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domain verification test

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This commit is contained in:
Zheyuan Wu
2025-07-10 13:53:04 -05:00
parent 1871fd820a
commit 9d3115f089
7 changed files with 169 additions and 205 deletions
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11
Dockerfile
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@@ -12,9 +12,6 @@ RUN apk add --no-cache libc6-compat git
WORKDIR /app
# Initialize git repository to prevent Nextra warnings
RUN git init
# Install dependencies based on the preferred package manager
COPY package.json yarn.lock* package-lock.json* pnpm-lock.yaml* .npmrc* ./
RUN \
@@ -30,8 +27,7 @@ FROM base AS builder
WORKDIR /app
COPY --from=deps /app/node_modules ./node_modules
COPY . .
# Initialize git repository in builder stage as well
RUN git init
# This will do the trick, use the corresponding env file for each environment.
# COPY .env.production.sample .env.production
RUN npm run build
@@ -52,11 +48,6 @@ COPY --from=builder /app/public ./public
COPY --from=builder --chown=nextjs:nodejs /app/.next/standalone ./
COPY --from=builder --chown=nextjs:nodejs /app/.next/static ./.next/static
# Initialize git repository in production stage
RUN apk add --no-cache git && \
git init && \
chown -R nextjs:nodejs .git
USER nextjs
EXPOSE 3000

2
app/layout.tsx
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@@ -11,6 +11,7 @@ export const metadata = {
title: {
template: '%s - NoteNextra'
},
description: 'A static note sharing site for minimum care',
applicationName: 'NoteNextra',
generator: 'Next.js',
@@ -18,6 +19,7 @@ export const metadata = {
title: 'NoteNextra'
},
other: {
'algolia-site-verification': '7303797A38EAD6FC',
'msapplication-TileImage': '/ms-icon-144x144.png',
'msapplication-TileColor': '#fff'
},

103
content/Math401/Math401_P1.md
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@@ -54,105 +54,7 @@ $$
\operatorname{Pr}[|f(x)-M|>\epsilon]\leq \exp(-\frac{C(n-1)\epsilon^2}{\eta^2})
$$
Decomposing the statement in detail,
#### $\eta$-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 $L$.
> This theorem is exactly the 5.1.4 on the _High-dimensional probability_ by Roman Vershynin.
#### 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_.
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
$$
\operatorname{Pr}[f(X)\leq M+t]\geq 1-2\exp(-ct^2)
$$
> 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
$$
\kappa_n(\epsilon)=\frac{\int_\epsilon^{\frac{\pi}{2}}\cos^{n-1}(t)dt}{\int_0^{\frac{\pi}{2}}\cos^{n-1}(t)dt}
$$
Hardcore computing may generates the bound but M. Gromov did not make the detailed explanation here.
[Decomposing the statement in detail](Math401_P1_3.md)
### Random states and random subspaces
@@ -164,6 +66,8 @@ $$
\mathbb{E}[H(\psi_A)] \geq \log_2(d_A)-\frac{1}{2\ln(2)}\frac{d_A}{d_B}
$$
[Decomposing the statement in detail](Math401_P1_2.md)
From the Levy's lemma, we have
If we define $\beta=\frac{d_A}{\log_2(d_B)}$, then we have
@@ -191,7 +95,6 @@ $B$ performs a measurement on the combined state of the one qubit and the entang
$B$ decodes the result and obtains the 2 classical bits sent by $A$.
### Consequences for mixed state entanglement measures
#### Quantum mutual information

45
content/Math401/Math401_P1_1.md
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@@ -1,5 +1,44 @@
# Math 401, Paper 1, Side note 1: Quantum information theory and Measure concentration
## Typicality
> The idea of typicality in high-dimensions is very important topic in understanding this paper and taking it to the next level of detail under language of mathematics. I'm trying to comprehend these material and write down my understanding in this note.
Let $X$ be the alphabet of our source of information.
Let $x^n=x_1,x_2,\cdots,x_n$ be a sequence with $x_i\in X$.
We say that $x^n$ is $\epsilon$-typical with respect to $p(x)$ if
- For all $a\in X$ with $p(a)>0$, we have
$$
\|\frac{1}{n}N(a|x^n)-p(a)|\leq \frac{\epsilon}{\|X\|}
$$
- For all $a\in X$ with $p(a)=0$, we have
$$
N(a|x^n)=0
$$
Here $N(a|x^n)$ is the number of times $a$ appears in $x^n$. That's basically saying that:
1. The difference between **the probability of $a$ appearing in $x^n$** and the **probability of $a$ appearing in the source of information $p(a)$** should be within $\epsilon$ divided by the size of the alphabet $X$ of the source of information.
2. The probability of $a$ not appearing in $x^n$ should be 0.
Here are two easy propositions that can be proved:
For $\epsilon>0$, the probability of a sequence being $\epsilon$-typical goes to 1 as $n$ goes to infinity.
If $x^n$ is $\epsilon$-typical, then the probability of $x^n$ is produced is $2^{-n[H(X)+\epsilon]}\leq p(x^n)\leq 2^{-n[H(X)-\epsilon]}$.
The number of $\epsilon$-typical sequences is at least $2^{n[H(X)+\epsilon]}$.
Recall that $H(X)=-\sum_{a\in X}p(a)\log_2 p(a)$ is the entropy of the source of information.
## Shannon theory in Quantum information theory
Shannon theory provides a way to quantify the amount of information in a message.
Practically speaking:
@@ -11,5 +50,11 @@ Practically speaking:
- Features from linear structure:
- Entanglement and non-orthogonality
## Partial trace and purification
### Partial trace
## MM space

11
content/Math401/Math401_P1_2.md Normal file
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@@ -0,0 +1,11 @@
# 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.
## Statement
Let $\mathcal{H}$ be a Hilbert space and $\mathcal{L}(\mathcal{H})$ be the set of linear operators on $\mathcal{H}$.
Let $\mathcal{E}: \mathcal{L}(\mathcal{H}) \to \mathcal{L}(\mathcal{H})$ be a quantum channel.
Then, for any $\rho \in \mathcal{L}(\mathcal{H})$, we have:

107
content/Math401/Math401_P1_3.md Normal file
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@@ -0,0 +1,107 @@
# Math 401, Paper 1, Side note 3: Levy's concentration theorem
## Levy's concentration theorem in _High-dimensional probability_ by Roman Vershynin
### Levy's concentration theorem (Vershynin's version)
#### $\eta$-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 $L$.
> This theorem is exactly the 5.1.4 on the _High-dimensional probability_ by Roman Vershynin.
#### 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_.
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 in _Metric Structures for Riemannian and Non-Riemannian Spaces_ by M. Gromov
### 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
$$
\kappa_n(\epsilon)=\frac{\int_\epsilon^{\frac{\pi}{2}}\cos^{n-1}(t)dt}{\int_0^{\frac{\pi}{2}}\cos^{n-1}(t)dt}
$$
Hardcore computing may generates the bound but M. Gromov did not make the detailed explanation here.

95
package-lock.json generated
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