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@@ -8,18 +8,255 @@ If you are familiar with the linear algebra defined before, you can jump right i
 
 ### Pure states
 
+#### Pure state and mixed state
+
+A pure state is a state that is represented by a unit vector in $\mathscr{H}^{\otimes N}$.
+
+A mixed state is a state that is represented by a density operator in $\mathscr{H}^{\otimes N}$. (convex combination of pure states)
+
+if $\rho_j=|\psi_j\rangle\langle\psi_j|$, then $\rho=\sum_{j=1}^N p_j\rho_j$ is a mixed state, where $p_j\geq 0$ and $\sum_{j=1}^N p_j=1$.
+
+#### Coset space
+
+Two non-zero vectors $u,v\in \mathscr{H}$ are said to represent the same state if $u=cv$ for some complex number $c$ with $|c|=1$.
+
+The set of states of a quantum system is called the **coset space** of $\mathscr{H}$, $u\sim v$ if $u=cv$ for some complex number $c$ with $|c|=1$.
+
+The coset space is called the projective space of $\mathscr{H}$, denoted by $P(\mathscr{H})\colon=(\mathscr{H}\setminus\{0\})/\sim$.
+
+Any vector in the form $e^{i\theta}|u\rangle$ for some $u\in \mathscr{H}$ and $\theta\in \mathbb{R}$ represents the same state as $|u\rangle$.
+
+Example: the system of a qubit has a Hilbert space $\mathbb{C}^2$, the coset space is $P(\mathbb{C}^2)\cong S^2$ is the Bloch sphere.
+
 ### Composite systems
 
+#### Tensor product
+
+The tensor product of two Hilbert spaces $\mathscr{H}_1$ and $\mathscr{H}_2$ is the Hilbert space $\mathscr{H}_1\otimes\mathscr{H}_2$ with the inner product $\langle u_1\otimes u_2,v_1\otimes v_2\rangle=\langle u_1,v_1\rangle\langle u_2,v_2\rangle$.
+
+The tensor product of two vectors $u_1\in \mathscr{H}_1$ and $u_2\in \mathscr{H}_2$ is the vector $u_1\otimes u_2\in \mathscr{H}_1\otimes\mathscr{H}_2$.
+
+#### Multipartite systems
+
+For each part in a multipartite quantum system, each part is associated a Hilbert space $\mathscr{H}_i$. The total system is associated a Hilbert space $\mathscr{H}=\mathscr{H}_1\otimes\mathscr{H}_2\otimes\cdots\otimes\mathscr{H}_n$.
+
+The state of the total system has the form $u_1\otimes u_2\otimes\cdots\otimes u_n$ for some $u_i\in \mathscr{H}_i$.
+
+#### Entanglement (talk later)
+
+A state $|\psi\rangle$ is entangled if it cannot be expressed as a product state $v_1\otimes v_2$ for any single-qubit states $|v_1\rangle$ and $|v_2\rangle$. In other words, an entangled state is non-separable.
+
+Example: the Bell state $|\psi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ is entangled.
+
+Assume it can be written as $|\psi\rangle=|\psi_1\rangle\otimes|\psi_2\rangle$ where $|\psi_1\rangle=a|0\rangle+b|1\rangle$ and $|\psi_2\rangle=c|0\rangle+d|1\rangle$. Then:
+$$|\psi\rangle=a|00\rangle+b|01\rangle+c|10\rangle+d|11\rangle$$
+Setting this equal to $|\psi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$ gives:
+$$ac|00\rangle+ad|01\rangle+bc|10\rangle+bd|11\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$$
+This requires:
+$$ac=bd=\frac{1}{2}$$
+$$ad=bc=0$$
+
+This is a contradiction, so $|\psi^+\rangle$ is entangled.
+
 ### Mixed states and density operators
 
+#### Density operator
+
+A density operator is a Hermitian, positive semi-definite operator with trace 1.
+
+The density operator of a pure state $|\psi\rangle$ is $\rho=|\psi\rangle\langle\psi|$.
+
+The density operator of a mixed state is given by the unit vector $u_1,u_2,\cdots,u_n$ in $\mathscr{H}$ with the probability $p_1,p_2,\cdots,p_n$, $p_i\geq 0$ such that $\sum_{i=1}^n p_i=1$.
+
+The density operator is $\rho=\sum_{i=1}^n p_i|u_i\rangle\langle u_i|$.
+
+#### Trace 1 proposition
+
+Density operator on the finite dimensional Hilbert space $\mathscr{H}$ are positive operators having trace equal to 1.
+
+#### Pure state lemma
+
+A state is pure if and only if $Tr(\rho^2)=1$.
+
+For any mixed state $\rho$, $Tr(\rho^2)<1$.
+
+[Proof ignored here]
+
+#### Unitary freedom in the ensemble for density operators theorem
+
+Let $v_1,v_2,\cdots,v_l$ and $w_1,w_2,\cdots,w_l$ be two collections of vectors in the finite dimensional Hilbert space $\mathscr{H}$, the vectors being arbitrary (can be zero) except for the requirement that they define the same density operator $\rho$.
+
+$$
+\sum_{i=1}^l |v_i\rangle\langle v_i|=\sum_{i=1}^l |w_i\rangle\langle w_i|
+$$
+
+Then there exists a unitary matrix $U=(\mu_{ij})_{1\leq i,j\leq l}$ such that:
+
+$$
+v_i=\sum_{j=1}^l \mu_{ij}w_j
+$$
+
+The converse is also true.
+
+If $\rho$ is a density operator on $\mathscr{H}$ given by: $\sum_{i=1}^l |w_i\rangle\langle w_i|$ and vector $v_i$ is given by: $v_i=\sum_{j=1}^l \mu_{ij}w_j$, then $\rho_1=\sum_{i=1}^l |v_i\rangle\langle v_i|$ is the density operator of the subsystem $\mathscr{H}_1$.
+
+[Proof ignored here]
+
 ### Density operator of subsystems
 
+#### Schmidt Decomposition theorem
+
+Let $|u\rangle\in \mathscr{H}_1\otimes\mathscr{H}_2$ be a unit vector (pure state), then there exists orthonormal bases $|v_i\rangle$ of $\mathscr{H}_1$ and $|w_j\rangle$ of $\mathscr{H}_2$ and $\{\lambda_k\},k\leq r$, where $r$ is the Schmidt rank of $|u\rangle$, such that:
+
+$$
+|u\rangle=\sum_{k=1}^r \lambda_k|v_k\rangle\otimes|w_k\rangle
+$$
+
+where $\lambda_k$ are **non-negative real numbers**. such that $\sum_{k=1}^r \lambda_k^2=1$.
+
+[Proof ignored here]
+
+**Remark**: non-zero vector $u\in \mathscr{H}_1\otimes\mathscr{H}_2$ decomposes as a tensor product $u=u_1\otimes u_2$ if and only if the Schmidt rank of $u$ is 1. **A state** that cannot be decomposed as a tensor product is called **entangled**.
+
+#### Reduced density operator
+
+In $\mathscr{H}_1\otimes\mathscr{H}_2$, the reduced density operator of the subsystem $\mathscr{H}_1$ is:
+
+$$
+\rho_1=\operatorname{Tr}_2(\rho)=\sum_{k=1}^r \lambda_k^2|v_k\rangle\langle v_k|
+$$
+
+where $\rho$ is the density operator in $\mathscr{H}_1\otimes\mathscr{H}_2$.
+
+Example:
+
+Let $\rho=\frac{1}{2}(|01\rangle+|10\rangle)\in \mathbb{C}^2\otimes\mathbb{C}^2$,
+
+Expand the expression of $\rho$ in the basis of $\mathbb{C}^2\otimes\mathbb{C}^2$:
+
+$$
+\rho=\frac{1}{2}(|01\rangle\langle 01|+|01\rangle\langle 10|+|10\rangle\langle 01|+|10\rangle\langle 10|)
+$$
+
+then the reduced density operator of the subsystem $\mathbb{C}^2$ in first qubit is:
+
+$$
+\begin{aligned}
+\rho_1&=\operatorname{Tr}_2(\rho)\\
+&=\frac{1}{2}(\langle 1|1\rangle|0\rangle\langle 0|+\langle 1|0\rangle|0\rangle\langle 1|+\langle 0|1\rangle|1\rangle\langle 0|+\langle 0|0\rangle|1\rangle\langle 1|)\\
+&=\frac{1}{2}(|0\rangle\langle 0|+|1\rangle\langle 1|)\\
+&=\frac{1}{2}I
+\end{aligned}
+$$
+
 ### State purification
 
+Every mixed state can be derived as the reduction of a pure state on an enlarged Hilbert space.
+
+#### State purification theorem
+
+Let $\rho$ be a mixed state in a finite dimensional Hilbert space $\mathscr{H}$, then there exists a unit vector $|w\rangle\in \mathscr{H}\otimes\mathscr{H}$ such that:
+
+$$
+\rho=\operatorname{Tr}_2(|w\rangle\langle w|)
+$$
+
+Hint of proof:
+
+Let $u_1,u_2,\cdots,u_d$ be an orthonormal basis of $\mathscr{H}$, $\sum_{i=1}^d p_i=1$, $p_i\geq 0$, then:
+
+$$
+\rho=\sum_{i=1}^d p_i|u_i\rangle\langle u_i|
+$$
+
+Let $w=\sum_{i=1}^d \sqrt{p_i}u_i\otimes u_i$.
+
 ### Observables
 
+The observables in the quantum theory are self-adjoint operators on the Hilbert space $\mathscr{H}$, denoted by $A\in \mathscr{O}$
+
+In finite dimensional Hilbert space, $A$ can be written as $\sum_{\lambda\in \operatorname{sp}{(A)}}\lambda P_\lambda$, where $P_\lambda$ is the projection operator onto the eigenspace of $A$ corresponding to the eigenvalue $\lambda$. $P_\lambda=P_\lambda^2=P_\lambda^*$.
+
+
 ### Effects and Busch's theorem for effect operators
 
+Below is a section on Topic 4, about Gleason's theorem and definition of states, and Born's rule for describing the states using density operators.
+
+#### Definition of states (non-commutative (_quantum_) probability theory)
+
+A state on $(\mathscr{B}(\mathscr{H}),\mathscr{P})$ is a map $\mu:\mathscr{P}\to[0,1]$ such that:
+
+1. $\mu(O)=0$, where $O$ is the zero projection.
+2. If $P_1,P_2,\cdots,P_n$ are pairwise disjoint orthogonal projections, then $\mu(P_1\lor P_2\lor\cdots\lor P_n)=\sum_{i=1}^n\mu(P_i)$.
+
+Where projections are disjoint if $P_iP_j=P_jP_i=O$.
+
+#### Definition of density operator (non-commutative (_quantum_) probability theory)
+
+A density operator $\rho$ on the finite-dimensional Hilbert space $\mathscr{H}$ is:
+
+1. self-adjoint ($A^*=A$, that is $\langle Ax,y\rangle=\langle x,Ay\rangle$ for all $x,y\in\mathscr{H}$)
+2. positive semi-definite (all eigenvalues are non-negative)
+3. $\operatorname{Tr}(\rho)=1$.
+
+If $(|\psi_1\rangle,|\psi_2\rangle,\cdots,|\psi_n\rangle)$ is an orthonormal basis of $\mathscr{H}$ consisting of eigenvectors of $\rho$, for the eigenvalue $p_1,p_2,\cdots,p_n$, then $p_j\geq 0$ and $\sum_{j=1}^n p_j=1$.
+
+We can write $\rho$ as
+
+$$
+\rho=\sum_{j=1}^n p_j|\psi_j\rangle\langle\psi_j|
+$$
+
+(under basis $|\psi_j\rangle$, it is a diagonal matrix with $p_j$ on the diagonal)
+
+Every basis of $\mathscr{H}$ can be decomposed to these forms.
+
+#### Theorem: Born's rule
+
+Let $\rho$ be a density operator on $\mathscr{H}$. then
+
+$$
+\mu(P)\coloneqq\operatorname{Tr}(\rho P)=\sum_{j=1}^n p_j\langle\psi_j|P|\psi_j\rangle
+$$
+
+Defines a probability measure on the space $\mathscr{P}$.
+
+[Proof ignored here]
+
+#### Theorem: Gleason's theorem (very important)
+
+Let $\mathscr{H}$ be a Hilbert space over $\mathbb{C}$ or $\mathbb{R}$ of dimension $n\geq 3$. Let $\mu$ be a state on the space $\mathscr{P}(\mathscr{H})$ of projections on $\mathscr{H}$. Then there exists a unique density operator $\rho$ such that
+
+$$
+\mu(P)=\operatorname{Tr}(\rho P)
+$$
+
+for all $P\in\mathscr{P}(\mathscr{H})$. $\mathscr{P}(\mathscr{H})$ is the space of all orthogonal projections on $\mathscr{H}$.
+
+[Proof ignored here]
+
+Extending the experimental procedure in quantum physics, **many of the outcome probabilities are expectation of effects instead of projections.** (POVMs)
+
+#### Definition of effect
+
+An effect is a positive (self-adjoint) operator $E$ on $\mathscr{H}$ such that $0\leq E\leq I$.
+
+The set of effects on $\mathscr{H}$ is denoted by $\mathscr{E}(\mathscr{H})$.
+
+An operator $E$ is said to be the **extreme point** of the convex set $\mathscr{E}(\mathscr{H})$ if it cannot be written as a convex combination of two other effects.
+
+That is, If $E$ is an extreme point, then $E=\lambda E_1+(1-\lambda)E_2$ for some $0\leq \lambda\leq 1$ and $E_1,E_2\in \mathscr{E}(\mathscr{H})$ implies $E=E_1=E_2$.
+
+#### Effect operator lemma
+
+The set of orthogonal projections on $\mathscr{H}$, $\mathscr{P}(\mathscr{H})$, is the set of extreme points of $\mathscr{E}(\mathscr{H})$.
+
+#### Projection operator and effect operator
+
+
+
+
+
 ### Measurements
 
 ### Quantum operations and CPTP maps