updates

content/Math401/Math401_P1.md

@@ -1,4 +1,4 @@
-# Math 401, Paper 1: Concentration of measure effects in quantum information (Patrick Hayden)
+# Math 401 Paper 1: Concentration of measure effects in quantum information (Patrick Hayden)
 
 [PDF](https://www.ams.org/books/psapm/068/2762144)
 

content/Math401/Math401_P1_1.md

@@ -1,4 +1,4 @@
-# Math 401, Paper 1, Side note 1: Quantum information theory and Measure concentration
+# Math 401 Paper 1, Side note 1: Quantum information theory and Measure concentration
 
 ## Typicality
 
@@ -56,7 +56,7 @@ Practically speaking:
 
 Recall that the bipartite state of a quantum system is a linear operator on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$, where $\mathscr{A}$ and $\mathscr{B}$ are finite-dimensional Hilbert spaces.
 
-#### Definition of partial trace
+#### Definition of partial trace for arbitrary linear operators
 
 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.
 
@@ -74,16 +74,71 @@ $$
 \operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i
 $$
 
+#### Partial trace for density operators
+
+Let $\rho$ be a density operator in $\mathscr{H}_1\otimes\mathscr{H}_2$, the partial trace of $\rho$ over $\mathscr{H}_2$ is the density operator in $\mathscr{H}_1$ (reduced density operator for the subsystem $\mathscr{H}_1$) given by:
+
+$$
+\rho_1\coloneqq\operatorname{Tr}_2(\rho)
+$$
+
+<details>
+<summary>Examples</summary>
+
+Let $\rho=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$ be a density operator on $\mathscr{H}=\mathbb{C}^2\otimes \mathbb{C}^2$. 
+
+Expand the expression of $\rho$ in the basis of $\mathbb{C}^2\otimes\mathbb{C}^2$ using linear combination of basis vectors:
+
+$$
+\rho=\frac{1}{2}(|01\rangle\langle 01|+|01\rangle\langle 10|+|10\rangle\langle 01|+|10\rangle\langle 10|)
+$$
+
+Note $\operatorname{Tr}_2(|ab\rangle\langle cd|)=|a\rangle\langle c|\cdot \langle b|d\rangle$.
+
+Then the reduced density operator of the subsystem $\mathbb{C}^2$ in first qubit is, note the $\langle 0|0\rangle=\langle 1|1\rangle=1$ and $\langle 0|1\rangle=\langle 1|0\rangle=0$:
+
+$$
+\begin{aligned}
+\rho_1&=\operatorname{Tr}_2(\rho)\\
+&=\frac{1}{2}(\langle 1|1\rangle |0\rangle\langle 0|+\langle 0|1\rangle |0\rangle\langle 1|+\langle 1|0\rangle |1\rangle\langle 0|+\langle 0|0\rangle |1\rangle\langle 1|)\\
+&=\frac{1}{2}(|0\rangle\langle 0|+|1\rangle\langle 1|)\\
+&=\frac{1}{2}I
+\end{aligned}
+$$
+
+is a mixed state.
+
+</details>
+
 ### Purification
 
-Let $\rho$ be any [state](https://notenextra.trance-0.com/Math401/Math401_T6#pure-states) (may not be pure) on the finite dimensional Hilbert space $\mathscr{H}$. then there exists a unit vector $w\in \mathscr{H}\otimes \mathscr{H}$ such that $\rho=\operatorname{Tr}+2(|w\rangle\langle w|)$ is a pure state.
+Let $\rho$ be any [state](https://notenextra.trance-0.com/Math401/Math401_T6#pure-states) (may not be pure) on the finite dimensional Hilbert space $\mathscr{H}$. then there exists a unit vector $w\in \mathscr{H}\otimes \mathscr{H}$ such that $\rho=\operatorname{Tr}_2(|w\rangle\langle w|)$ is a pure state.
 
 <details>
 <summary>Proof</summary>
 
+Let $(u_1,u_2,\cdots,u_n)$ be an orthonormal basis of $\mathscr{H}$ consisting of eigenvectors of $\rho$ for the eigenvalues $p_1,p_2,\cdots,p_n$. As $\rho$ is a states, $p_i\geq 0$ for all $i$ and $\sum_{i=1}^n p_i=1$.
 
+We can write $\rho$ as
 
+$$
+\rho=\sum_{i=1}^n p_i |u_i\rangle\langle u_i|
+$$
+
+Let $w=\sum_{i=1}^n \sqrt{p_i} u_i\otimes u_i$, note that $w$ is a unit vector (pure state). Then
+
+$$
+\begin{aligned}
+\operatorname{Tr}_2(|w\rangle\langle w|)&=\operatorname{Tr}_2(\sum_{i=1}^n \sum_{j=1}^n \sqrt{p_ip_j} |u_i\otimes u_i\rangle \langle u_j\otimes u_j|)\\
+&=\sum_{i=1}^n \sum_{j=1}^n \sqrt{p_ip_j} \operatorname{Tr}_2(|u_i\otimes u_i\rangle \langle u_j\otimes u_j|)\\
+&=\sum_{i=1}^n \sum_{j=1}^n \sqrt{p_ip_j} \langle u_i|u_j\rangle |u_i\rangle\langle u_i|\\
+&=\sum_{i=1}^n \sum_{j=1}^n \sqrt{p_ip_j} \delta_{ij} |u_i\rangle\langle u_i|\\
+&=\sum_{i=1}^n p_i |u_i\rangle\langle u_i|\\
+&=\rho
+\end{aligned}
+$$
+
+is a pure state.
+
+QED
 </details>
-
-## MM space
-

content/Math401/Math401_P1_2.md

@@ -1,7 +1,75 @@
-# Math 401, Paper 1, Side note 2: Page's lemma
+# Math 401 Paper 1, Side note 2: Page's lemma
 
 The page's lemma is a fundamental result in quantum information theory that provides a lower bound on the probability of error in a quantum channel.
 
+## Basic definitions
+
+### $SO(n)$
+
+The special orthogonal group $SO(n)$ is the set of all **distance preserving** linear transformations on $\mathbb{R}^n$.
+
+It is the group of all $n\times n$ orthogonal matrices ($A^T A=I_n$) on $\mathbb{R}^n$ with determinant $1$.
+
+$$
+SO(n)=\{A\in \mathbb{R}^{n\times n}: A^T A=I_n, \det(A)=1\}
+$$
+
+<details>
+<summary>Extensions</summary>
+
+In [The random Matrix Theory of the Classical Compact groups](https://case.edu/artsci/math/esmeckes/Haar_book.pdf), the author gives a more general definition of the Haar measure on the compact group $SO(n)$,
+
+$O(n)$ (the group of all $n\times n$ **orthogonal matrices** over $\mathbb{R}$),
+
+$$
+O(n)=\{A\in \mathbb{R}^{n\times n}: AA^T=A^T A=I_n\}
+$$
+
+$U(n)$ (the group of all $n\times n$ **unitary matrices** over $\mathbb{C}$), 
+
+$$
+U(n)=\{A\in \mathbb{C}^{n\times n}: A^*A=AA^*=I_n\}
+$$
+
+Recall that $A^*$ is the complex conjugate transpose of $A$.
+
+$SU(n)$ (the group of all $n\times n$ unitary matrices over $\mathbb{C}$ with determinant $1$), 
+
+$$
+SU(n)=\{A\in \mathbb{C}^{n\times n}: A^*A=AA^*=I_n, \det(A)=1\}
+$$
+
+$Sp(2n)$ (the group of all $2n\times 2n$ symplectic matrices over $\mathbb{C}$),
+
+$$
+Sp(2n)=\{U\in U(2n): U^T J U=UJU^T=J\}
+$$
+
+where $J=\begin{pmatrix}
+0 & I_n \\
+-I_n & 0
+\end{pmatrix}$ is the standard symplectic matrix.
+
+</details>
+
+### Haar measure
+
+Let $(SO(n), \| \cdot \|, \mu)$ be a metric measure space where $\| \cdot \|$ is the [Hilbert-Schmidt norm](https://notenextra.trance-0.com/Math401/Math401_T2#definition-of-hilbert-schmidt-norm) and $\mu$ is the measure function.
+
+The Haar measure on $SO(n)$ is the unique probability measure that is invariant under the action of $SO(n)$ on itself.
+
+That is also called _translation-invariant_.
+
+That is, fixing $B\in SO(n)$, $\forall A\in SO(n)$, $\mu(A\cdot B)=\mu(B\cdot A)=\mu(B)$.
+
+The Haar measure is the unique probability measure that is invariant under the action of $SO(n)$ on itself.
+
+_The existence and uniqueness of the Haar measure is a theorem in compact lie group theory. For this research topic, we will not prove it._
+
+### Sub-Gaussian concentration
+
+### Random sampling on the $CP^n$
+
 ## Statement
 
 Choosing a random pure quantum state $\rho$ from the bi-partite pure state space $\mathcal{H}_A\otimes\mathcal{H}_B$ with the uniform distribution, the expected entropy of the reduced state $\rho_A$ is:
@@ -19,7 +87,7 @@ unless $A$ and $B$ are separable.
 In the original paper, the entropy of the average state taken under the unitary invariant Haar measure is:
 
 $$
-S_{m,n}=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n}
+S_{m,n}=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n}\simeq \ln m-\frac{m}{2n}
 $$
 
 
@@ -27,6 +95,8 @@ $$
 
 ## References
 
+- [The random Matrix Theory of the Classical Compact groups](https://case.edu/artsci/math/esmeckes/Haar_book.pdf)
+
 - [Page's conjecture](https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.71.1291)
 
 - [Page's conjecture simple proof](https://journals.aps.org/pre/pdf/10.1103/PhysRevE.52.5653)

content/Math401/Math401_P1_3.md

@@ -1,10 +1,9 @@
-# Math 401, Paper 1, Side note 3: Levy's concentration theorem
+# Math 401 Paper 1, Side note 3: Levy's concentration theorem
 
 ## Basic definitions
 
 ### Lipschitz function
 
-
 #### $\eta$-Lipschitz function
 
 Let $(X,\operatorname{dist}_X)$ and $(Y,\operatorname{dist}_Y)$ be two metric spaces. A function $f:X\to Y$ is said to be $\eta$-Lipschitz if there exists a constant $L\in \mathbb{R}$ such that
@@ -17,12 +16,6 @@ for all $x,y\in X$. And $\eta=\|f\|_{\operatorname{Lip}}=\inf_{L\in \mathbb{R}}L
 
 That basically means that the function $f$ should not change the distance between any two pairs of points in $X$ by more than a factor of $L$.
 
-### Sub-Gaussian concentration
-
-### Random sampling on the $CP^n$
-
-
-
 ## Levy's concentration theorem in _High-dimensional probability_ by Roman Vershynin
 
 ### Levy's concentration theorem (Vershynin's version)

content/Math401/Math401_T1.md

@@ -1,4 +1,4 @@
-# Topic 1: Probability under language of measure theory
+# Math401 Topic 1: Probability under language of measure theory
 
 ## Section 1: Uniform Random Numbers
 

content/Math401/Math401_T2.md

@@ -1,4 +1,4 @@
-# Topic 2: Finite-dimensional Hilbert spaces
+# Math401 Topic 2: Finite-dimensional Hilbert spaces
 
 Recall the complex number is a tuple of two real numbers, $z=(a,b)$ with addition and multiplication defined by
 
@@ -461,7 +461,7 @@ $$
 
 The inner product is the standard Hermitian inner product in $\mathbb{C}^{n\times n}$.
 
-#### Definition of Hilbert-Schmidt norm
+#### Definition of Hilbert-Schmidt norm (also called Frobenius norm)
 
 The Hilbert-Schmidt norm of a linear operator $T: \mathscr{H}\to \mathscr{H}$ is defined by
 

content/Math401/Math401_T3.md

@@ -1,4 +1,4 @@
-# Topic 3: Separable Hilbert spaces
+# Math401 Topic 3: Separable Hilbert spaces
 
 ## Infinite-dimensional Hilbert spaces
 

content/Math401/Math401_T4.md

@@ -1,4 +1,4 @@
-# Topic 4: The quantum version of probabilistic concepts
+# Math401 Topic 4: The quantum version of probabilistic concepts
 
 > In mathematics, on often speaks of non-commutative instead of quantum constructions.
 
@@ -130,7 +130,8 @@ $$
 \operatorname{Prob}(P_1=1,P_3=0)\leq \operatorname{Prob}(P_1=1,P_2=0)+\operatorname{Prob}(P_2=1,P_3=0)
 $$
 
-Proof:
+<details>
+<summary>Proof</summary>
 
 By the law of total probability, (The event that the photon passes through the first filter but not the third filter is the union of the event that the photon did not pass through the second filter and the event that the photon passed the second filter and did not pass through the third filter) we have
 
@@ -168,6 +169,11 @@ $$
 \end{aligned}
 $$
 
+This is a contradiction, so Bell's inequality is violated.
+
+QED
+</details>
+
 Other revised experiments (eg. Aspect's experiment, Calcium entangled photon experiment) are also conducted and the inequality is still violated.
 
 #### The true model of light polarization

content/Math401/Math401_T5.md

@@ -1,4 +1,4 @@
-# Topic 5: Introducing dynamics: classical and non-commutative
+# Math401 Topic 5: Introducing dynamics: classical and non-commutative
 
 ## Section 1: Dynamics in classical probability
 

content/Math401/Math401_T6.md

@@ -1,4 +1,4 @@
-# Math 401, Topic 6: Postulates of quantum theory and measurement operations
+# Math401 Topic 6: Postulates of quantum theory and measurement operations
 
 ## Section 1: Postulates of quantum theory
 
@@ -118,6 +118,42 @@ If $\rho$ is a density operator on $\mathscr{H}$ given by: $\sum_{i=1}^l |w_i\ra
 
 ### Density operator of subsystems
 
+#### Partial trace for density operators
+
+Let $\rho$ be a density operator in $\mathscr{H}_1\otimes\mathscr{H}_2$, the partial trace of $\rho$ over $\mathscr{H}_2$ is the density operator in $\mathscr{H}_1$ (reduced density operator for the subsystem $\mathscr{H}_1$) given by:
+
+$$
+\rho_1\coloneqq\operatorname{Tr}_2(\rho)=\sum_{k=1}^r \lambda_k^2|v_k\rangle\langle v_k|
+$$
+
+<details>
+<summary>Examples</summary>
+
+Let $\rho=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle)$ be a density operator on $\mathscr{H}=\mathbb{C}^2\otimes \mathbb{C}^2$. 
+
+Expand the expression of $\rho$ in the basis of $\mathbb{C}^2\otimes\mathbb{C}^2$ using linear combination of basis vectors:
+
+$$
+\rho=\frac{1}{2}(|01\rangle\langle 01|+|01\rangle\langle 10|+|10\rangle\langle 01|+|10\rangle\langle 10|)
+$$
+
+Note $\operatorname{Tr}_2(|ab\rangle\langle cd|)=|a\rangle\langle c|\cdot \langle b|d\rangle$.
+
+Then the reduced density operator of the subsystem $\mathbb{C}^2$ in first qubit is, note the $\langle 0|0\rangle=\langle 1|1\rangle=1$ and $\langle 0|1\rangle=\langle 1|0\rangle=0$:
+
+$$
+\begin{aligned}
+\rho_1&=\operatorname{Tr}_2(\rho)\\
+&=\frac{1}{2}(\langle 1|1\rangle |0\rangle\langle 0|+\langle 0|1\rangle |0\rangle\langle 1|+\langle 1|0\rangle |1\rangle\langle 0|+\langle 0|0\rangle |1\rangle\langle 1|)\\
+&=\frac{1}{2}(|0\rangle\langle 0|+|1\rangle\langle 1|)\\
+&=\frac{1}{2}I
+\end{aligned}
+$$
+
+is a mixed state.
+
+</details>
+
 #### 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:

content/Math401/Math401_T7.md

@@ -1 +1 @@
-# Math 401, Topic 7: Basic of quantum circuits
+# Math401 Topic 7: Basic of quantum circuits