## The Analysis of Evolution Process in a Time-Inhomogeneous Two-State Quantum Walk

Lin Yunguo,

 基金资助: 福建省自然科学基金.  2016J01283福建农林大学科技创新专项基金.  CXZX2020108A

 Fund supported: the Science Foundation of the Fujian Province.  2016J01283the Special Fund Project for Technology Innovation of Fujian Agriculture and Forestry University.  CXZX2020108A

Abstract

In this paper, we establish a mathematical model for a time-inhomogeneous two-state quantum walk and give a calculation of the position probability distribution. By calculating spectral values and spectral vectors, we analyze the evolution of quantum states. Furthermore, we derive an Itô formula and get a matrix decomposition and its interpretation.

Keywords： Quantum walk ; Inhomogeneous ; Probability distribution ; Spectral value ; Itô lemma

Lin Yunguo. The Analysis of Evolution Process in a Time-Inhomogeneous Two-State Quantum Walk. Acta Mathematica Scientia[J], 2021, 41(4): 1097-1110 doi:

## 2 模型及相关概念

$$$|\psi_{t}\rangle = \boldsymbol{ {U} }_t\boldsymbol{ {U} }_{t-1}\cdots \boldsymbol{ {U} }_1|\psi_{0}\rangle = \prod\limits^t_{k = 1} \boldsymbol{ {U} }_k|\psi_{0}\rangle = \sum\limits_{x\in {\Bbb Z}}|\psi_{t}(x)\rangle.$$$

$$$\widehat{\psi}_{t}(\xi) = \left( \begin{array}{c} \widehat{\psi}^L_t(\xi) \\ \widehat{\psi}^R_t(\xi) \\ \end{array} \right) = \left( \begin{array}{cc} \cos\theta_{t}{\rm e}^{-{\rm i}\xi} & \sin\theta_{t}{\rm e}^{-{\rm i}\xi} \\ \sin\theta_{t}{\rm e}^{{\rm i}\xi} & -\cos\theta_{t}{\rm e}^{{\rm i}\xi} \\ \end{array} \right)\left( \begin{array}{c} \widehat{\psi}^L_{t-1}(\xi) \\ \widehat{\psi}^R_{t-1}(\xi)\\ \end{array} \right) = \boldsymbol{ {C} }_{t}(\xi)\widehat{\psi}_{t-1}(\xi),$$$

$$$\widehat{\psi}_{t}(\xi) = \boldsymbol{ {C} }_t(\xi)\boldsymbol{ {C} }_{t-1}(\xi) \cdots\boldsymbol{ {C} }_1(\xi)\widehat{\psi}_{0}(\xi) = \prod\limits^t_{k = 1}\boldsymbol{ {C} }_k(\xi)\widehat{\psi}_{0}(\xi).$$$

## 3 位置概率分布

$$$i = u(N)2^{N-1}+u(N-1)2^{N-2}+\cdots+u(2)2^{1}+u(1)2^{0}\in\{0, 1, 2, \cdots, 2^N-1\}.$$$

$$$\boldsymbol{ {C} }_0 = \boldsymbol{ {E} }, \boldsymbol{ {C} }_1 = \boldsymbol{ {P} }_1+\boldsymbol{ {Q} }_1, \cdots, \prod\limits_{t = 1}^N\boldsymbol{ {C} }_t = \sum\limits_{i = 0}^{2^N-1}\boldsymbol{ {P} }_{N}^{v_i} = \sum\limits_{i = 0}^{2^N-1}\prod\limits_{t = 1}^N\boldsymbol{ {P} }_{N}^{v_i(t)}.$$$

$$$P(X = x) = \||\psi_{N}(x)\rangle\|^2 = |\psi^L_{N}(x)|^2+|\psi^R_{N}(x)|^2, x\in{\Bbb Z},$$$

$$$P(X = x) = \int^{\pi}_{-\pi} \frac{{\rm d}\zeta}{2\pi}{\rm e}^{{\rm i}\zeta x}(\prod\limits^N_{t = 1} \boldsymbol{ {C} }_t(\zeta)\widehat{\psi}_{0}(\zeta))^{*}\int^{\pi}_{-\pi} \frac{{\rm d}\xi}{2\pi} {\rm e}^{-{\rm i}\xi x}\prod\limits^N_{t = 1}\boldsymbol{ {C} }_t(\xi)\widehat{\psi}_{0}(\xi).$$$

(3.3)式是将$N$个酉算子$\boldsymbol{ U }_t$依次作用在量子初态上, 得到量子终态后进行量子测量计算出系统处于每个位置的概率. (3.4)式是利用傅里叶变换将概率分布的计算转化为积分计算问题. 在第5节中, 将提出伊藤公式对$\prod\limits^N_{t = 1}\boldsymbol{ C }_t({\xi})$进行分解计算. 系统到达某个位置是由一系列具有共同特征的硬币演化序列叠加作用在量子初态上得到的, 它包括具有相同的向左、向右演化算子个数. 基于此, 下面对硬币演化序列进行归纳计算, 由此给出位置概率分布的计算公式.

$\nabla^2\omega = (\nabla\omega(1)-\nabla\omega(0), \nabla\omega(2) -\nabla\omega(1), \cdots , \nabla\omega(n)-\nabla\omega(n-1))^T, \nabla\omega(0) = 0.$

$$$\boldsymbol{ {P} }_{N}^{v} = (-1)^{\lfloor\frac{s}{2}\rfloor}(\cot\theta_1)^{Mod(s, 2)}\prod\limits_{t = 1}^{N}\cos\theta_t (\tan\theta_t)^{\delta(\nabla^2\omega(t))}\frac{\boldsymbol{ {P} }_{N}^{v(N)}\boldsymbol{ {P} }_{1}^{v(1)}}{tr(\boldsymbol{ {P} }_{N}^{v(N)}\boldsymbol{ {P} }_{1}^{v(1)})},$$$

路径$\boldsymbol{ P }_{N}^{v}$可分为4类. 下面将对每一类路径进行归纳计算, 其中记$l$为每条路径中所包含的向右演化算子个数.

(1) $\boldsymbol{ P }_{N}^{v} = {\cal P}_m{\cal Q}_{m}{\cal P}_{m-1}{\cal Q}_{n-1}\cdots {\cal P}_1{\cal Q}_{1}\in{\cal P}_N$.

$\begin{eqnarray} \boldsymbol{ {P} }_{N}^{v}& = &(-1)^{l-m}\prod\limits_{t = 1}^{N}\cos\theta_t \tan^{\delta(\nabla^2\omega(t))}\theta_t\left( \begin{array}{cc} 1 & -\cot\theta_1\\ 0 & 0 \\ \end{array} \right){}\\ & = &(-1)^{l-m}\prod\limits_{t = 1}^{N}\cos\theta_t \tan^{\delta(\nabla^2\omega(t))}\theta_t\frac{\boldsymbol{ {P} }_{N}^{v(N)}\boldsymbol{ {P} }_{1}^{v(1)}}{tr(\boldsymbol{ {P} }_{N}^{v(N)}\boldsymbol{ {P} }_{1}^{v(1)})}. \end{eqnarray}$

(2) $\boldsymbol{ P }_{N}^{v} = {\cal Q}_{m+1}{\cal P}_m{\cal Q}_{m}{\cal P}_{m-1}{\cal Q}_{m-1}\cdots {\cal P}_1{\cal Q}_{1}\in{\cal P}_N$.

$\begin{eqnarray} \boldsymbol{ {P} }_{N}^{v}& = &(-1)^{l-m}\prod\limits_{t = 1}^{N}\cos\theta_t \tan^{\delta(\nabla^2\omega(t))}\theta_t\left( \begin{array}{cc} 0 & 0\\ \cot\theta_1 &1\\ \end{array} \right){}\\& = &(-1)^{l-m}\prod\limits_{t = 1}^{N}\cos\theta_t \tan^{\delta(\nabla^2\omega(t))}\theta_t\frac{\boldsymbol{ {P} }_{N}^{v(N)}\boldsymbol{ {P} }_{1}^{v(1)}}{tr(\boldsymbol{ {P} }_{N}^{v(N)}\boldsymbol{ {P} }_{1}^{v(1)})}. \end{eqnarray}$

(4) $\boldsymbol{ P }_{N}^{v} = {\cal P}_{m+1}{\cal Q}_{m}{\cal P}_{m} {\cal Q}_{m-1}{\cal P}_{m-1} \cdots{\cal Q}_{1}{\cal P}_{1}\in{\cal P}_N$.

$$$P(X = x) = \left\{ \begin{array}{ll} \cos^{2N}\theta\|\boldsymbol{ {P} }_{00}|\psi_0\rangle\|^2, & x = -N, \\ \cos^{2N}\theta(\|(k_{1}\boldsymbol{ {P} }_{00} + k_3\boldsymbol{ {P} }_{01})|\psi_0 \rangle\|^2+\|(k_{2}\boldsymbol{ {P} }_{11}+k_3\boldsymbol{ {P} }_{10})|\psi_0\rangle\|^2), & -N<x<N, \\ \cos^{2N}\theta\|\boldsymbol{ {P} }_{11}|\psi_0\rangle\|^2, & x = N. \end{array} \right.$$$

根据定理3.1, 系统处于位置$x$的概率为

## 4 谱分析

$t = 0$时, 称$\tilde{\psi}_{0}(\xi) = (0\cdots0 \widehat{\psi}_{0}(\xi)^T(0) 0\cdots0)^T$为一个初始完全状态波函数. 所有完全状态波函数组成的集合对于向量的普通加法和数乘运算构成数域${\Bbb C}^2$上一个有限维线性空间$V_{4N+2}$. 取该线性空间的一组基为$\tilde{\varepsilon}_0(-N), \tilde{\varepsilon}_1(-N), \tilde{\varepsilon}_0(-N+1), \tilde{\varepsilon}_1(-N+1), \cdots, \tilde{\varepsilon}_0(N), \tilde{\varepsilon}_1(N)$, 其中$\tilde{\varepsilon}_0(x) = (0, \cdots, 0, \widehat{\varepsilon}_0(\xi)^T(x), 0, \cdots, 0)^T, \tilde{\varepsilon}_1(x) = (0, \cdots, 0, \widehat{\varepsilon}_1(\xi)^T(x), 0, \cdots, 0)^T, \widehat{\varepsilon}_0(\xi)(x) = (1, 0)^T, \widehat{\varepsilon}_1(\xi)(x) = (0, 1)^T, x\in\{-N, -N+1, \cdots, N\}$. 任取一个向量$\tilde{\psi}_t(\xi)\in V_{4N+2}$, 则该向量可以表示为$\tilde{\psi}_t(\xi) = (\widehat{\psi}_t(\xi)(x)) = \sum\limits_{x = -N}^{N}(\widehat{\psi}_t^L(\xi) \tilde{\varepsilon}_0(x) +\widehat{\psi}_t^R(\xi)\tilde{\varepsilon}_1(x)),$其中$0\leq\|\widehat{\psi}_t(\xi)(x)\|\leq 1, t = 0, 1, \cdots, N.$

$\tilde{\psi}_{t}(\xi) $$\tilde{\psi}_{t-1}(\xi) 关系为 $$\left\{ \begin{array}{lll} \widehat{\psi}_{t}(\xi)(-x) & = \widehat{\psi}_{t}(\xi)(x) = 0, & |x|>t, \\ \widehat{\psi}_{t}(\xi)(-x) & = {\rm e}^{-{\rm i}\xi}\boldsymbol{ {P} }_{t}\widehat{\psi}_{t-1}(\xi)(-x+1), & x = -t, \\ \widehat{\psi}_{t}(\xi)(x) & = {\rm e}^{{\rm i}\xi}\boldsymbol{ {Q} }_{t}\widehat{\psi}_{t-1}(\xi)(x-1), & x = t, \\ \widehat{\psi}_{t}(\xi)(x) & = {\rm e}^{-{\rm i}\xi}\boldsymbol{ {P} }_{t}\widehat{\psi}_{t-1}(\xi)(x+1)+{\rm e}^{{\rm i}\xi}\boldsymbol{ {Q} }_{t}\widehat{\psi}_{t-1}(\xi)(x-1), & |x|<t.\\ \end{array} \right.$$ (4.1)式用矩阵表示为 \tilde{\psi}_{t}(\xi) = \tilde{\boldsymbol{ C }}(\xi)\tilde{\psi}_{t-1}(\xi), t = 0, 1, \cdots, N , 其中 $$\widetilde{\boldsymbol{ {C} }}(\xi) = \left( \begin{array}{cccccccccc} 0 & {\rm e}^{-{\rm i}\xi}\boldsymbol{ {P} }_{N-1} & \cdots & 0 & 0 & 0 & \cdots & 0 & {\rm e}^{{\rm i}\xi}\boldsymbol{ {Q} }_N \\ {\rm e}^{{\rm i}\xi}\boldsymbol{ {Q} }_{N} & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & 0 \\ 0 & {\rm e}^{{\rm i}\xi}\boldsymbol{ {Q} }_{N-1} & \cdots & 0 & 0 & 0 & \cdots & 0 & 0\\ \vdots & \vdots & \cdots & \vdots & \vdots & \vdots & \cdots & \vdots & \vdots \\ 0 & 0 & \cdots & {\rm e}^{-{\rm i}\xi}\boldsymbol{ {P} }_{2}& 0 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \cdots & 0 & {\rm e}^{-{\rm i}\xi}\boldsymbol{ {P} }_1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \cdots & {\rm e}^{{\rm i}\xi}\boldsymbol{ {Q} }_{2}& 0 & {\rm e}^{-{\rm i}\xi}\boldsymbol{ {P} }_{2} & \cdots & 0 & 0 \\ 0 & 0 & \cdots & 0 & {\rm e}^{{\rm i}\xi}\boldsymbol{ {Q} }_1 & 0 & \cdots & 0 & 0 \\ 0 & 0 & \cdots & 0 & 0 & {\rm e}^{{\rm i}\xi}\boldsymbol{ {Q} }_{2} & \cdots & 0 & 0 \\ \vdots & \vdots & \cdots & \vdots & \vdots & \vdots & \cdots & 0 & \vdots \\ 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & {\rm e}^{-{\rm i}\xi}\boldsymbol{ {P} }_{N-1} & 0 \\ 0 & 0 & \cdots & 0 & 0 & 0 & \cdots & 0 & {\rm e}^{-{\rm i}\xi}\boldsymbol{ {P} }_{N} \\ {\rm e}^{-{\rm i}\xi}\boldsymbol{ {P} }_{N} & 0 & \cdots & 0 & 0 & 0 & \cdots & {\rm e}^{{\rm i}\xi}\boldsymbol{ {Q} }_{N-1} & 0 \\ \end{array} \right).$$ 定义4.2 称 \widetilde{\boldsymbol{ C }}(\xi) 是一个完全硬币翻转算子. 运用算子 \widetilde{\boldsymbol{ C }}(\xi) , 有 \tilde{\psi}_{t}(\xi) = \tilde{\boldsymbol{ C }}^t(\xi)\tilde{\psi}_{0}(\xi) , 当 \xi = 0 时, 有 \tilde{\psi}_{t}(0) = \tilde{\boldsymbol{ C }}^t(0)\tilde{\psi}_{0}(0) . 定义4.3 定义完全硬币翻转算子 \tilde{\boldsymbol{ C }}(\xi) 的谱值 \lambda$$ \tilde{\boldsymbol{ C }}(\xi)\hat{\psi}_t(\xi) = \lambda\hat{\psi}_t(\xi), \hat{\psi}_t(\xi)\in V_{4N+2}$; 称所有$\tilde{\boldsymbol{ C }}(\xi)$的谱值组成的集合$\sigma(\tilde{\boldsymbol{ C }}(\xi)) $$\tilde{\boldsymbol{ C }}(\xi) 的谱; 定义 \tilde{\boldsymbol{ C }}(\xi) 的谱半径为 性质4.1 完全硬币翻转算子的谱是闭的单位圆盘, 即 r(\tilde{\boldsymbol{ C }}(\xi))\leq 1 . 设 \tilde{\boldsymbol{ C }}(\xi) 是作用在 V_{4N+2} 上的一个有界线性算子, t 为系统演化次数, t = 0, 1, \cdots, N . \tilde{\boldsymbol{ C }}(\xi) 的谱值表示为 \tilde{\boldsymbol{ C }}(\xi)\tilde{\psi}_t(\xi) = \lambda\tilde{\psi}_t(\xi), \tilde{\psi}_t(\xi)\in V_{4N+2} . \tilde{\boldsymbol{ C }}(\xi) 的第 x 行为 \tilde{\boldsymbol{ C }}(\xi)(x) , 则有 将该式子展开 对位置 x 累加得到 经化简有 所以有 其中 x = 1-t, 2-t, \cdots, t-1 . 对于任意的 x\in[-t, t] , 有 0\leq \|\hat{\psi}_t(\xi)(x)\|\leq 1 , 所以存在 [-t, t] 的子集 X 使得 \forall x\in X , 有 \|\hat{\psi}_t(\xi)(x)\|\neq 0 . 因而有 |{\rm e}^{-{\rm i}\xi}\boldsymbol{ P }_t+{\rm e}^{{\rm i}\xi}Q_t-\lambda\boldsymbol{ I }| = 0, |{\rm e}^{-{\rm i}\xi}\boldsymbol{ P }_t-\lambda\boldsymbol{ I }| = 0, |{\rm e}^{{\rm i}\xi}\boldsymbol{ Q }_{t}-\lambda\boldsymbol{ I }| = 0 . 计算解得 \lambda^2+{\rm e}^{{\rm i}\xi}\cos\theta_t\lambda = 0, \lambda^2-{\rm e}^{-{\rm i}\xi}\cos\theta_{t}\lambda = 0, \lambda^2+({\rm e}^{{\rm i}\xi}-{\rm e}^{-{\rm i}\xi})\cos\theta_t\lambda-1 = 0 . 得到谱值为 0, -{\rm e}^{{\rm i}\xi}\cos\theta_t; 0, {\rm e}^{-{\rm i}\xi}\cos\theta_{t}; -{\rm i}(\sin(\xi)\cos\theta_t\pm\sqrt{1 - \sin^2(\xi)\cos^2\theta_t}i)\triangleq (-{\rm i}){\rm e}^{\pm {\rm i}\eta_t} , 其中 \eta_t = \arccos(\sin(\xi)\cos(\theta_t)) . 因而有 \max|\lambda_{1, 2}| = 1 . 所以 r(\tilde{\boldsymbol{ C }}(\xi)) = 1 . 证毕. 性质4.1给出完全硬币翻转算子 \tilde{\boldsymbol{ C }}(\xi) 的谱值和谱半径. 其中, 谱值为 0,-\mathrm{e}^{\mathrm{i} \xi} \cos \theta_{t} ; 0, \mathrm{e}^{-\mathrm{i} \xi} \cos \theta_{t} ;(-\mathrm{i}) \mathrm{e}^{\pm \mathrm{i} \eta_{t}}, t=0,1, \cdots, N 同时, 对谱向量讨论如下. (1) 当谱值 \lambda = (-{\rm i}){\rm e}^{\pm {\rm i}\eta_t} 时, 谱向量为 $$\hat{\psi}_t(\xi)(x) = \left\{ \begin{array}{ll} 0, & |x|>t,\\ { } \frac{{\rm e}^{-{\rm i}\xi}}{\lambda}\rm{\bf{ {P} }}_t{\psi}(\xi)(-x+1), & x = -t,\\ { } \frac{{\rm e}^{{\rm i}\xi}}{\lambda}\rm{\bf{ {Q} }}_t{\psi}(\xi)(x-1), & x = t,\\ \rm{\bf{ {T} }}^x{\psi}(\xi)(0),& |x|<t.\\ \end{array} \right.$$ 其中 $$\boldsymbol{ {T} } = \frac{{\rm e}^{{\rm i}\xi}}{\lambda \cos\theta_t}\left( \begin{array}{cc} \lambda^2-\sin\theta^2_t {\quad} & \sin\theta_t \cos\theta_{t} \\ \sin\theta_{t}\cos\theta_{t}{\quad} & -\cos\theta^2_t \\ \end{array} \right).$$ (2) 当谱值 \lambda = -{\rm e}^{{\rm i}\xi}\cos\theta_t, {\rm e}^{-{\rm i}\xi}\cos\theta_t 时, 谱向量为 $$\hat{\psi}_t(\xi)(x) = \left\{ \begin{array}{ll} 0, & |x|>t, \\ { } \frac{{\rm e}^{-{\rm i}\xi}}{\lambda}\boldsymbol{ {P} }_t\hat{\psi}_t(\xi)(-x+1), & x = -t, \\ { } \frac{{\rm e}^{{\rm i}\xi}}{\lambda}\boldsymbol{ {Q} }_t\hat{\psi}_t(\xi)(x-1), & x = t, \\ \boldsymbol{ {T} }^x{\psi}(\xi)(0), & |x|<t.\\ \end{array} \right.$$ (3) 当谱值 \lambda = 0 时, 谱向量为 $$\hat{\psi}_t(\xi)(x) = \left\{ \begin{array}{ll} 0, & |x|\neq t-1, \\ (a, b)^T, & x = -t+1, \\ (c, d)^T, & x = t-1, \\ \end{array} \right.$$ 其中 \cos\theta a +\sin\theta b = 0, \sin\theta c-\cos\theta d = 0. ## 5 伊藤过程、伊藤方程 定义5.1[19] 设随机过程 \{X(t), t\geq 0\} 满足 $$\begin{array}{l} {\rm d}(X(t)) = \mu(t, X(t)){\rm d}t+\sigma(t, X(t)){\rm d}W_t, \\ { } X(t) = X(t_0)+\int_{t_0}^t\mu(t, X(t)){\rm d}t+\int_{t_0}^t\sigma(t, X(t)){\rm d}W_t, \end{array}$$ 则称它为一个伊藤过程, 称(5.1)式为伊藤随机微分方程. 伊藤过程也称为扩散过程. 在该定义中, X(t) 表示位移随机变量, {\rm d}(X(t)) = X(t)-X(t-1) 表示位移增量, \mu(t, X(t)) 是漂移率, \mu(t, X(t)){\rm d}t 表示每单位时间内的漂移量, \sigma^2(t, X(t)) 表示方差率, {\rm d}W_t 表示噪声服从标准布朗过程, \sigma(t, X(t)){\rm d}W_t 表示噪声引起的随机波动量. 引理5.1 设 X(t) 是一个伊藤过程, 给定一个具有二阶连续可微的二元函数 Y(t) = f(t, X(t))\in C^2([0, \infty]) , 则随机过程 Y(t) = f(t, X(t)) 满足 \begin{eqnarray} {\rm d}Y(t)& = & \bigg(\frac{\partial f(t, X(t))}{\partial t}+\mu(t, X(t))\frac{\partial f(t, X(t))}{\partial x}+\frac{\sigma^2(t, X(t))}{2}\frac{\partial^2 f(t, X(t))}{\partial x^2} \bigg){\rm d}t {}\\ &&+\sigma(t, X(t))\frac{\partial f(t, X(t))}{\partial x}{\rm d}W_t. \end{eqnarray} 该引理称为伊藤引理. 其中, \frac{\partial f(t, X(t))}{\partial t}+\mu(t, X(t))\frac{\partial f(t, X(t))}{\partial x}$$ +\frac{\sigma^2(t, X(t))}{2}\frac{\partial^2 f(t, X(t))}{\partial x^2}$表示漂移率, $(\sigma(t, X(t))\frac{\partial f(t, X(t))}{\partial x})^2$表示方差率. 文献[19-21]研究了齐次量子游荡的伊藤引理和伊藤公式, 下面给出非齐次量子游荡的伊藤公式.

$\begin{eqnarray} \prod\limits^N_{t = 1}\boldsymbol{ {C} }_t({\xi})& = &\sum\limits^{2^N-1}_{k = 0}{\rm e}^{{\rm i}\xi\omega_k(t)}\boldsymbol{ {P} }_{N}^{v_{k}} {}\\ & = &\prod\limits^N_{t = 1}\boldsymbol{ {C} }_t+ {\rm i}\sin{\xi}\sum\limits^{2^N-1}_{k = 0}\sum\limits^{N-1}_{t = 0}{\rm e}^{{\rm i}\xi\omega_k(t)}(\omega_k(t+1)-\omega_k(t))\boldsymbol{ {P} }_{N}^{v_{k}}{}\\ &&+(\cos{\xi}-1)\sum\limits^{2^N-1}_{k = 0}\sum\limits^{N-1}_{t = 0}{\rm e}^{{\rm i}\xi\omega_k(t)}\boldsymbol{ {P} }_{N}^{v_{k}}, \end{eqnarray}$

首先, 取$X(t) = \omega_k(t)$表示第$k$条位置路径在$t$时刻的位置. 令

$t = 0, 1, \cdots, N-1$进行累加, 则有

$$$\prod\limits^N_{t = 1}\boldsymbol{ {C} }_t({\xi}) = \sum\limits^{2^N-1}_{k = 0}\boldsymbol{ {C} }^T_0{\rm e}^{{\rm i}\xi\omega_k}\boldsymbol{ {P} }_{N}^{v_{k}} +\sum\limits^{2^N-1}_{k = 0}\boldsymbol{ {C} }_{1}^T{\rm e}^{{\rm i}\xi\omega_k}\boldsymbol{ {P} }_{N}^{v_{k}} -\sum\limits^{2^N-1}_{k = 0}\boldsymbol{ {C} }_{-1}^T{\rm e}^{{\rm i}\xi\omega_k}\boldsymbol{ {P} }_{N}^{v_{k}}.$$$

$$$\widetilde{\boldsymbol{ {C} }}^N(0)\widetilde{\psi}_0(\xi) +\widetilde{\boldsymbol{ {C} }}^N_{\leftarrow}(\xi)\widetilde{\psi}_0(\xi) -\widetilde{\boldsymbol{ {C} }}^N_{\rightarrow }(\xi)\widetilde{\psi}_0(\xi).$$$

(1) $\widetilde{\boldsymbol{ C }}^N(0)\widetilde{\psi}_0(\xi)(x)$, 表示未演化前系统处于位置$x$的初始信息;

(2) $\widetilde{\boldsymbol{ C }}^N_{\leftarrow}(\xi)\widetilde{\psi}_0(\xi)(x)$, 表示系统在演化过程中所有移入到位置$x$的路径信息. 这体现在式子$\sum\limits_{t' = 0}^{N-1}\sum\limits_{k = 0}^{2^N-1}{\rm e}^{{\rm i}\xi(\omega_k(t') +\nabla\omega_k(t'))}\boldsymbol{ P }_N^{v_{k}}\hat{\psi}_0(\xi)$中的$\omega_k(t')+\nabla\omega_k(t')$, 它表示从当前位置开始向前移入一个单位. 在图论角度来看, 可以用下式计算它的入度$\delta^2_+(x)$

$$$\delta^2_+(x) = \int^{\pi}_{-\pi} \frac{{\rm d}\zeta}{2\pi}{\rm e}^{{\rm i}\zeta x}(\widetilde{\boldsymbol{ {E} }}^N_{\leftarrow}(\zeta)\widehat{\psi}_{0}(\zeta))^{*}\int^{\pi}_{-\pi} \frac{{\rm d}\xi}{2\pi}{\rm e}^{-{\rm i}\xi x}\widetilde{\boldsymbol{ {E} }}^N_{\leftarrow}(\xi)\widehat{\psi}_{0}(\xi),$$$

$N = 3$, 令$\boldsymbol{ C }_t = \boldsymbol{ P }_t+\boldsymbol{ Q }_t, t = 1, 2, 3$, 分别计算$\widetilde{\boldsymbol{ C }}^N(0)\widetilde{\psi}_0(\xi)(x) $$\widetilde{\boldsymbol{ C }}^N_{\leftarrow}(\xi)\widetilde{\psi}_0(\xi)(x)$$ \widetilde{\boldsymbol{ C }}^N_{\rightarrow }(\xi)\widetilde{\psi}_0(\xi)(x)$如下.

(1)

(2)

(3)

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