)

U x + L = [ − 1 + g + 2 ℏ + 2 1 − g + 2 0 0 0 0 0 0 ] , U x + O = [ 0 0 0 ℏ + 2 g + ℏ + 2 0 0 0 ] , U x + R = [ 0 0 0 0 0 0 1 − g + 2 ℏ + 2 − 1 + g + 2 ] . (2.10)

U x − L = [ − 1 + g − 2 ℏ − 2 1 − g − 2 0 0 0 0 0 0 ] , U x − O = [ 0 0 0 ℏ − 2 g − ℏ − 2 0 0 0 ] , U x − R = [ 0 0 0 0 0 0 1 − g − 2 ℏ − 2 − 1 + g − 2 ] . (2.11)

U 0 = ξ 3 [ − 1 2 2 0 0 0 0 0 0 ] , U 0 = ξ 3 [ 0 0 0 2 − 1 2 0 0 0 ] , U 0 = ξ 3 [ 0 0 0 0 0 0 2 2 − 1 ] . (2.12)

In the present section, we first recall some fundamental notions and facts about stationary measure. Firstly, we introduced a mapping ϕ : ( ℂ 3 ) ℤ → ℝ + ℤ

ϕ ( x ) = ( ⋯ , ‖ Ψ ( − 1 ) ‖ 2 , ‖ Ψ ( 0 ) ‖ 2 , ‖ Ψ ( 1 ) ‖ 2 , ⋯ ) T ∈ ℝ + ℤ ,

where ℂ is the set of complex number and ‖   ⋅   ‖ is the norm in ℂ 3 and Ψ = ( ⋯ , Ψ ( − 1 ) , Ψ ( 0 ) , Ψ ( 1 ) , ⋯ ) T ∈ ( ℂ 3 ) ℤ . For every x ∈ ℤ , we note that

ϕ ( Ψ ) ( x ) = ‖ Ψ ( x ) ‖ 2 , Ψ ∈ ( ℂ 3 ) ℤ . (3.1)

Then, the function x → ϕ ( Ψ ) ( x ) gives a measure μ on ℤ by μ ( ⋅ ) = ϕ ( Ψ ) ( ⋅ ) for Ψ .

Definition 3.1. Let

M s = { ϕ ( Ψ 0 ) ∈ ℝ + ℤ \ { 0 } : there   exists   Ψ 0   such   that                         ϕ ( ( U ( s ) ) n Ψ 0 ) = ϕ ( Ψ 0 )   for   any   n ≥ 0 } (3.2)

where 0 is the zero vector. We call the element of M s the stationary measure of quantum walk. If μ ∈ M s , then μ n = μ , where μ n ( x ) = ϕ ( Ψ n ( x ) ) is the measure of the quantum walk at position x ∈ ℤ and at time n ∈ ℕ .

Next we consider the eigenvalue problem:

U ( s ) Ψ = λ Ψ , ( λ ∈ ℂ , | λ | = 1 ) (3.3)

In this section, we obtain the stationary measure of the three-state quantum walk with one defect by following lemma.

Lemma 4.1. [17] Let { U y } y ∈ ℤ be the set of y-parameterized unitary matrices of the three-state inhomogeneous quantum walk, and Ψ ( x ) = [ Ψ L ( x ) , Ψ O ( x ) , Ψ R ( x ) ] T be the probability amplitude. Note that there is a restriction for the initial state Ψ ( 0 ) [18] Then the solutions for U ( s ) Ψ = λ Ψ ( Ψ ∈ M a p ( ℤ , ℂ 3 ) , λ ∈ S 1 ) , where S 1 = { z ∈ ℂ : | z | = 1 } , are

Ψ x = { ∏ y = 1 x T y ( + ) Ψ ( 0 ) , x ≥ 1 , Ψ ( 0 ) , x = 0 , ∏ y = − 1 x T y ( − ) Ψ ( 0 ) , x ≤ − 1 , (4.1)

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 − ] , (4.2)

with

t 11 + = ( λ − e y ) ( λ 2 − g y − 1 c y ) − g y − 1 b y f y λ [ a y ( λ − e y ) + b y d y ] , t 12 + = − h y − 1 [ b y f y + c y ( λ − e y ) ] λ [ a y ( λ − e y ) + b y d y ]

t 13 + = − i y − 1 [ b y f y + c y ( λ − e y ) ] λ [ a y ( λ − e y ) + b y d y ] , t 21 + = λ 2 d y + g y − 1 ( a y f y − c y d y ) λ [ a y ( λ − e y ) + b y d y ]

t 22 + = h y − 1 ( a y f y − c y d y ) λ [ a y ( λ − e y ) + b y d y ] , t 23 + = i y − 1 ( a y f y − c y d y ) λ [ a y ( λ − e y ) + b y d y ]

t 31 + = g y − 1 λ , t 32 + = h y − 1 λ , t 33 + = i y − 1 λ ,

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 2 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 2 − g y c y + 1 ) − h y c y + 1 d y λ [ h y f y + i y ( λ − e y ) ] .

We now state the stationary measure of one-phase model with one defect.

Theorem 4.1. Let Ψ n ( x ) = ( Ψ n ( x ) L , Ψ n ( x ) O , Ψ n ( x ) R ) be the wave function of probability amplitude, and Ψ 0 L = α , Ψ 0 O = β , Ψ 0 R = γ be the initial state. We take α = − γ , β = 0 and λ = − 1 . Then through the definition (2.1)

U x = e i θ x 3 [ − 1 2 2 2 − 1 2 2 2 − 1 ] , x ∈ ℤ

where

θ x = { 0 , x = ± 1 , ± 2 , ⋯ , 2 π τ , x = 0.

We obtain the stationary measure

μ ( x ) = { ( 4 − 2 ℜ ξ ) | α | 2 , x = ± 1 , 2 | α | 2 , others . (4.3)

Proof. Put α = Ψ L ( 0 ) , β = Ψ O ( 0 ) and γ = Ψ R ( 0 ) . Now we take α = − γ , β = 0 and λ = − 1 , then the solutions for U ( s ) Ψ = λ Ψ are

Ψ x = { ∏ y = 1 x T y ( + ) Ψ ( 0 ) , x ≥ 1 , Ψ ( 0 ) , x = 0 , ∏ y = − 1 x T y ( − ) Ψ ( 0 ) , x ≤ − 1 ,

where T y ± are

T 1 ( + ) = [ 1 0 0 2 ξ 3 − 1 2 ξ 3 − ξ 3 − 2 ξ 3 − 2 ξ 3 ξ 3 ] , T 1 ( − ) = [ ξ 3 − 2 ξ 3 − 2 ξ 3 − ξ 3 2 ξ 3 2 ξ 3 − 1 0 0 1 ] .

T y ( + ) = [ 1 0 0 − 1 3 2 3 − 1 3 − 2 3 − 2 3 1 3 ] , T y ( − ) = [ 1 3 − 2 3 − 2 3 − 1 3 2 3 − 1 3 0 0 1 ] ( | y | ≥ 2 ) .

Then through the formula (4.1), we can obtain

Ψ ( x ) = { [ α 0 − α ] , | x | ≠ 1 , [ α ( ξ − 1 ) α − ξ α ] , x = 1 , [ ξ α ( 1 − ξ ) α − α ] , x = − 1. (4.4)

Therefore the corresponding stationary measure is given by

μ ( x ) = { ( 4 − 2 ℜ ξ ) | α | 2 , x = ± 1 , 2 | α | 2 , others .

¨

Next we state the stationary measure of two-phase model by transfer matrices method.

Theorem 4.2. Let Ψ n ( x ) = ( Ψ n ( x ) L , Ψ n ( x ) O , Ψ n ( x ) R ) be the wave function of probability amplitude, and Ψ 0 L = α , Ψ 0 O = β , Ψ 0 R = γ be the initial state. We take α = − γ , β = 0 and λ = − 1 . Then through the definition (2.2)

U x = { U + , x ≥ 1 , U 0 , x = 0 , U − , x ≤ − 1 ,

where

U ± = [ − 1 + g ± 2 ℏ ± 2 1 − g ± 2 ℏ ± 2 g ± ℏ ± 2 1 − g ± 2 ℏ ± 2 − 1 + g ± 2 ] , U 0 = ξ 3 [ − 1 2 2 2 − 1 2 2 2 − 1 ] ,

where g ± = cos γ ± , ℏ ± = sin γ ± , γ ± ∈ [ 0 , 2 π ) , ξ = e 2 π i τ , τ ∈ ( 0 , 1 ) . Then the stationary measure is

μ ( x ) = ( 2 + Δ ) | α | 2 (4.5)

where

Δ = { ℏ − 2 2 , x < 1 , ℏ − 2 ( 1 − ℜ ξ ) ( 1 + g − ) 2 , x = − 1 , 0 , x = 0 , ℏ + 2 ( 1 − ℜ ξ ) ( 1 + g + ) 2 , x = 1 , ℏ + 2 2 , x < 1 (4.6)

Proof. Put α = Ψ L ( 0 ) , β = Ψ O ( 0 ) and γ = Ψ R ( 0 ) . Now we take α = − γ , β = 0 and λ = − 1 , then the solutions for U ( s ) Ψ = λ Ψ are

Ψ x = { ∏ y = 1 x T y ( + ) Ψ ( 0 ) , x ≥ 1 , Ψ ( 0 ) , x = 0 , ∏ y = − 1 x T y ( − ) Ψ ( 0 ) , x ≤ − 1 ,

where T y ± are

T 1 ( + ) = [ 1 0 0 − ℏ + ( 3 − 2 ξ ) 3 2 ( 1 + g + ) 2 ℏ + ξ 3 2 ( 1 + g + ) − ℏ + ξ 3 2 ( 1 + g + ) − 2 ξ 3 − 2 ξ 3 ξ 3 ] ,

T 1 ( − ) = [ ξ 3 − 2 ξ 3 − 2 ξ 3 − ℏ − ξ 3 2 ( 1 + g − ) 2 ℏ − ξ 3 2 ( 1 + g − ) − ℏ − ( 3 − 2 ξ ) 3 2 ( 1 + g − ) 0 0 1 ] .

T y ( + ) = [ 1 0 0 ℏ + 2 2 1 − g + 2 − ℏ + 2 2 − 1 − g + 2 − ℏ + 2 1 + g + 2 ] , T y ( − ) = [ 1 + g − 2 − ℏ − 2 − 1 − g − 2 ℏ − 2 2 1 − g − 2 − ℏ − 2 2 0 0 1 ] ( | y | ≥ 2 ) .

Then through the formula (4.1), we can obtain

Ψ ( x ) = { [ α ℏ − 2 α − α ] , x < − 1 , [ ξ α ℏ − ( 1 − ξ ) 2 ( 1 + g − ) α − α ] , x = − 1 , [ α 0 − α ] , x = 0 , [ α ℏ + 2 ( 1 + g + ) α − ξ α ] , x = 1 , [ α ℏ + 2 α − α ] , x > 1.

Therefore we obtain the stationary measure

μ ( x ) = ( 2 + Δ ) | α | 2

where

Δ = { ℏ − 2 2 , x < 1 , ℏ − 2 ( 1 − ℜ ξ ) ( 1 + g − ) 2 , x = − 1 , 0 , x = 0 , ℏ + 2 ( 1 − ℜ ξ ) ( 1 + g + ) 2 , x = 1 , ℏ + 2 2 , x < 1

¨

In this paper, we derive the stationary measure of three-state walks with one dimension via transfer matrices. As a future work, we would investigate spectral theory and localization of three-state quantum walks.

The authors declare no conflicts of interest regarding the publication of this paper.

Gao, J.L. and Zhang, M.J. (2023) Stationary Measures of Three-State Quantum Walks with Defect on the One-Dimension Lattice. Open Journal of Applied Sciences, 13, 473-482. https://doi.org/10.4236/ojapps.2023.134038