update

pages/Math401/Math401_T2.md

@@ -170,7 +170,7 @@ $$
 
 Let $u,v\in \mathscr{H}$.
 
-$||v||$ is the length of $v$.
+$\|v\|$ is the length of $v$.
 
 $v$ is a unit vector if $\|v\|=1$.
 
@@ -335,3 +335,437 @@ $$
 
 is a linear map from $\mathscr{H}$ to $\mathscr{H}$.
 
+### The spectral theorem for self-adjoint operators
+
+### Spectral theorem for self-adjoint operators
+
+#### Definition of spectral theorem
+
+Let $\mathscr{H}$ be a Hilbert space. A self-adjoint operator $T: \mathscr{H}\to \mathscr{H}$ is a linear operator that is equal to its adjoint.
+
+Then all the eigenvalues of $T$ are real and there exists an orthonormal basis of $\mathscr{H}$ consisting of eigenvectors of $T$.
+
+#### Definition of spectrum
+
+The spectrum of a linear operator on finite-dimensional Hilbert space $T: \mathscr{H}\to \mathscr{H}$ is the set of all distinct eigenvalues of $T$.
+
+$$
+\operatorname{sp}(T)=\{\lambda: \lambda\text{ is an eigenvalue of } T\}\subset \mathbb{C}
+$$
+
+#### Definition of Eigenspace
+
+If $\lambda$ is an eigenvalue of $T$, the eigenspace of $T$ corresponding to $\lambda$ is the set of all eigenvectors of $T$ corresponding to $\lambda$.
+
+$$
+E_\lambda(T)=\{v\in \mathscr{H}: Tv=\lambda v\}
+$$
+
+We denote $P_\lambda(T):\mathscr{H}\to E_\lambda(T)$ the orthogonal projection onto $E_\lambda(T)$.
+
+#### Definition of Operator norm
+
+The operator norm of a linear operator $T: \mathscr{H}\to \mathscr{H}$ is the largest eigenvalue of $T$.
+
+$$
+\|T\|=\max_{\|v\|=1} \|Tv\|
+$$
+
+We say $T$ is **bounded** if $\|T\|<\infty$.
+
+We denote $B(\mathscr{H})$ the set of all bounded linear operators on $\mathscr{H}$.
+
+### Partial trace
+
+#### Definition of trace
+
+Let $T$ be a linear operator on $\mathscr{H}$, $(e_1,e_2,\cdots,e_n)$ be a basis of $\mathscr{H}$ and $(\epsilon_1,\epsilon_2,\cdots,\epsilon_n)$ be a basis of dual space $\mathscr{H}^*$. Then the trace of $T$ is defined by
+
+$$
+\operatorname{Tr}(T)=\sum_{i=1}^n \epsilon_i(T(e_i))=\sum_{i=1}^n \langle e_i,T(e_i)\rangle
+$$
+
+This is equivalent to the sum of the diagonal elements of $T$.
+
+> Check the rest of the section defining the partial trace by viewing the tensor product section first.
+
+#### Definition of partial trace
+
+Let $T$ be a linear operator on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$, where $\mathscr{A}$ and $\mathscr{B}$ are finite-dimensional Hilbert spaces.
+
+An operator $T$ on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$ can be written as
+
+$$
+T=\sum_{i=1}^n a_i A_i\otimes B_i
+$$
+
+where $A_i$ is a linear operator on $\mathscr{A}$ and $B_i$ is a linear operator on $\mathscr{B}$.
+
+The partial trace of $T$ is the linear operator on $\mathscr{B}$ defined by
+
+$$
+\operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i
+$$
+
+Or we can define the map $L_v: \mathscr{A}\to \mathscr{A}\otimes \mathscr{B}$ by
+
+$$
+L_v(u)=u\otimes v
+$$
+
+Note that $\langle u,L_v^*(u')\otimes v'\rangle=\langle u,u'\rangle \langle v,v'\rangle=\langle u\otimes v,u'\otimes v'\rangle=\langle L_v(u),u'\otimes v'\rangle$.
+
+Therefore, $L_v^*\sum_{j} u_j\otimes v_j=\sum_{j} \langle v,v_j\rangle u_j$.
+
+Then the partial trace of $T$ can also be defined by
+
+**Let $\{v_j\}$ be a set of orthonormal basis of $\mathscr{B}$.**
+
+$$
+\operatorname{Tr}_{\mathscr{B}}(T)=\sum_{j} L^*_{v_j}(T)L_{v_j}
+$$
+
+#### Definition of partial trace with respect to a state
+
+Let $T$ be a linear operator on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$, where $\mathscr{A}$ and $\mathscr{B}$ are finite-dimensional Hilbert spaces.
+
+Let $\rho$ be a state on $\mathscr{B}$ consisting of orthonormal basis $\{v_j\}$ and eigenvalue $\{\lambda_j\}$.
+
+The partial trace of $T$ with respect to $\rho$ is the linear operator on $\mathscr{A}$ defined by
+
+$$
+\operatorname{Tr}_{\mathscr{A}}(T)=\sum_{j} \lambda_j L^*_{v_j}(T)L_{v_j}
+$$
+
+### Space of Bounded Linear Operators
+
+> Recall the trace of a matrix is the sum of its diagonal elements.
+
+#### Hilbert-Schmidt inner product
+
+Let $T,S\in B(\mathscr{H})$. The Hilbert-Schmidt inner product of $T$ and $S$ is defined by
+
+$$
+\langle T,S\rangle=\operatorname{Tr}(T^*S)
+$$
+
+> Note here, $T^*$ is the complex conjugate transpose of $T$.
+
+If we introduce the basis $\{e_i\}$ in $\mathscr{H}$, then we can write the the space of bounded linear operators as $n\times n$ complex-valued matrices $M_n(\mathbb{C})$.
+
+For $T=(a_{ij})$, $S=(b_{ij})$, we have
+
+$$
+\operatorname{Tr}(A^*B)=\sum_{i=1}^n \sum_{j=1}^n \overline{a_{ij}}b_{ij}
+$$
+
+The inner product is the standard Hermitian inner product in $\mathbb{C}^{n\times n}$.
+
+#### Definition of Hilbert-Schmidt norm
+
+The Hilbert-Schmidt norm of a linear operator $T: \mathscr{H}\to \mathscr{H}$ is defined by
+
+$$
+\|T\|=\sqrt{\sum_{i=1}^n \sum_{j=1}^n |a_{ij}|^2}
+$$
+
+**[The trace of operator does not depend on the basis.](https://notenextra.trance-0.com/Math429/Math429_L38#theorem-850)**
+
+### Tensor products of finite-dimensional Hilbert spaces
+
+Let $X=X_1\times X_2\times \cdots \times X_n$ be a Cartesian product of $n$ sets.
+
+Let $x=(x_1,x_2,\cdots,x_n)$ be a vector in $X$.
+$x_j\in X_j$ for $j=1,2,\cdots,n$.
+
+Let $a\in X_j$ for $j=1,2,\cdots,n$.
+
+Let's denote the space of all functions from $X$ to $\mathbb{C}$ by $\mathscr{H}$ and the space of all functions from $X_j$ to $\mathbb{C}$ by $\mathscr{H}_j$.
+
+$$
+\epsilon_{a}^{(j)}(x_j)=\begin{cases}
+1 & \text{if } x_j=a \\
+0 & \text{if } x_j\neq a
+\end{cases}
+$$
+
+Then we can define a basis of $\mathscr{H}_j$ by $\{\epsilon_{a}^{(j)}(x_j)\}_{a\in X_j}$.
+
+_Any function $f:X_j\to \mathbb{C}$ can be written as a linear combination of the basis vectors._
+
+$$
+f(x_j)=\sum_{a\in X_j} f(a)\epsilon_{a}^{(j)}(x_j)
+$$
+
+
+Now, let $a=(a_1,a_2,\cdots,a_n)$ be a vector in $X$, and $x=(x_1,x_2,\cdots,x_n)$ be a vector in $X$. Note that $a_j,x_j\in X_j$ for $j=1,2,\cdots,n$.
+
+Define
+
+$$
+\epsilon_a(x)=\prod_{j=1}^n \epsilon_{a_j}^{(j)}(x_j)=\begin{cases}
+1 & \text{if } a_j=x_j \text{ for all } j=1,2,\cdots,n \\
+0 & \text{otherwise}
+\end{cases}
+$$
+
+Then we can define a basis of $\mathscr{H}$ by $\{\epsilon_a\}_{a\in X}$.
+
+_Any function $f:X\to \mathbb{C}$ can be written as a linear combination of the basis vectors._
+
+$$
+f(x)=\sum_{a\in X} f(a)\epsilon_a(x)
+$$
+
+**The tensor product of basis elements** is defined by
+
+$$
+\epsilon_a=\epsilon_{a_1}^{(1)}\otimes \epsilon_{a_2}^{(2)}\otimes \cdots \otimes \epsilon_{a_n}^{(n)}
+$$
+
+**The tensor product of two finite-dimensional Hilbert spaces** (in $\mathscr{H}$) is defined by
+
+Let $\mathscr{H}_1$ and $\mathscr{H}_2$ be two finite dimensional Hilbert spaces. Let $u_1\in \mathscr{H}_1$ and $v_1\in \mathscr{H}_2$.
+
+$$
+u_1\otimes v_1
+$$
+
+is a bi-anti-linear map from $\mathscr{H}_1\otimes \mathscr{H}_2$ to $\mathbb{F}$ (in this case, $\mathbb{C}$). And $\forall u\in \mathscr{H}_1, v\in \mathscr{H}_2$,
+
+$$
+(u_1\otimes v_1)(u, v)=\langle u,u_1\rangle \langle v,v_1\rangle
+$$
+
+We call such forms **decomposable**. The tensor product of two finite-dimensional Hilbert spaces, denoted by $\mathscr{H}_1\otimes \mathscr{H}_2$, is the set of all linear combinations of decomposable forms. Represented by the following:
+
+$$
+(\sum_{i=1}^n a_i u_i\otimes v_i)(u, v)=\sum_{i=1}^n a_i \langle v,u_i\rangle \langle v_i,u\rangle
+$$
+
+Note that $a_i\in \mathbb{C}$ for complex-vector spaces.
+
+This is a linear space of dimension $\dim \mathscr{H}_1\times \dim \mathscr{H}_2$.
+
+We define the inner product of two elements of $\mathscr{H}_1\otimes \mathscr{H}_2$ ($u_1\otimes v_1:(\mathscr{H}_1\otimes \mathscr{H}_2)\to \mathbb{C}$, $u_2\otimes v_2:(\mathscr{H}_1\otimes \mathscr{H}_2)\to \mathbb{C}$ $\in \mathscr{H}_1\otimes \mathscr{H}_2$) by
+
+$$
+\langle u_1\otimes v_1, u_2\otimes v_2\rangle=\langle u_1,u_2\rangle \langle v_1,v_2\rangle=(u_1\otimes v_1)(u_2,v_2)
+$$
+
+### Tensor products of linear operators
+
+Let $T_1$ be a linear operator on $\mathscr{H}_1$ and $T_2$ be a linear operator on $\mathscr{H}_2$, where $\mathscr{H}_1$ and $\mathscr{H}_2$ are finite-dimensional Hilbert spaces. The tensor product of $T_1$ and $T_2$ (denoted by $T_1\otimes T_2$) on $\mathscr{H}_1\otimes \mathscr{H}_2$, such that **on decomposable elements** is defined by
+
+$$
+(T_1\otimes T_2)(v_1\otimes v_2)=T_1(v_1)\otimes T_2(v_2)=\langle v_1,T_1(v_1)\rangle \langle v_2,T_2(v_2)\rangle
+$$
+
+for all $v_1\in \mathscr{H}_1$ and $v_2\in \mathscr{H}_2$.
+
+The tensor product of two linear operators $T_1$ and $T_2$ is a linear combination in the form as follows:
+
+$$
+\sum_{i=1}^n a_i T_1(u_i)\otimes T_2(v_i)
+$$
+
+for all $u_i\in \mathscr{H}_1$ and $v_i\in \mathscr{H}_2$.
+
+Such tensor product of linear operators is well defined.
+
+If $\sum_{i=1}^n a_i u_i\otimes v_i=\sum_{j=1}^m b_j u_j\otimes v_j$, then $a_i=b_j$ for all $i=1,2,\cdots,n$ and $j=1,2,\cdots,m$.
+
+Then $\sum_{i=1}^n a_i T_1(u_i)\otimes T_2(v_i)=\sum_{j=1}^m b_j T_1(u_j)\otimes T_2(v_j)$.
+
+#### Tensor product of linear operators on Hilbert spaces
+
+Let $T_1$ be a linear operator on $\mathscr{H}_1$ and $T_2$ be a linear operator on $\mathscr{H}_2$, where $\mathscr{H}_1$ and $\mathscr{H}_2$ are finite-dimensional Hilbert spaces. The tensor product of $T_1$ and $T_2$ (denoted by $T_1\otimes T_2$) on $\mathscr{H}_1\otimes \mathscr{H}_2$, such that **on decomposable elements** is defined by
+
+$$
+(T_1\otimes T_2)(v_1\otimes v_2)=T_1(v_1)\otimes T_2(v_2)=\langle v_1,T_1(v_1)\rangle \langle v_2,T_2(v_2)\rangle
+$$
+
+#### Extended Dirac notation
+
+Suppose $\mathscr{H}=\mathbb{C}^n$ with the standard basis $\{e_i\}$.
+
+$e_j=|j\rangle$ and
+
+$$
+|j_1\dots j_n\rangle=e_{j_1}\otimes e_{j_2}\otimes \cdots \otimes e_{j_n}=
+$$
+
+#### The Hadamard Transform
+
+Let $\mathscr{H}=\mathbb{C}^2$ with the standard basis $\{e_1,e_2\}=\{|0\rangle,|1\rangle\}$.
+
+The linear operator $H_2$ is defined by
+
+$$
+H_2=\frac{1}{\sqrt{2}}\begin{pmatrix}
+1 & 1 \\
+1 & -1
+\end{pmatrix}=\frac{1}{\sqrt{2}}(|0\rangle\langle 0|+|1\rangle\langle 0|+|0\rangle\langle 1|-|1\rangle\langle 1|)
+$$
+
+The Hadamard transform is the linear operator $H_2$ on $\mathbb{C}^2$.
+
+### Singular value and Schmidt decomposition
+
+#### Definition of SVD (Singular Value Decomposition)
+
+Let $T:\mathscr{U}\to \mathscr{V}$ be a linear operator between two finite-dimensional Hilbert spaces $\mathscr{U}$ and $\mathscr{V}$.
+
+We denote the inner product of $\mathscr{U}$ and $\mathscr{V}$ by $\langle \cdot, \cdot \rangle$.
+
+Then there exists a decomposition of $T$
+
+$$
+T=d_1 T_1+d_2 T_2+\cdots +d_n T_n
+$$
+
+with $d_1>d_2>\cdots >d_n>0$ and $T_i:\mathscr{U}\to \mathscr{V}$ such that:
+
+1. $T_iT_j^*=0$, $T_i^*T_j=0$ for $i\neq j$(
+2. $T_i|_{\mathscr{R}(T_i^*)}:\mathscr{R}(T_i^*)\to \mathscr{R}(T_i)$ is an isomorphism with inverse $T_i^*$ where $\mathscr{R}(\cdot)$ is the range of the operator.
+
+The $d_i$ are called the singular values of $T$.
+
+[Gram-Schmidt Decomposition](https://notenextra.trance-0.com/Math429/Math429_L27#theorem-632-gram-schmidt)
+
+## Basic Group Theory
+
+### Finite groups
+
+#### Definition of group
+
+A group is a set $G$ with a binary operation $\cdot$ that satisfies the following axioms:
+
+1. **Closure**: For all $a,b\in G$, $a\cdot b\in G$.
+2. **Associativity**: For all $a,b,c\in G$, $(a\cdot b)\cdot c=a\cdot (b\cdot c)$.
+3. **Identity**: There exists an element $e\in G$ such that for all $a\in G$, $a\cdot e=e\cdot a=a$.
+4. **Inverses**: For all $a\in G$, there exists an element $b\in G$ such that $a\cdot b=b\cdot a=e$.
+
+#### Symmetric group $S_n$
+
+The symmetric group $S_n$ is the group of all permutations of $n$ elements.
+
+$$
+S_n=\{f: \{1,2,\cdots,n\}\to \{1,2,\cdots,n\} \text{ is a bijection}\}
+$$
+
+#### Unitary group $U(n)$
+
+The unitary group $U(n)$ is the group of all $n\times n$ unitary matrices.
+
+Such that $A^*=A$, where $A^*$ is the complex conjugate transpose of $A$. $A^*=(\overline{A})^T$.
+
+#### Cyclic group $\mathbb{Z}_n$
+
+The cyclic group $\mathbb{Z}_n$ is the group of all integers modulo $n$.
+
+$$
+\mathbb{Z}_n=\{0,1,2,\cdots,n-1\}
+$$
+
+#### Definition of group homomorphism
+
+A group homomorphism is a function $\varPhi:G\to H$ between two groups $G$ and $H$ that satisfies the following axiom:
+
+$$
+\varPhi(a\cdot b)=\varPhi(a)\cdot \varPhi(b)
+$$
+
+A bijective group homomorphism is called group isomorphism.
+
+#### Homomorphism sends identity to identity, inverses to inverses
+
+Let $\varPhi:G\to H$ be a group homomorphism. $e_G$ and $e_H$ are the identity elements of $G$ and $H$ respectively. Then
+
+1. $\varPhi(e_G)=e_H$
+2. $\varPhi(a^{-1})=\varPhi(a)^{-1}$. $\forall a\in G$
+
+### More on the symmetric group
+
+#### General linear group over $\mathbb{C}$
+
+The general linear group over $\mathbb{C}$ is the group of all $n\times n$ invertible complex matrices.
+
+$$
+GL(n,\mathbb{C})=\{A\in M_n(\mathbb{C}) \text{ is invertible}\}
+$$
+
+The map $T: S_n\to GL(n,\mathbb{C})$ is a group homomorphism.
+
+#### Definition of sign of a permutation
+
+Let $T:S_n\to GL(n,\mathbb{C})$ be the group homomorphism. The sign of a permutation $\sigma\in S_n$ is defined by
+
+$$
+\operatorname{sgn}(\sigma)=\det(T(\sigma))
+$$
+
+We say $\sigma$ is even if $\operatorname{sgn}(\sigma)=1$ and odd if $\operatorname{sgn}(\sigma)=-1$.
+
+### Fourier Transform in $\mathbb{Z}_N$.
+
+The vector space $L^2(\mathbb{Z}_N)$ is the set of all complex-valued functions on $\mathbb{Z}_N$ with the inner product
+
+$$
+\langle f,g\rangle=\sum_{k=0}^{N-1} \overline{f(k)}g(k)
+$$
+
+An orthonormal basis of $L^2(\mathbb{Z}_N)$ is given by $\delta_y,y\in \mathbb{Z}_N$.
+
+$$
+\delta_y(k)=\begin{cases}
+1 & \text{if } k=y \\
+0 & \text{otherwise}
+\end{cases}
+$$
+
+in Dirac notation, we have
+
+$$
+\delta_y=|y\rangle=|y+N\rangle
+$$
+
+#### Definition of Fourier transform
+
+Define $\varphi_k(x)=\frac{1}{\sqrt{N}}e^{2\pi i kx/N}$ for $k\in \mathbb{Z}_N$. $\varphi_k:\mathbb{Z}\to \mathbb{C}$ is a function.
+
+The Fourier transform of a function $F\in L^2(\mathbb{Z}_N)$ such that $(Ff)(k)=\langle \varphi_k,f\rangle$ is defined by
+
+$$
+F=\frac{1}{\sqrt{N}}\sum_{j=0}^{N-1} \sum_{k=0}^{N-1} e^{2\pi i kj/N}|k\rangle\langle j|
+$$
+
+### Symmetric and anti-symmetric tensors
+
+Let $\mathscr{H}^{\otimes n}$ be the $n$-fold tensor product of a Hilbert space $\mathscr{H}$.
+
+We define the $S_n$ on $\mathscr{H}^{\otimes n}$ by
+
+Let $\eta\in S_n$ be a permutation.
+
+$$
+\prod(\eta)v_1\otimes v_2\otimes \cdots \otimes v_n=v_{\eta^{-1}(1)}\otimes v_{\eta^{-1}(2)}\otimes \cdots \otimes v_{\eta^{-1}(n)}
+$$
+
+And extend to $\mathscr{H}^{\otimes n}$ by linearity.
+
+This gives the property that $\zeta,\eta\in S_n$, $\prod(\zeta\eta)=\prod(\zeta)\prod(\eta)$.
+
+#### Definition of symmetric and anti-symmetric tensors
+
+Let $\mathscr{H}$ be a finite-dimensional Hilbert space.
+
+An element in $\mathscr{H}^{\otimes n}$ is called symmetric if it is invariant under the action of $S_n$. Let $\alpha\in \mathscr{H}^{\otimes n}$
+
+$$\prod(\eta)\alpha=\alpha \text{ for all } \eta\in S_n.$$
+
+It is called anti-symmetric if
+
+$$
+\prod(\eta)\alpha=\operatorname{sgn}(\eta)\alpha \text{ for all } \eta\in S_n.
+$$

pages/Math401/Math401_T3.md

@@ -0,0 +1,71 @@
+# Topic 3: Separable Hilbert spaces
+
+## Infinite-dimensional Hilbert spaces
+
+Recall from Topic 1.
+
+[$L^2$ space](https://notenextra.trance-0.com/Math401/Math401_T1#section-3-further-definitions-in-measure-theory-and-integration)
+
+Let $\lambda$ be a measure on $\mathbb{R}$, or any other field you are interested in.
+
+A function is square integrable if
+
+$$
+\int_\mathbb{R} |f(x)|^2 d\lambda(x)<\infty
+$$
+
+### $L^2$ space and general Hilbert spaces
+
+#### Definition of $L^2(\mathbb{R},\lambda)$
+
+The space $L^2(\mathbb{R},\lambda)$ is the space of all square integrable, measurable functions on $\mathbb{R}$ with respect to the measure $\lambda$ (The Lebesgue measure).
+
+The Hermitian inner product is defined by
+
+$$
+\langle f,g\rangle=\int_\mathbb{R} \overline{f(x)}g(x) d\lambda(x)
+$$
+
+The norm is defined by
+
+$$
+\|f\|=\sqrt{\int_\mathbb{R} |f(x)|^2 d\lambda(x)}
+$$
+
+The space $L^2(\mathbb{R},\lambda)$ is complete.
+
+[Proof ignored here]
+
+> Recall the definition of [complete metric space](https://notenextra.trance-0.com/Math4111/Math4111_L17#definition-312).
+
+The inner product space $L^2(\mathbb{R},\lambda)$ is complete.
+
+#### Definition of general Hilbert space
+
+A Hilbert space is a complete inner product space.
+
+#### General Pythagorean theorem
+
+Let $u_1,u_2,\cdots,u_N$ be an orthonormal set in an inner product space $\mathscr{V}$ (may not be complete). Then for all $v\in \mathscr{V}$,
+
+$$
+\|v\|^2=\sum_{i=1}^N |\langle v,u_i\rangle|^2+\left\|v-\sum_{i=1}^N \langle v,u_i\rangle u_i\right\|^2
+$$
+
+[Proof ignored here]
+
+#### Bessel's inequality
+
+Let $u_1,u_2,\cdots,u_N$ be an orthonormal set in an inner product space $\mathscr{V}$ (may not be complete). Then for all $v\in \mathscr{V}$,
+
+$$
+\sum_{i=1}^N |\langle v,u_i\rangle|^2\leq \|v\|^2
+$$
+
+Immediate from the general Pythagorean theorem.
+
+### Orthonormal bases
+
+#### Definition of orthonormal basis
+
+An orthonormal basis of a Hilbert space $\mathscr{H}$ is a set of orthonormal vectors that spans $\mathscr{H}$.

pages/Math401/_meta.js

@@ -8,4 +8,5 @@ export default {
     Math401_N3: "Math 401, Notes 3",
     Math401_T1: "Math 401, Topic 1: Probability under language of measure theory",
     Math401_T2: "Math 401, Topic 2: Finite-dimensional Hilbert spaces",
+    Math401_T3: "Math 401, Topic 3: Separable Hilbert spaces",
 }