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+# Math 401, Fall 2025: Thesis notes, R1, Non-commutative probability theory
+
+> Progress: 0/NaN=NaN% (denominator and enumerator may change)
+
+## Notations and definitions
+
+This part will cover the necessary notations and definitions for the remaining parts of the recollection.
+
+### Notations of Hilbert space
+
+A Hilbert space is a vector space equipped with an inner product.
+
+### 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
+
+$$
+\operatorname{dist}_Y(f(x),f(y))\leq L\operatorname{dist}_X(x,y)
+$$
+
+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 $L$.
+
+### Operations on Hilbert space and Measurements
+
+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._
+
+### Random sampling on the $\mathbb{C}P^n$
+
+Note that the space of pure state in bipartite system
+
+## Non-commutative probability theory
+
+### Partial trace and purification
+
+#### Partial trace
+
+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 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.
+
+An operator $T$ on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$ can be written as (by the definition of [tensor product of linear operators](https://notenextra.trance-0.com/Math401/Math401_T2#tensor-products-of-linear-operators))
+
+$$
+T=\sum_{i=1}^n a_i A_i\otimes B_i
+$$
+
+where $A_i$ is a linear operator on $\mathscr{A}$ and $B_i$ is a linear operator on $\mathscr{B}$.
+
+The $\mathscr{B}$-partial trace of $T$ ($\operatorname{Tr}_{\mathscr{B}}(T):\mathcal{L}(\mathscr{A}\otimes \mathscr{B})\to \mathcal{L}(\mathscr{A})$) is the linear operator on $\mathscr{A}$ defined by
+
+$$
+\operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i
+$$
+
+#### Definition of 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.
+
+<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.
+
+</details>
+
+## Drawing the connection between the space $S^{2n+1}$, $CP^n$, and $\mathbb{R}$
+
+A pure quantum state of size $N$ can be identified with a **Hopf circle** on the sphere $S^{2N-1}$.
+
+A random pure state $|\psi\rangle$ of a bipartite $N\times K$ system such that $K\geq N\geq 3$.
+
+The partial trace of such system produces a mixed state $\rho(\psi)=\operatorname{Tr}_K(|\psi\rangle\langle \psi|)$, with induced measure $\mu_K$. When $K=N$, the induced measure $\mu_K$ is the Hilbert-Schmidt measure.
+
+Consider the function $f:S^{2N-1}\to \mathbb{R}$ defined by $f(x)=S(\rho(\psi))$, where $S(\cdot)$ is the von Neumann entropy. The Lipschitz constant of $f$ is $\sim \ln N$.
\ No newline at end of file
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+# Math 401, Fall 2025: Thesis notes, R2, Levy's concentration theorem and Levy's family
+
+> Progress: 2/5=40% (denominator and enumerator may change)
+
+## Levy's concentration theorem
+
+> [!TIP]
+>
+> This version of Levy's concentration theorem can be found in [Geometry of Quantum states](https://www.cambridge.org/core/books/geometry-of-quantum-states/46B62FE3F9DA6E0B4EDDAE653F61ED8C) 15.84 and 15.85.
+
+Our goal is to prove the generalized version of Levy's concentration theorem used in Hayden's work for $\eta$-Lipschitz functions.
+
+Let $f:S^{n-1}\to \mathbb{R}$ be a $\eta$-Lipschitz function. Let $M_f$ denote the median of $f$ and $\langle f\rangle$ denote the mean of $f$. (Note this can be generalized to many other manifolds.)
+
+Select a random point $x\in S^{n-1}$ with $n>2$ according to the uniform measure (Haar measure). Then the probability of observing a value of $f$ much different from the reference value is exponentially small.
+
+$$
+\operatorname{Pr}[|f(x)-M_f|>\epsilon]\leq \exp(-\frac{n\epsilon^2}{2\eta^2})
+$$
+$$
+\operatorname{Pr}[|f(x)-\langle f\rangle|>\epsilon]\leq 2\exp(-\frac{(n-1)\epsilon^2}{2\eta^2})
+$$
+
+### Levy's concentration theorem via sub-Gaussian concentration
+
+> [!TIP]
+>
+> This version of Levy's concentration theorem can be found in [High-dimensional probability](https://www.math.uci.edu/~rvershyn/papers/HDP-book/HDP-2.pdf) 5.1.4.
+
+#### Isoperimetric inequality on $\mathbb{R}^n$
+
+Among all subsets $A\subset \mathbb{R}^n$ with a given volume, the Euclidean ball has the minimal area.
+
+That is, for any $\epsilon>0$, Euclidean balls minimize the volume of the $\epsilon$-neighborhood of $A$.
+
+Where the volume of the $\epsilon$-neighborhood of $A$ is defined as
+
+$$
+A_\epsilon(A)\coloneqq \{x\in \mathbb{R}^n: \exists y\in A, \|x-y\|_2\leq \epsilon\}=A+\epsilon B_2^n
+$$
+
+Here the $\|\cdot\|_2$ is the Euclidean norm. (The theorem holds for both geodesic metric on sphere and Euclidean metric on $\mathbb{R}^n$.)
+
+#### Isoperimetric inequality on the sphere
+
+Let $\sigma_n(A)$ denotes the normalized area of $A$ on $n$ dimensional sphere $S^n$. That is $\sigma_n(A)\coloneqq\frac{\operatorname{Area}(A)}{\operatorname{Area}(S^n)}$.
+
+Let $\epsilon>0$. Then for any subset $A\subset S^n$, given the area $\sigma_n(A)$, the spherical caps minimize the volume of the $\epsilon$-neighborhood of $A$.
+
+> The above two inequalities is not proved in the Book _High-dimensional probability_. But you can find it in the Appendix C of Gromov's book _Metric Structures for Riemannian and Non-Riemannian Spaces_.
+
+To continue prove the theorem, we use sub-Gaussian concentration *(Chapter 3 of _High-dimensional probability_ by Roman Vershynin)* of sphere $\sqrt{n}S^n$. 
+
+This will leads to some constant $C>0$ such that the following lemma holds:
+
+#### The "Blow-up" lemma
+
+Let $A$ be a subset of sphere $\sqrt{n}S^n$, and $\sigma$ denotes the normalized area of $A$. Then if $\sigma\geq \frac{1}{2}$, then for every $t\geq 0$,
+
+$$
+\sigma(A_t)\geq 1-2\exp(-ct^2)
+$$
+
+where $A_t=\{x\in S^n: \operatorname{dist}(x,A)\leq t\}$ and $c$ is some positive constant.
+
+#### Proof of the Levy's concentration theorem
+
+Proof:
+
+Without loss of generality, we can assume that $\eta=1$. Let $M$ denotes the median of $f(X)$.
+
+So $\operatorname{Pr}[|f(X)\leq M|]\geq \frac{1}{2}$, and $\operatorname{Pr}[|f(X)\geq M|]\geq \frac{1}{2}$.
+
+Consider the sub-level set $A\coloneqq \{x\in \sqrt{n}S^n: |f(x)|\leq M\}$.
+
+Since $\operatorname{Pr}[X\in A]\geq \frac{1}{2}$, by the blow-up lemma, we have
+
+$$
+\operatorname{Pr}[X\in A_t]\geq 1-2\exp(-ct^2)
+$$
+
+And since
+
+$$
+\operatorname{Pr}[X\in A_t]\leq \operatorname{Pr}[f(X)\leq M+t]
+$$
+
+Combining the above two inequalities, we have
+
+$$
+\operatorname{Pr}[f(X)\leq M+t]\geq 1-2\exp(-ct^2)
+$$
+
+## Levy's concentration theorem via Levy family
+
+> [!TIP]
+>
+> This version of Levy's concentration theorem can be found in:
+> - [Metric Structures for Riemannian and Non-Riemannian Spaces by M. Gromov](https://www.amazon.com/Structures-Riemannian-Non-Riemannian-Progress-Mathematics/dp/0817638989/ref=tmm_hrd_swatch_0?_encoding=UTF8&dib_tag=se&dib=eyJ2IjoiMSJ9.Tp8dXvGbTj_D53OXtGj_qOdqgCgbP8GKwz4XaA1xA5PGjHj071QN20LucGBJIEps.9xhBE0WNB0cpMfODY5Qbc3gzuqHnRmq6WZI_NnIJTvc&qid=1750973893&sr=8-1)
+> - [Metric Measure Geometry by Takashi Shioya](https://arxiv.org/pdf/1410.0428)
+
+
+### Levy's concentration theorem (Gromov's version)
+
+> The Levy's lemma can also be found in _Metric Structures for Riemannian and Non-Riemannian Spaces_ by M. Gromov. $3\frac{1}{2}.19$ The Levy concentration theory.
+
+#### Theorem $3\frac{1}{2}.19$ Levy concentration theorem:
+
+An arbitrary 1-Lipschitz function $f:S^n\to \mathbb{R}$ concentrates near a single value $a_0\in \mathbb{R}$ as strongly as the distance function does.
+
+That is
+
+$$
+\mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp(-\frac{(n-1)\epsilon^2}{2})
+$$
+
+where 
+
+$$
+\kappa_n(\epsilon)=\frac{\int_\epsilon^{\frac{\pi}{2}}\cos^{n-1}(t)dt}{\int_0^{\frac{\pi}{2}}\cos^{n-1}(t)dt}
+$$
+
+$a_0$ is the **Levy mean** of function $f$, that is the level set of $f^{-1}:\mathbb{R}\to S^n$ divides the sphere into equal halves, characterized by the following equality:
+
+$$
+\mu(f^{-1}(-\infty,a_0])\geq \frac{1}{2} \text{ and } \mu(f^{-1}[a_0,\infty))\geq \frac{1}{2}
+$$
+
+Hardcore computing may generates the bound but M. Gromov did not make the detailed explanation here.
+
+> Detailed proof by Takashi Shioya.
+>
+> The central idea is to draw the connection between the given three topological spaces, $S^{2n+1}$, $CP^n$ and $\mathbb{R}$.
+
+First, we need to introduce the following distribution and lemmas/theorems:
+
+**OBSERVATION**
+
+consider the orthogonal projection from $\mathbb{R}^{n+1}$, the space where $S^n$ is embedded, to $\mathbb{R}^k$, we denote the restriction of the projection as $\pi_{n,k}:S^n(\sqrt{n})\to \mathbb{R}^k$. Note that $\pi_{n,k}$ is a 1-Lipschitz function (projection will never increase the distance between two points).
+
+We denote the normalized Riemannian volume measure on $S^n(\sqrt{n})$ as $\sigma^n(\cdot)$, and $\sigma^n(S^n(\sqrt{n}))=1$.
+
+#### Definition of Gaussian measure on $\mathbb{R}^k$
+
+We denote the Gaussian measure on $\mathbb{R}^k$ as $\gamma^k$.
+
+$$
+d\gamma^k(x)\coloneqq\frac{1}{\sqrt{2\pi}^k}\exp(-\frac{1}{2}\|x\|^2)dx
+$$
+
+$x\in \mathbb{R}^k$, $\|x\|^2=\sum_{i=1}^k x_i^2$ is the Euclidean norm, and $dx$ is the Lebesgue measure on $\mathbb{R}^k$.
+
+Basically, you can consider the Gaussian measure as the normalized Lebesgue measure on $\mathbb{R}^k$ with standard deviation $1$.
+
+#### Maxwell-Boltzmann distribution law
+
+> It is such a wonderful fact for me, that the projection of $n+1$ dimensional sphere with radius $\sqrt{n}$ to $\mathbb{R}^k$ is a Gaussian distribution as $n\to \infty$.
+
+For any natural number $k$,
+
+$$
+\frac{d(\pi_{n,k})_*\sigma^n(x)}{dx}\to \frac{d\gamma^k(x)}{dx}
+$$
+
+where $(\pi_{n,k})_*\sigma^n$ is the push-forward measure of $\sigma^n$ by $\pi_{n,k}$.
+
+In other words,
+
+$$
+(\pi_{n,k})_*\sigma^n\to \gamma^k\text{ weakly as }n\to \infty
+$$
+
+<details>
+<summary>Proof</summary>
+
+We denote the $n$ dimensional volume measure on $\mathbb{R}^k$ as $\operatorname{vol}_k$.
+
+Observe that $\pi_{n,k}^{-1}(x),x\in \mathbb{R}^k$ is isometric to $S^{n-k}(\sqrt{n-\|x\|^2})$, that is, for any $x\in \mathbb{R}^k$, $\pi_{n,k}^{-1}(x)$ is a sphere with radius $\sqrt{n-\|x\|^2}$ (by the definition of $\pi_{n,k}$).
+
+So,
+
+$$
+\begin{aligned}
+\frac{d(\pi_{n,k})_*\sigma^n(x)}{dx}&=\frac{\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(x))}{\operatorname{vol}_k(S^n(\sqrt{n}))}\\
+&=\frac{(n-\|x\|^2)^{\frac{n-k}{2}}}{\int_{\|x\|\leq \sqrt{n}}(n-\|x\|^2)^{\frac{n-k}{2}}dx}\\
+\end{aligned}
+$$
+
+as $n\to \infty$.
+
+note that $\lim_{n\to \infty}{(1-\frac{a}{n})^n}=e^{-a}$ for any $a>0$.
+
+$(n-\|x\|^2)^{\frac{n-k}{2}}=\left(n(1-\frac{\|x\|^2}{n})\right)^{\frac{n-k}{2}}\to n^{\frac{n-k}{2}}\exp(-\frac{\|x\|^2}{2})$
+
+So
+
+$$
+\begin{aligned}
+\frac{(n-\|x\|^2)^{\frac{n-k}{2}}}{\int_{\|x\|\leq \sqrt{n}}(n-\|x\|^2)^{\frac{n-k}{2}}dx}&=\frac{e^{-\frac{\|x\|^2}{2}}}{\int_{x\in \mathbb{R}^k}e^{-\frac{\|x\|^2}{2}}dx}\\
+&=\frac{1}{(2\pi)^{\frac{k}{2}}}e^{-\frac{\|x\|^2}{2}}\\
+&=\frac{d\gamma^k(x)}{dx}
+\end{aligned}
+$$
+
+QED
+
+</details>
+
+#### Proof of the Levy's concentration theorem via the Maxwell-Boltzmann distribution law
+
+We use the Maxwell-Boltzmann distribution law and Levy's isoperimetric inequality to prove the Levy's concentration theorem.
+
+The goal is the same as the Gromov's version, first we bound the probability of the sub-level set of $f$ by the $\kappa_n(\epsilon)$ function by Levy's isoperimetric inequality. Then we claim that the $\kappa_n(\epsilon)$ function is bounded by the Gaussian distribution.
+
+Note, this section is not rigorous enough in sense of mathematics and the author should add sections about Levy family and observable diameter to make the proof more rigorous and understandable.
+
+<details>
+<summary>Proof</summary>
+
+Let $f:S^n\to \mathbb{R}$ be a 1-Lipschitz function.
+
+Consider the two sets of points on the sphere $S^n$ with radius $\sqrt{n}$:
+
+$$
+\Omega_+=\{x\in S^n: f(x)\leq a_0-\epsilon\}, \Omega_-=\{x\in S^n: f(x)\geq a_0+\epsilon\}
+$$
+
+Note that $\Omega_+\cup \Omega_-$ is the whole sphere $S^n(\sqrt{n})$.
+
+By the Levy's isoperimetric inequality, we have
+
+$$
+\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\epsilon))\leq \operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\Omega_+))+\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\Omega_-))
+$$
+
+We define $\kappa_n(\epsilon)$ as the following:
+
+$$
+\kappa_n(\epsilon)=\frac{\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(\epsilon))}{\operatorname{vol}_k(S^n(\sqrt{n}))}=\frac{\int_\epsilon^{\frac{\pi}{2}}\cos^{n-1}(t)dt}{\int_0^{\frac{\pi}{2}}\cos^{n-1}(t)dt}
+$$
+
+By the Levy's isoperimetric inequality, and the Maxwell-Boltzmann distribution law, we have
+
+$$
+\mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp(-\frac{(n-1)\epsilon^2}{2})
+$$
+</details>
+
+## Levy's Isoperimetric inequality
+
+> This section is from the Appendix $C_+$ of Gromov's book _Metric Structures for Riemannian and Non-Riemannian Spaces_.
+
+Not very edible for undergraduates.
+
+## Crash course on Riemannian manifolds
+
+> This part might be extended to a separate note, let's check how far we can go from this part.
+>
+> References:
+>
+> - [Riemannian Geometry by John M. Lee](https://www.amazon.com/Introduction-Riemannian-Manifolds-Graduate-Mathematics/dp/3319917544?dib=eyJ2IjoiMSJ9.88u0uIXulwPpi3IjFn9EdOviJvyuse9V5K5wZxQEd6Rto5sCIowzEJSstE0JtQDW.QeajvjQEbsDmnEMfPzaKrfVR9F5BtWE8wFscYjCAR24&dib_tag=se&keywords=riemannian+manifold+by+john+m+lee&qid=1753238983&sr=8-1)
+
+### Riemannian manifolds
+
+A Riemannian manifold is a smooth manifold equipped with a **Riemannian metric**, which is a smooth assignment of an inner product to each tangent space $T_pM$ of the manifold.
+
+An example of Riemannian manifold is the sphere $\mathbb{C}P^n$.
+
+### Riemannian metric
+
+A Riemannian metric is a smooth assignment of an inner product to each tangent space $T_pM$ of the manifold.
+
+An example of Riemannian metric is the Euclidean metric on $\mathbb{R}^n$.
+
+### Notion of Connection
+
+A connection is a way to define the directional derivative of a vector field along a curve on a Riemannian manifold.
+
+For every $p\in M$, where $M$ denote the manifold, suppose $M=\mathbb{R}^n$, then let $X=(f_1,\cdots,f_n)$ be a vector field on $M$. The directional derivative of $X$ along the point $p$ is defined as
+
+$$
+D_VX=\lim_{h\to 0}\frac{X(p+h)-X(p)}{h}
+$$
+
+### Nabla notation and Levi-Civita connection
+
+
+### Ricci curvature
+
+
diff --git a/content/Math401/Extending_thesis/Math401_R3.md b/content/Math401/Extending_thesis/Math401_R3.md
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+# Math 401, Fall 2025: Thesis notes, R3, Page's lemma
+
+> Progress: 0/4=0% (denominator and enumerator may change)
+
+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.
+
+## 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:
+
+$$
+\mathbb{E}[H(\rho_A)]\geq \ln d_A -\frac{1}{2\ln 2} \frac{d_A}{d_B}
+$$
+
+## Page's conjecture
+
+A quantum system $AB$ with the Hilbert space dimension $mn$ in a pure state $\rho_{AB}$ has entropy $0$ but the entropy of the reduced state $\rho_A$, assume $m\leq n$, then entropy of the two subsystem $A$ and $B$ is greater than $0$.
+
+unless $A$ and $B$ are separable.
+
+In the original paper, the entropy of the average state taken under the unitary invariant Haar measure is:
+
+$$
+S_{m,n}=\sum_{k=n+1}^{mn}\frac{1}{k}-\frac{m-1}{2n}\simeq \ln m-\frac{m}{2n}
+$$
+
+## References to begin with
+
+- [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)
+
+- [Geometry of quantum states an introduction to quantum entanglement second edition](https://www.cambridge.org/core/books/geometry-of-quantum-states/46B62FE3F9DA6E0B4EDDAE653F61ED8C)
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diff --git a/content/Math401/Extending_thesis/Math401_R4.md b/content/Math401/Extending_thesis/Math401_R4.md
new file mode 100644
index 0000000..ba4e57c
--- /dev/null
+++ b/content/Math401/Extending_thesis/Math401_R4.md
@@ -0,0 +1,17 @@
+# Math 401, Fall 2025: Thesis notes, R4, Superdense coding and Quantum error correcting codes
+
+> Progress: 0/NaN=NaN% (denominator and enumerator may change)
+
+This part may not be a part of "mathematical" research. But that's what I initially begin with.
+
+## Superdense coding
+
+> [!TIP]
+>
+> A helpful resource is [The Functional Analysis of Quantum Information Theory](https://arxiv.org/pdf/1410.7188) Section 2.2
+>
+> Or another way in quantum computing [Quantum Computing and Quantum Information](https://www.cambridge.org/highereducation/books/quantum-computation-and-quantum-information/01E10196D0A682A6AEFFEA52D53BE9AE#overview) Section 2.3
+
+## Quantum error correcting codes
+
+This part is intentionally left blank and may be filled near the end of the semester, by assignments given in CSE5313.
diff --git a/content/Math401/Extending_thesis/Math401_S1.md b/content/Math401/Extending_thesis/Math401_S1.md
new file mode 100644
index 0000000..ae08885
--- /dev/null
+++ b/content/Math401/Extending_thesis/Math401_S1.md
@@ -0,0 +1,14 @@
+# Math 401, Fall 2025: Thesis notes, S1, Complex projective space.
+
+> [!CAUTION]
+>
+> In this section, without explicitly stated, all dimensions are in the complex field.
+
+A complex projective space is a space that is the set of all lines through the origin in a complex vector space.
+
+Described by that nature, there exists a natural definition of the complex projective space given as follows:
+
+$$
+\mathbb{C}P^n=\frac{\mathbb{C}^{n+1}\setminus\{0\}}{\sim}
+$$
+
diff --git a/content/Math401/Extending_thesis/Math401_S2.md b/content/Math401/Extending_thesis/Math401_S2.md
new file mode 100644
index 0000000..9496552
--- /dev/null
+++ b/content/Math401/Extending_thesis/Math401_S2.md
@@ -0,0 +1,25 @@
+# Math 401, Fall 2025: Thesis notes, S2, Majorana stellar representation of quantum states.
+
+## Majorana representation of quantum states
+
+> [!TIP]
+>
+> A helpful resource is [Geometry of Quantum states](https://www.cambridge.org/core/books/geometry-of-quantum-states/46B62FE3F9DA6E0B4EDDAE653F61ED8C) Section 4.4 and Chapter 7.
+
+Vectors in $\mathbb{C}^{n+1}$ can be represented by a set of $n$ degree polynomials.
+
+$$
+\vec{Z}=(Z_1,\cdots,Z_n)\sim w(z)=Z_0+Z_1z+\cdots+Z_nz^n
+$$
+
+If $Z_0\neq 0$, then we can rescale the polynomial to make $Z_0=1$.
+
+Therefore, points in $\mathbb{C}P^{n}$ will be one-to-one corresponding to the set of $n$ degree polynomials with $n$ complex roots.
+
+$$
+Z_0+Z_1z+\cdots+Z_nz^n=0=Z_0(z-z_1)(z-z_2)\cdots(z-z_n)
+$$
+
+If $Z_0=0$, then count $\infty$ as root.
+
+Using stereographic projection of each root we can get a unordered collection of $S^2$. Example: $\mathbb{C}P=S^2$, $\mathbb{C}p^2=S^2\times S^2\setminus S_2$ where $S_2$ is symmetric group.
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diff --git a/content/Math401/Extending_thesis/_meta.js b/content/Math401/Extending_thesis/_meta.js
new file mode 100644
index 0000000..8728ea8
--- /dev/null
+++ b/content/Math401/Extending_thesis/_meta.js
@@ -0,0 +1,3 @@
+export default {
+    Math401_S1: "Math 401, Fall 2025: Thesis notes, Section 1",
+}
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diff --git a/content/Math401/Extending_thesis/index.md b/content/Math401/Extending_thesis/index.md
new file mode 100644
index 0000000..790c27d
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+++ b/content/Math401/Extending_thesis/index.md
@@ -0,0 +1,33 @@
+# Math 401, Fall 2025: Overview of thesis
+
+This is a note base on first discussion with Prof. Feres on 2025-09-02
+
+Due to time constraint, our goal for this semester is to extend the study of concentration of measure effects described by Hayden's paper to Majorana **stellar representation of quantum states**.
+
+That is, we want to build connection between the system described by follows:
+
+## Bounding the entropy of the state via Levy's concentration theorem and Page's lemma
+
+Recall that the bipartite quantum states of $\mathcal{P}(A\otimes B)$. Assume $A$ has $\dim A=d_A$, $\dim B=d_B$, the the system is isomorphic to the complex projective space $\mathbb{C}P^{d_A d_B-1}$.
+
+Then over partial trace operations over $B$, we can obtain a mixed quantum state denoted by $S_A$ on the Hilbert space $A$.
+
+Then we measure the von Neumann entropy of $S_A$ to get the entropy of the state.
+
+From the Hayden's work, using analysis of Levy's concentration theorem, and Page's lemma, we can find that the entropy of the state is concentrated around a certain value which is close to maximally entangled state.
+
+---
+
+This project is incomplete due to several critical missing parts that I don't have comprehensive knowledge to fill in.
+
+One goal for this section of study is to fully investigate the missing parts and fill in the gaps. It is irrelevant to any one except me for trivial reasons. But I don't want to speak anything that I don't have a good understanding of.
+
+To achieve this goal, I will set up few side project that continue to investigate the missing parts, and the notes will start with letter `R`, for recollections.
+
+To make these sections self-contained. Some materials will be borrowed from other notes.
+
+## Bounding the entropy of the state via exploring Majorana stellar representation of quantum states
+
+As Professor Feres mentioned, we can further explore the Majorana stellar representation of quantum states to bound the entropy of the state.
+
+The new topics discovered will be noted with letter `S`. for stellar representation.
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diff --git a/content/Math401/Math401_P1.md b/content/Math401/Freiwald_summer/Math401_P1.md
similarity index 87%
rename from content/Math401/Math401_P1.md
rename to content/Math401/Freiwald_summer/Math401_P1.md
index adf294f..1adea2f 100644
--- a/content/Math401/Math401_P1.md
+++ b/content/Math401/Freiwald_summer/Math401_P1.md
@@ -1,6 +1,8 @@
 # Math 401 Paper 1: Concentration of measure effects in quantum information (Patrick Hayden)
 
-[PDF](https://www.ams.org/books/psapm/068/2762144)
+[Concentration of measure effects in quantum information](https://www.ams.org/books/psapm/068/2762144)
+
+A more comprehensive version of this paper is in [Aspect of generic entanglement](https://arxiv.org/pdf/quant-ph/0407049).
 
 ## Quantum codes
 
@@ -42,7 +44,9 @@ $\phi_{AB}=(I_{A'}\otimes \eta)\circ\omega_{AB}$
 
 (above formula is from the presentation of Patrick Hayden.)
 
-For now we ignore this part if we don't consider the application of the following sections. The detailed explanation will be added later.
+For now we ignore this part if we don't consider the application of the following sections. The detailed explanation will be added later (hopefully very soon).
+
+---
 
 ### Surprise in high-dimensional quantum systems
 
@@ -54,7 +58,7 @@ $$
 \operatorname{Pr}[|f(x)-M|>\epsilon]\leq \exp(-\frac{C(n-1)\epsilon^2}{\eta^2})
 $$
 
-[Decomposing the statement in detail](Math401_P1_3.md)
+[Decomposing the statement in detail as side note 3](Math401_P1_3.md)
 
 ### Random states and random subspaces
 
@@ -66,7 +70,7 @@ $$
 \mathbb{E}[H(\psi_A)] \geq \log_2(d_A)-\frac{1}{2\ln(2)}\frac{d_A}{d_B}
 $$
 
-[Decomposing the statement in detail](Math401_P1_2.md)
+[Decomposing the statement in detail as side note 2](Math401_P1_2.md)
 
 From the Levy's lemma, we have
 
@@ -77,6 +81,8 @@ $$
 $$
 where $C$ is a small constant and $d_B\geq d_A\geq 3$.
 
+> Noted in [Aspect of generic entanglement](https://arxiv.org/pdf/quant-ph/0407049) $C_3=(8\pi^2\ln(2))^{-1}$.
+
 #### ebits and qbits
 
 ### Superdense coding of quantum states
diff --git a/content/Math401/Math401_P1_1.md b/content/Math401/Freiwald_summer/Math401_P1_1.md
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rename from content/Math401/Math401_T6.md
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diff --git a/content/Math401/Math401_T7.md b/content/Math401/Freiwald_summer/Math401_T7.md
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diff --git a/content/Math401/Freiwald_summer/_meta.js b/content/Math401/Freiwald_summer/_meta.js
new file mode 100644
index 0000000..2a0df01
--- /dev/null
+++ b/content/Math401/Freiwald_summer/_meta.js
@@ -0,0 +1,16 @@
+export default {
+    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",
+    Math401_T4: "Math 401, Topic 4: The quantum version of probabilistic concepts",
+    Math401_T5: "Math 401, Topic 5: Introducing dynamics: classical and non-commutative",
+    Math401_T6: "Math 401, Topic 6: Postulates of quantum theory and measurement operations",
+    Math401_T7: "Math 401, Topic 7: Basic of quantum circuits",
+    "---":{
+        type: 'separator'
+    },
+    Math401_P1: "Math 401, Paper 1: Concentration of measure effects in quantum information (Patrick Hayden)",
+    Math401_P1_1: "Math 401, Paper 1, Side note 1: Quantum information theory and Measure concentration",
+    Math401_P1_2: "Math 401, Paper 1, Side note 2: Page's lemma",
+    Math401_P1_3: "Math 401, Paper 1, Side note 3: Levy's concentration theorem",
+}
\ No newline at end of file
diff --git a/content/Math401/_meta.js b/content/Math401/_meta.js
index 0df50d8..d7ae839 100644
--- a/content/Math401/_meta.js
+++ b/content/Math401/_meta.js
@@ -6,21 +6,6 @@ export default {
     Math401_N1: "Math 401, Notes 1",
     Math401_N2: "Math 401, Notes 2",
     Math401_N3: "Math 401, Notes 3",
-    "---":{
-        type: 'separator'
-    },
-    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",
-    Math401_T4: "Math 401, Topic 4: The quantum version of probabilistic concepts",
-    Math401_T5: "Math 401, Topic 5: Introducing dynamics: classical and non-commutative",
-    Math401_T6: "Math 401, Topic 6: Postulates of quantum theory and measurement operations",
-    Math401_T7: "Math 401, Topic 7: Basic of quantum circuits",
-    "---":{
-        type: 'separator'
-    },
-    Math401_P1: "Math 401, Paper 1: Concentration of measure effects in quantum information (Patrick Hayden)",
-    Math401_P1_1: "Math 401, Paper 1, Side note 1: Quantum information theory and Measure concentration",
-    Math401_P1_2: "Math 401, Paper 1, Side note 2: Page's lemma",
-    Math401_P1_3: "Math 401, Paper 1, Side note 3: Levy's concentration theorem",
+    Freiwald_summer: "Math 401, Summer 2025: Freiwald research project notes",
+    Extending_thesis: "Math 401, Fall 2025: Thesis notes",
 }
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