update
This commit is contained in:
Zheyuan Wu
2
Jenkinsfile
vendored
2
Jenkinsfile
vendored
@@ -37,7 +37,7 @@ pipeline {
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echo "Removing existing container"
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sh 'docker rm notenextra-jenkins || true'
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echo "Running new docker container"
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sh "docker run -d -p 13000:3000 --name notenextra-jenkins ${imageTag}"
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sh "docker run -d -p 13000:3000 --restart=on-failure:10 --name notenextra-jenkins ${imageTag}"
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}
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}
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}
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@@ -48,7 +48,7 @@ A quantum coin can be represented mathematically by linear combination of $|0\ra
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#### Definition of states (classical probability)
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[definition of states continue here.]
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A state in classical probability is a probability distribution on the set of all possible outcomes. We can list as $(p_1,p_2,\cdots,p_n)$.
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To each event $A\in \Omega$, we associate the operator on $\mathscr{H}$ of multiplication by the indicator function $P_A\coloneqq M_{\mathbb{I}_A}:f\mapsto \mathbb{I}_A f$ that projects onto the subspace of $\mathscr{H}$ corresponding to the event $A$.
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@@ -193,3 +193,166 @@ With the following properties:
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(b) $P(\bigcap_{i=1}^n A_i)=\bigwedge_{i=1}^n P(A_i)$
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(c) $P(A^c)=I-P(A)$
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(d) If $A_j$ are mutually disjoint (that is $P(A_i)P(A_j)=P(A_j)P(A_i)=O$ for $i\neq j$), then $P(\bigcup_{j=1}^n A_j)=\sum_{j=1}^n P(A_j)$
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#### Definition of probability of a random variable
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For a system prepared in state $\rho$, the probability of the random variable by the projection-valued measure $P$ is in the Borel set $A$ is $\operatorname{Tr}(\rho P(A))$.
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### Expectation of an random variable and projective measurement
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Notice that if we set $\lambda$ is _observed_ with probability $p_\lambda=\operatorname{Tr}(\rho P_\lambda)$. $\rho'\coloneqq\sum_{\lambda\in sp(T)}P_\lambda \rho P_\lambda$ is a density operator.
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#### Definition of expectation of operators
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Let $T$ be a self-adjoint operator on $\mathscr{H}$. The expectation of $T$ relative to the density operator $\rho$ is given by
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$$
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\mathbb{E}_\rho(T)=\operatorname{Tr}(\rho T)=\sum_{\lambda\in sp(T)}\lambda \operatorname{Tr}(\rho P(\lambda))
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$$
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if we set $T=\sum_{\lambda\in sp(T)}\lambda P_\lambda$, then $\mathbb{E}_\rho(T)=\sum_{\lambda\in sp(T)}\lambda \operatorname{Tr}(\rho P(\lambda))$.
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### The uncertainty principle
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Let $A,B$ be two self-adjoint operators on $\mathscr{H}$. Then we define the following two self-adjoint operators:
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$$
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i[A,B]\coloneqq i(AB-BA)
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$$
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$$
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A\circ B\coloneqq \frac{AB+BA}{2}
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$$
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Note that $A\circ B$ satisfies Jordan's identity.
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$$
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(A\circ B)\circ (A\circ A)=A\circ (B\circ (A\circ A))
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$$
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#### Definition of variance
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Given a state $\rho$, the variance of $A$ is given by
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$$
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\operatorname{Var}_\rho(A)\coloneqq\mathbb{E}_\rho(A^2)-\mathbb{E}_\rho(A)^2=\operatorname{Tr}(\rho A^2)-\operatorname{Tr}(\rho A)^2
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$$
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#### Definition of covariance
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Given a state $\rho$, the covariance of $A$ and $B$ is given by the Jordan product of $A$ and $B$.
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$$
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\operatorname{Cov}_\rho(A,B)\coloneqq\mathbb{E}_\rho(A\circ B)-\mathbb{E}_\rho(A)\mathbb{E}_\rho(B)=\operatorname{Tr}(\rho A\circ B)-\operatorname{Tr}(\rho A)\operatorname{Tr}(\rho B)
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$$
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#### Robertson-Schrödinger uncertainty relation in finite dimensional Hilbert space
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Let $\rho$ be a state on $\mathscr{H}$, $A,B$ be two self-adjoint operators on $\mathscr{H}$. Then
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$$
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\operatorname{Var}_\rho(A)\operatorname{Var}_\rho(B)\geq\operatorname{Cov}_\rho(A,B)^2+\frac{1}{4}|\mathbb{E}_\rho([A,B])|^2
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$$
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If $\rho$ is a pure state ($\rho=|\psi\rangle\langle\psi|$ for some unit vector $|\psi\rangle\in\mathscr{H}$), and the equality holds, then $A$ and $B$ are collinear (i.e. $A=c B$ for some constant $c\in\mathbb{R}$).
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When $A$ and $B$ commute, the classical inequality is recovered (Cauchy-Schwarz inequality).
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$$
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\operatorname{Var}_\rho(A)\operatorname{Var}_\rho(B)\geq\operatorname{Cov}_\rho(A,B)^2
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$$
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[Proof ignored here]
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### The uncertainty relation for unbounded symmetric operators
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#### Definition of symmetric operator
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An operator $A$ is symmetric if for all $\phi,\psi\in\mathscr{H}$, we have
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$$
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\langle A\phi,\psi\rangle=\langle\phi,A\psi\rangle
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$$
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An example of symmetric operator is $T(\psi)=i\hbar\frac{d\psi}{dx}$. If we let $\mathscr{H}=\mathscr{D}(T)$, $\hbar$ is the Planck constant.
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$\mathscr{D}(T)$ be the space of all square integrable, differentiable, and it's derivative is also square integrable functions on $\mathbb{R}$.
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#### Definition of joint domain of two operators
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Let $(A,\mathscr{D}(A)),(B,\mathscr{D}(B))$ be two symmetric operators on their corresponding domains. The domain of $AB$ is defined as
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$$
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\mathscr{D}(AB)\coloneqq\{\psi\in\mathscr{D}(B):B\psi\in\mathscr{D}(A)\}
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$$
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Since $(AB)\psi=A(B\psi)$, the variance of an operator $A$ relative to a pure state $\rho=|\psi\rangle\langle\psi|$ is given by
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$$
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\operatorname{Var}_\rho(A)=\operatorname{Tr}(\rho A^2)-\operatorname{Tr}(\rho A)^2=\langle\psi,A^2\psi\rangle-\langle\psi,A\psi\rangle^2
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$$
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If $A$ is symmetric, then $\operatorname{Var}_\rho(A)=\langle A\psi,A\psi\rangle-\langle \psi, A\psi\rangle^2$.
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#### Robertson-Schrödinger uncertainty relation for unbounded symmetric operators
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Let $(A,\mathscr{D}(A)),(B,\mathscr{D}(B))$ be two symmetric operators on their corresponding domains. Then
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$$
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\operatorname{Var}_\rho(A)\operatorname{Var}_\rho(B)\geq\operatorname{Cov}_\rho(A,B)^2+\frac{1}{4}|\mathbb{E}_\rho([A,B])|^2
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$$
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If $\rho$ is a pure state ($\rho=|\psi\rangle\langle\psi|$ for some unit vector $|\psi\rangle\in\mathscr{H}$), and the equality holds, then $A\psi$ and $B\psi$ are collinear (i.e. $A\psi=c B\psi$ for some constant $c\in\mathbb{R}$).
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[Proof ignored here]
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### Summary of analog of classical probability theory and non-commutative (_quantum_) probability theory
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| Classical probability | Non-commutative (_Quantum_) probability |
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| --------- | ------- |
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| Sample space $\Omega$, cardinality $\vert\Omega\vert=n$, example: $\Omega=\{0,1\}$ | Complex Hilbert space $\mathscr{H}$, dimension $\dim\mathscr{H}=n$, example: $\mathscr{H}=\mathbb{C}^2$ |
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|Common algebra of $\mathbb{C}$ valued functions| Algebra of bounded operators $\mathscr{B}(\mathscr{H})$|
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|$f\mapsto \bar{f}$ complex conjugation| $P\mapsto P^*$ adjoint|
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|Events: indicator functions of sets| Projections: space of orthogonal projections $\mathscr{P}\subseteq\mathscr{B}(\mathscr{H})$|
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|functions $f$ such that $f^2=f=\overline{f}$| orthogonal projections $P$ such that $P^*=P=P^2$|
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|$\mathbb{R}$-valued functions $f=\overline{f}$| self-adjoint operators $A=A^*$|
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| $\mathbb{I}_{f^{-1}(\{\lambda\})}$ is the indicator function of the set $f^{-1}(\{\lambda\})$| $P(\lambda)$ is the orthogonal projection to eigenspace|
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|$f=\sum_{\lambda\in \operatorname{Range}(f)}\lambda \mathbb{I}_{f^{-1}(\{\lambda\})}$|$A=\sum_{\lambda\in \operatorname{sp}(A)}\lambda P(\lambda)$|
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|Probability measure $\mu$ on $\Omega$| Density operator $\rho$ on $\mathscr{H}$|
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|Delta measure $\delta_\omega$| Pure state $\rho=\vert\psi\rangle\langle\psi\vert$|
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|$\mu$ is non-negative measure and $\sum_{i=1}^n\mu(\{i\})=1$| $\rho$ is positive semi-definite and $\operatorname{Tr}(\rho)=1$|
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|Expected value of random variable $f$ is $\mathbb{E}_{\mu}(f)=\sum_{i=1}^n f(i)\mu(\{i\})$| Expected value of operator $A$ is $\mathbb{E}_\rho(A)=\operatorname{Tr}(\rho A)$|
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|Variance of random variable $f$ is $\operatorname{Var}_\mu(f)=\sum_{i=1}^n (f(i)-\mathbb{E}_\mu(f))^2\mu(\{i\})$| Variance of operator $A$ is $\operatorname{Var}_\rho(A)=\operatorname{Tr}(\rho A^2)-\operatorname{Tr}(\rho A)^2$|
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|Covariance of random variables $f$ and $g$ is $\operatorname{Cov}_\mu(f,g)=\sum_{i=1}^n (f(i)-\mathbb{E}_\mu(f))(g(i)-\mathbb{E}_\mu(g))\mu(\{i\})$| Covariance of operators $A$ and $B$ is $\operatorname{Cov}_\rho(A,B)=\operatorname{Tr}(\rho A\circ B)-\operatorname{Tr}(\rho A)\operatorname{Tr}(\rho B)$|
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|Composite system is given by Cartesian product of the sample spaces $\Omega_1\times\Omega_2$| Composite system is given by tensor product of the Hilbert spaces $\mathscr{H}_1\otimes\mathscr{H}_2$|
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|Product measure $\mu_1\times\mu_2$ on $\Omega_1\times\Omega_2$| Tensor product of space $\rho_1\otimes\rho_2$ on $\mathscr{H}_1\otimes\mathscr{H}_2$|
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|Marginal distribution $\pi_*v$|Partial trace $\operatorname{Tr}_2(\rho)$|
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### States of two dimensional system and the complex projective space (_Bloch sphere_)
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Let $v=e^{i\theta}u$, then the space of pure states ($\rho=|u\rangle\langle u|$) is the complex projective space $\mathbb{C}P^1$.
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$\alpha=x_i+iy_i,\beta=x_2+iy_2$ must satisfy $|\alpha|^2+|\beta|^2=1$, that is $x_1^2+x_2^2+y_1^2+y_2^2=1$.
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The set of unit vectors in $\mathbb{C}^2$ is the unit sphere in $\mathbb{R}^3$.
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So the space of pure states is the unit circle in $\mathbb{R}^2$.
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#### Mapping between the space of pure states and the complex projective space
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Any two dimensional pure state can be written as $e^{i\theta}u$, where $u$ is a unit vector in $\mathbb{R}^2$. There exists a bijective map $P:S^2\to\mathscr{P}_1\subseteq M_2(\mathbb{C})$ such that $P(u)=|u\rangle\langle u|$.
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$$
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P(\vec{x})=\frac{1}{2}(I+\vec{a}\cdot\vec{\sigma})=\frac{1}{2}\begin{pmatrix}
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1&0\\
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0&1
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\end{pmatrix}+\frac{a_x}{2}\begin{pmatrix}
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0&1\\
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1&0
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\end{pmatrix}+\frac{a_y}{2}\begin{pmatrix}
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0&-i\\
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i&0
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\end{pmatrix}+\frac{a_z}{2}\begin{pmatrix}
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1&0\\
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0&-1
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\end{pmatrix}
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37
pages/Math401/Math401_T5.md
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37
pages/Math401/Math401_T5.md
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@@ -0,0 +1,37 @@
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# Topic 5: Introducing dynamics: classical and non-commutative
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## Section 1: Dynamics in classical probability
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### Basic definitions
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#### Definition of orbit
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Let $T:\Omega\to\Omega$ be a map (may not be invertible) generating a dynamical system on $\Omega$. Given $\omega\in \Omega$, the (forward) orbit of $\omega$ is the set $\mathscr{O}(\omega)=\{T^n(\omega)\}_{n\in\mathbb{Z}}$.
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The theory of dynamics is the study of properties of orbits.
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#### Definition of measure-preserving map
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Let $P$ be a probability measure on a $\sigma$-algebra $\mathscr{F}$ of subsets of $\Omega$. A measurable transformation $T:\Omega\to\Omega$ is said to be measure-preserving if for all random variables $\psi:\Omega\to\mathbb{R}$, we have $\mathbb{E}(\psi\circ T)=\mathbb{E}(\psi)$, that is:
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$$
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\int_\Omega (\psi\circ T)(\omega)dP(\omega)=\int_\Omega \psi(\omega)dP(\omega)
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$$
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#### Definition of ergodic map
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A measurable transformation $T:\Omega\to\Omega$ is said to be ergodic if for all random variables $\psi:\Omega\to\mathbb{R}$, we have $\mathbb{E}(\psi\circ T)=\mathbb{E}(\psi)$, that is:
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$$
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\int_\Omega (\psi\circ T)(\omega)dP(\omega)=\int_\Omega \psi(\omega)dP(\omega)
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$$
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#### Definition of isometry
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The composition operator $\psi\mapsto U\psi=\psi\circ T$, where $T$ is a measure preserving map defined on $\mathscr{H}=L^2(\Omega,\mathscr{F},P)$ is isometry of $\mathscr{H}$ if $\langle U\psi,U\phi\rangle=\langle\psi,\phi\rangle$ for all $\psi,\phi\in\mathscr{H}$.
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#### Definition of unitary
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The composition operator $\psi\mapsto U\psi=\psi\circ T$, where $T$ is a measure preserving map defined on $\mathscr{H}=L^2(\Omega,\mathscr{F},P)$ is unitary of $\mathscr{H}$ if $U$ is an isometry and $T$ is invertible with measurable inverse.
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## Section 2: Continuous time (classical) dynamical systems
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@@ -10,4 +10,5 @@ export default {
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Math401_T2: "Math 401, Topic 2: Finite-dimensional Hilbert spaces",
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Math401_T3: "Math 401, Topic 3: Separable Hilbert spaces",
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Math401_T4: "Math 401, Topic 4: The quantum version of probabilistic concepts",
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Math401_T5: "Math 401, Topic 5: Introducing dynamics: classical and non-commutative",
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}
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