partial updates

latex/.gitignore

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+# ---> TeX
+## Core latex/pdflatex auxiliary files:
+*.aux
+*.lof
+*.log
+*.lot
+*.fls
+*.out
+*.toc
+*.fmt
+*.fot
+*.cb
+*.cb2
+.*.lb
+
+## Intermediate documents:
+*.dvi
+*.xdv
+*-converted-to.*
+# these rules might exclude image files for figures etc.
+# *.ps
+# *.eps
+# *.pdf
+
+## Generated if empty string is given at "Please type another file name for output:"
+.pdf
+
+## Bibliography auxiliary files (bibtex/biblatex/biber):
+*.bbl
+*.bbl-SAVE-ERROR
+*.bcf
+*.blg
+*-blx.aux
+*-blx.bib
+*.run.xml
+
+## Build tool auxiliary files:
+*.fdb_latexmk
+*.synctex
+*.synctex(busy)
+*.synctex.gz
+*.synctex.gz(busy)
+*.pdfsync
+*.rubbercache
+rubber.cache
+
+## Build tool directories for auxiliary files
+# latexrun
+latex.out/
+
+## Auxiliary and intermediate files from other packages:
+# algorithms
+*.alg
+*.loa
+
+# achemso
+acs-*.bib
+
+# amsthm
+*.thm
+
+# beamer
+*.nav
+*.pre
+*.snm
+*.vrb
+
+# changes
+*.soc
+
+# comment
+*.cut
+
+# cprotect
+*.cpt
+
+# elsarticle (documentclass of Elsevier journals)
+*.spl
+
+# endnotes
+*.ent
+
+# fixme
+*.lox
+
+# feynmf/feynmp
+*.mf
+*.mp
+*.t[1-9]
+*.t[1-9][0-9]
+*.tfm
+
+#(r)(e)ledmac/(r)(e)ledpar
+*.end
+*.?end
+*.[1-9]
+*.[1-9][0-9]
+*.[1-9][0-9][0-9]
+*.[1-9]R
+*.[1-9][0-9]R
+*.[1-9][0-9][0-9]R
+*.eledsec[1-9]
+*.eledsec[1-9]R
+*.eledsec[1-9][0-9]
+*.eledsec[1-9][0-9]R
+*.eledsec[1-9][0-9][0-9]
+*.eledsec[1-9][0-9][0-9]R
+
+# glossaries
+*.acn
+*.acr
+*.glg
+*.glo
+*.gls
+*.glsdefs
+*.lzo
+*.lzs
+*.slg
+*.slo
+*.sls
+
+# uncomment this for glossaries-extra (will ignore makeindex's style files!)
+# *.ist
+
+# gnuplot
+*.gnuplot
+*.table
+
+# gnuplottex
+*-gnuplottex-*
+
+# gregoriotex
+*.gaux
+*.glog
+*.gtex
+
+# htlatex
+*.4ct
+*.4tc
+*.idv
+*.lg
+*.trc
+*.xref
+
+# hypdoc
+*.hd
+
+# hyperref
+*.brf
+
+# knitr
+*-concordance.tex
+# TODO Uncomment the next line if you use knitr and want to ignore its generated tikz files
+# *.tikz
+*-tikzDictionary
+
+# listings
+*.lol
+
+# luatexja-ruby
+*.ltjruby
+
+# makeidx
+*.idx
+*.ilg
+*.ind
+
+# minitoc
+*.maf
+*.mlf
+*.mlt
+*.mtc[0-9]*
+*.slf[0-9]*
+*.slt[0-9]*
+*.stc[0-9]*
+
+# minted
+_minted*
+*.pyg
+
+# morewrites
+*.mw
+
+# newpax
+*.newpax
+
+# nomencl
+*.nlg
+*.nlo
+*.nls
+
+# pax
+*.pax
+
+# pdfpcnotes
+*.pdfpc
+
+# sagetex
+*.sagetex.sage
+*.sagetex.py
+*.sagetex.scmd
+
+# scrwfile
+*.wrt
+
+# svg
+svg-inkscape/
+
+# sympy
+*.sout
+*.sympy
+sympy-plots-for-*.tex/
+
+# pdfcomment
+*.upa
+*.upb
+
+# pythontex
+*.pytxcode
+pythontex-files-*/
+
+# tcolorbox
+*.listing
+
+# thmtools
+*.loe
+
+# TikZ & PGF
+*.dpth
+*.md5
+*.auxlock
+
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+*.ptc
+
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+*.tdo
+
+# vhistory
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+
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+*.lod
+
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+*.xcp
+
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+*.xmpi
+
+# xindy
+*.xdy
+
+# xypic precompiled matrices and outlines
+*.xyc
+*.xyd
+
+# endfloat
+*.ttt
+*.fff
+
+# Latexian
+TSWLatexianTemp*
+
+## Editors:
+# WinEdt
+*.bak
+*.sav
+
+# Texpad
+.texpadtmp
+
+# LyX
+*.lyx~
+
+# Kile
+*.backup
+
+# gummi
+.*.swp
+
+# KBibTeX
+*~[0-9]*
+
+# TeXnicCenter
+*.tps
+
+# auto folder when using emacs and auctex
+./auto/*
+*.el
+
+# expex forward references with \gathertags
+*-tags.tex
+
+# standalone packages
+*.sta
+
+# Makeindex log files
+*.lpz
+
+# xwatermark package
+*.xwm
+
+# REVTeX puts footnotes in the bibliography by default, unless the nofootinbib
+# option is specified. Footnotes are the stored in a file with suffix Notes.bib.
+# Uncomment the next line to have this generated file ignored.
+#*Notes.bib
+
+# additional trash files
+*.bcf-*
+
+# python
+__pycache__

latex/LICENSE

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+MIT License
+
+Copyright (c) 2026 Trance-0
+
+Permission is hereby granted, free of charge, to any person obtaining a copy of this software and 
+associated documentation files (the "Software"), to deal in the Software without restriction, including 
+without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell 
+copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the 
+following conditions:
+
+The above copyright notice and this permission notice shall be included in all copies or substantial 
+portions of the Software.
+
+THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT 
+LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO 
+EVENT SHALL THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER LIABILITY, WHETHER 
+IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE 
+USE OR OTHER DEALINGS IN THE SOFTWARE.

latex/chapters/chap1.tex

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+% chapters/chap1.tex
+\documentclass[../main.tex]{subfiles}
+
+% If this chapter is compiled *by itself*, we must load only its own .bib
+% and print its bibliography at the end of the chapter.
+\ifSubfilesClassLoaded{
+  \addbibresource{../main.bib}
+}
+
+\usepackage{amsmath, amsfonts, amsthm}
+\usepackage{fancyhdr,parskip}
+\usepackage{fullpage}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% add special notation supports
+\usepackage[mathscr]{euscript}
+\usepackage{mathtools}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% add image package and directory
+\usepackage{graphicx}
+\usepackage{tikz}
+\graphicspath{{../images/}}
+
+
+\begin{document}
+
+
+\chapter{Concentration of Measure And Quantum Entanglement}
+
+
+First, we will build the mathematical model describing the behavior of quantum system and why they makes sense for physicists and meaningful for general publics.
+
+\section{Motivation}
+
+First, we introduce a motivation for introducing non-commutative probability theory to the study of quantum mechanics. This section is mainly based on the book~\cite{kummer1998elements}.
+
+\subsection{Light polarization and the violation of Bell's inequality}
+
+The light which comes through a polarizer is polarized in a certain direction. If we fix the first filter and rotate the second filter, we will observe the intensity of the light will change.
+
+The light intensity decreases with $\alpha$ (the angle between the two filters). The light should vanish when $\alpha=\pi/2$.
+
+However, for a system of 3 polarizing filters $F_1,F_2,F_3$, having directions $\alpha_1,\alpha_2,\alpha_3$, if we put them on the optical bench in pairs, then we will have three random variables $P_1,P_2,P_3$.
+
+\begin{figure}[h]
+	\centering
+	\includegraphics[width=0.7\textwidth]{Filter_figure.png}
+	\caption{The light polarization experiment, image from \cite{kummer1998elements}}
+	\label{fig:Filter_figure}
+\end{figure}
+
+\begin{theorem}
+	\label{theorem:Bell's_3_variable_inequality}
+	Bell's 3 variable inequality:
+
+	For any three random variables $P_1,P_2,P_3$ in a classical probability space, we have
+
+	$$
+	\operatorname{Prob}(P_1=1,P_3=0)\leq \operatorname{Prob}(P_1=1,P_2=0)+\operatorname{Prob}(P_2=1,P_3=0)
+	$$
+\end{theorem}
+
+\begin{proof}
+	By the law of total probability there are only two possibility if we don't observe any light passing the filter pair $F_i,F_j$, it means the photon is either blocked by $F_i$ or $F_j$, it means
+
+    $$
+    \begin{aligned}
+    \operatorname{Prob}(P_1=1,P_3=0)&=\operatorname{Prob}(P_1=1,P_2=0,P_3=0)\\
+    &+\operatorname{Prob}(P_1=1,P_2=1,P_3=0)\\
+    &\leq\operatorname{Prob}(P_1=1,P_2=0)+\operatorname{Prob}(P_2=1,P_3=0)
+    \end{aligned}
+    $$
+\end{proof}
+
+However, according to our experimental measurement, for any pair of polarizers $F_i,F_j$, by the complement rule, we have
+$$
+\begin{aligned}
+\operatorname{Prob}(P_i=1,P_j=0)&=\operatorname{Prob}(P_i=1)-\operatorname{Prob}(P_i=1,P_j=1)\\
+&=\frac{1}{2}-\frac{1}{2}\cos^2(\alpha_i-\alpha_j)\\
+&=\frac{1}{2}\sin^2(\alpha_i-\alpha_j)
+\end{aligned}
+$$
+
+This leads to a contradiction if we apply the inequality to the experimental data.
+
+$$
+\frac{1}{2}\sin^2(\alpha_1-\alpha_3)\leq\frac{1}{2}\sin^2(\alpha_1-\alpha_2)+\frac{1}{2}\sin^2(\alpha_2-\alpha_3)
+$$
+
+If $\alpha_1=0,\alpha_2=\frac{\pi}{6},\alpha_3=\frac{\pi}{3}$, then
+
+$$
+\begin{aligned}
+\frac{1}{2}\sin^2(-\frac{\pi}{3})&\leq\frac{1}{2}\sin^2(-\frac{\pi}{6})+\frac{1}{2}\sin^2(\frac{\pi}{6}-\frac{\pi}{3})\\
+\frac{3}{8}&\leq\frac{1}{8}+\frac{1}{8}\\
+\frac{3}{8}&\leq\frac{1}{4}
+\end{aligned}
+$$
+
+Other revised experiments (e.g., Aspect's experiment, calcium entangled photon experiment) are also conducted and the inequality is still violated.
+
+\subsection{The true model of light polarization}
+    
+The full description of the light polarization is given below:
+
+State of polarization of a photon: $\psi=\alpha|0\rangle+\beta|1\rangle$, where $|0\rangle$ and $|1\rangle$ are the two orthogonal polarization states in $\mathbb{C}^2$.
+
+Polarization filter (generalized 0,1 valued random variable): orthogonal projection $P_\alpha$ on $\mathbb{C}^2$ corresponding to the direction $\alpha$ (operator satisfies $P_\alpha^*=P_\alpha=P_\alpha^2$).
+
+The matrix representation of $P_\alpha$ is given by
+
+$$
+P_\alpha=\begin{pmatrix}
+\cos^2(\alpha) & \cos(\alpha)\sin(\alpha)\\
+\cos(\alpha)\sin(\alpha) & \sin^2(\alpha)
+\end{pmatrix}
+$$
+
+Probability of a photon passing through the filter $P_\alpha$ is given by $\langle P_\alpha\psi,\psi\rangle$; this is $\cos^2(\alpha)$ if we set $\psi=|0\rangle$.
+
+Since the probability of a photon passing through the three filters is not commutative, it is impossible to discuss $\operatorname{Prob}(P_1=1,P_3=0)$ in the classical setting.
+
+We now show how the experimentally observed probability
+$$
+\frac{1}{2}\sin^2(\alpha_i-\alpha_j)
+$$
+arises from the operator model.
+
+Assume the incoming light is \emph{unpolarized}. It is therefore described by
+the density matrix
+$$
+\rho=\frac{1}{2} I .
+$$
+
+Let $P_{\alpha_i}$ and $P_{\alpha_j}$ be the orthogonal projections corresponding
+to the two polarization filters with angles $\alpha_i$ and $\alpha_j$.
+
+The probability that a photon passes the first filter $P_{\alpha_i}$ is given by the Born rule:
+
+$$
+\operatorname{Prob}(P_i=1)
+=\operatorname{tr}(\rho P_{\alpha_i})
+=\frac{1}{2} \operatorname{tr}(P_{\alpha_i})
+=\frac{1}{2}
+$$
+
+If the photon passes the first filter, the post-measurement state is given by the L\"uders rule:
+
+$$
+\rho \longmapsto
+\rho_i
+=\frac{P_{\alpha_i}\rho P_{\alpha_i}}{\operatorname{tr}(\rho P_{\alpha_i})}
+= P_{\alpha_i}.
+$$
+
+The probability that the photon then passes the second filter is
+
+$$
+\operatorname{Prob}(P_j=1 \mid P_i=1)
+=\operatorname{tr}(P_{\alpha_i} P_{\alpha_j})
+=\cos^2(\alpha_i-\alpha_j).
+$$
+
+Hence, the probability that the photon passes $P_{\alpha_i}$ and is then blocked by $P_{\alpha_j}$ is
+
+$$
+\begin{aligned}
+\operatorname{Prob}(P_i=1, P_j=0)
+&= \operatorname{Prob}(P_i=1)
+   - \operatorname{Prob}(P_i=1, P_j=1) \\
+&= \frac12 - \frac12 \cos^2(\alpha_i-\alpha_j) \\
+&= \frac12 \sin^2(\alpha_i-\alpha_j).
+\end{aligned}
+$$
+
+This agrees with the experimentally observed transmission probabilities, but it should be emphasized that this quantity corresponds to a \emph{sequential measurement} rather than a joint probability in the classical sense.
+
+\section{Concentration of measure phenomenon}
+
+\begin{defn}
+	$\eta$-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 $\eta=\|f\|_{\operatorname{Lip}}=\inf_{L\in \mathbb{R}}L$.
+\end{defn}
+
+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 is a stronger condition than continuity, every Lipschitz function is continuous, but not every continuous function is Lipschitz.
+
+\begin{lemma}
+	\label{lemma:isoperimetric_inequality_on_sphere}
+	Isoperimetric inequality on the sphere:
+
+    Let $\sigma_n(A)$ denote the normalized area of $A$ on the $n$-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 $A$.
+
+    Suppose $\sigma^n(\cdot)$ is the normalized volume measure on the sphere $S^n(1)$, then for any closed subset $\Omega\subset S^n(1)$, we take a metric ball $B_\Omega$ of $S^n(1)$ with $\sigma^n(B_\Omega)=\sigma^n(\Omega)$. Then we have
+
+    $$
+    \sigma^n(U_r(\Omega))\geq \sigma^n(U_r(B_\Omega))
+    $$
+
+    where $U_r(A)=\{x\in X:d(x,A)< r\}$
+\end{lemma}
+
+Intuitively, the lemma means that the spherical caps are the most efficient way to cover the sphere.
+
+Here, the efficiency is measured by the epsilon-neighborhood of the boundary of the spherical cap.
+
+To prove the lemma, we need to have a good understanding of the Riemannian geometry of the sphere. For now, let's just take the lemma for granted.
+
+\subsection{Levy's concentration theorem}
+
+\begin{theorem}
+	\label{theorem:Levy's_concentration_theorem}
+	Levy's 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,
+    $$
+    \mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp\left(-\frac{(n-1)\epsilon^2}{2}\right)
+    $$
+    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 \textbf{Levy mean} of function $f$, that is, the level set $f^{-1}:\mathbb{R}\to S^n$ divides the sphere into equal halves, characterized by the following equality:
+    $$
+    \mu(f^{-1}(-\infty,a_0])\geq \frac{1}{2} \text{ and } \mu(f^{-1}[a_0,\infty))\geq \frac{1}{2}
+    $$
+\end{theorem}
+
+We will prove the theorem via the Maxwell-Boltzmann distribution law in this section for simplicity. ~\cite{shioya2014metricmeasuregeometry} The theorem will be discussed later in more general cases.
+
+\begin{defn}
+	\label{defn:Gaussian_measure}
+	Gaussian measure:
+
+    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 $\mathbb{R}^k$.
+    
+\end{defn}
+
+Basically, you can consider the Gaussian measure as the normalized Lebesgue measure on $\mathbb{R}^k$ with standard deviation $1$.
+
+It also has another name, the Projective limit theorem.~\cite{romanvershyni}
+
+If $X\sim \operatorname{Unif}(S^n(\sqrt{n}))$, then for any fixed unit vector $x$ we have $\langle X,x\rangle\to N(0,1)$ in distribution as $n\to \infty$.
+
+\begin{figure}[h]
+    \centering
+    \includegraphics[width=0.8\textwidth]{../images/maxwell.png}
+    \caption{Maxwell-Boltzmann distribution law, image from \cite{romanvershyni}}
+    \label{fig:Maxwell-Boltzmann_distribution_law}
+\end{figure}
+
+\begin{lemma}
+	\label{lemma:Maxwell-Boltzmann_distribution_law}
+    Maxwell-Boltzmann distribution law:
+    
+    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,
+    $$
+    (\pi_{n,k})_*\sigma^n\to \gamma^k\text{ weakly as }n\to \infty
+    $$
+\end{lemma}
+
+\begin{proof}
+    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
+    $$
+    \begin{aligned}
+    \frac{(n-\|x\|^2)^{\frac{n-k}{2}}}{\int_{\|x\|\leq \sqrt{n}}(n-\|x\|^2)^{\frac{n-k}{2}}dx}&=\frac{e^{-\frac{\|x\|^2}{2}}}{\int_{x\in \mathbb{R}^k}e^{-\frac{\|x\|^2}{2}}dx}\\
+    &=\frac{1}{(2\pi)^{\frac{k}{2}}}e^{-\frac{\|x\|^2}{2}}\\
+    &=\frac{d\gamma^k(x)}{dx}
+    \end{aligned}
+    $$
+\end{proof}
+
+Now we can prove Levy's concentration theorem, the proof is from~\cite{shioya2014metricmeasuregeometry}.
+
+\begin{proof}
+    Let $f_n:S^n(\sqrt{n})\to \mathbb{R}$, $n=1,2,\ldots$, be 1-Lipschitz functions.
+
+    Let $x$ and $x'$ be two given real numbers and $\gamma^1(-\infty,x]=\overline{\sigma}_\infty[-\infty,x']$, suppose $\sigma_\infty\{x'\}=0$, where $\{\sigma_i\}$ is a sequence of Borel probability measures on $\mathbb{R}$.
+
+    We want to show that, for all non-negative real numbers $\epsilon_1$ and $\epsilon_2$.
+
+    $$
+    \sigma_\infty[x'-\epsilon_1,x'+\epsilon_2]\geq \gamma^1[x-\epsilon_1,x+\epsilon_2]
+    $$
+    
+    Consider the two spherical cap $\Omega_+\coloneq \{f_{n_i}\geq x'\}$ and $\Omega_-\coloneq \{f_{n_i}\leq x\}$. Note that $\Omega_+\cup \Omega_-=S^{n_i}(\sqrt{n_i})$.
+
+    It is sufficient to show that,
+
+    $$
+    U_{\epsilon_1}(\Omega_+)\cup U_{\epsilon_2}(\Omega_-)\subset \{x'-\epsilon_1\leq f_{n_i}\leq x'+\epsilon_2\}
+    $$
+
+    By 1-Lipschitz continuity of $f_{n_i}$, we have for all $\zeta\in U_{\epsilon_1}(\Omega_+)$, there is a point $\xi\in \Omega_+$ such that $d(\zeta,\xi)\leq \epsilon_1$. So $U_{\epsilon_1}(\Omega_+)\subset \{f_{n_i}\geq x'-\epsilon_1\}$. With the same argument, we have $U_{\epsilon_2}(\Omega_-)\subset \{f_{n_i}\leq x+\epsilon_2\}$.
+
+    So the push-forward measure of $(f_{n_i})_*\sigma^{n_i}$ of $[x'-\epsilon_1,x'+\epsilon_2]$ is
+
+    $$
+    \begin{aligned}
+    (f_{n_i})_*\sigma^{n_i}[x'-\epsilon_1,x'+\epsilon_2]&=\sigma^{n_i}(x'-\epsilon_1\leq f_{n_i}\leq x'+\epsilon_2)\\
+    &\geq \sigma^{n_i}(U_{\epsilon_1}(\Omega_+)\cap U_{\epsilon_2}(\Omega_-))\\
+    &=\sigma^{n_i}(U_{\epsilon_1}(\Omega_+))+\sigma^{n_i}(U_{\epsilon_2}(\Omega_-))-1\\
+    \end{aligned}
+    $$
+
+    By the lemma~\ref{lemma:isoperimetric_inequality_on_sphere}, we have
+
+    $$
+    \sigma^{n_i}(U_{\epsilon_1}(\Omega_+))\geq \sigma^{n_i}(U_{\epsilon_1}(B_{\Omega_+}))\quad \text{and} \quad \sigma^{n_i}(U_{\epsilon_2}(\Omega_-))\geq \sigma^{n_i}(U_{\epsilon_2}(B_{\Omega_-}))
+    $$
+
+    By the lemma~\ref{lemma:Maxwell-Boltzmann_distribution_law}, we have
+
+    $$
+    \sigma^{n_i}(U_{\epsilon_1}(\Omega_+))+\sigma^{n_i}(U_{\epsilon_2}(\Omega_-))\to \gamma^1[x'-\epsilon_1,x'+\epsilon_2]+\gamma^1[x-\epsilon_1,x+\epsilon_2]
+    $$
+
+    Therefore,
+
+    $$
+    \begin{aligned}
+    \sigma_\infty[x'-\epsilon_1,x'+\epsilon_2]&\geq \liminf_{i\to \infty}(f_{n_i})_*\sigma^{n_i}[x'-\epsilon_1,x'+\epsilon_2]\\
+    &\geq \gamma^1[x'-\epsilon_1,\infty)\cap \gamma^1(-\infty,x+\epsilon_2]-1\\
+    &=\gamma^1[x-\epsilon_1,x+\epsilon_2]
+    \end{aligned}
+    $$
+
+\end{proof}
+
+The full proof of Levy's concentration theorem requires more digestion for cases where $\overline{\sigma}_\infty\neq \delta_{\pm\infty}$ but I don't have enough time to do so. This section may be filled in the next semester.
+
+\section{The application of the concentration of measure phenomenon in non-commutative probability theory}
+
+In quantum communication, we can pass classical bits by sending quantum states. However, by the indistinguishability (Proposition~\ref{prop:indistinguishability}) of quantum states, we cannot send an infinite number of classical bits over a single qubit. There exists a bound for zero-error classical communication rate over a quantum channel.
+
+\begin{theorem}
+	\label{theorem:Holevo_bound}
+	Holevo bound:
+
+	The maximal amount of classical information that can be transmitted by a quantum system is given by the Holevo bound. $\log_2(d)$ is the maximum amount of classical information that can be transmitted by a quantum system with $d$ levels (that is, basically, the number of qubits).
+\end{theorem}
+
+The proof of the Holevo bound can be found in~\cite{Nielsen_Chuang_2010}. In current state of the project, this theorem is not heavily used so we will not make annotated proof here.
+
+\subsection{Quantum communication}
+
+To surpass the Holevo bound, we need to use the entanglement of quantum states.
+
+\begin{defn}
+	\label{defn:Bell_state}
+	Bell state:
+        
+        The Bell states are the following four states:
+        
+        $$
+        |\Phi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle),\quad |\Phi^-\rangle=\frac{1}{\sqrt{2}}(|00\rangle-|11\rangle)
+        $$
+        $$
+        |\Psi^+\rangle=\frac{1}{\sqrt{2}}(|01\rangle+|10\rangle),\quad |\Psi^-\rangle=\frac{1}{\sqrt{2}}(|01\rangle-|10\rangle)
+        $$
+        These are a basis of the 2-qubit Hilbert space.
+\end{defn}
+
+
+\subsection{Superdense coding and entanglement}
+
+The description of the superdense coding can be found in~\cite{gupta2015functionalanalysisquantuminformation} and~\cite{Hayden}.
+
+Suppose $A$ and $B$ share a Bell state (or other maximally entangled state) $|\Phi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$, where $A$ holds the first part and $B$ holds the second part.
+        
+$A$ wishes to send 2 \textbf{classical bits} to $B$.
+
+$A$ performs one of four Pauli unitaries (some fancy quantum gates named X, Y, Z, I) on the combined state of entangled qubits $\otimes$ one qubit. Then $A$ sends the resulting one qubit to $B$.
+
+This operation extends the initial one entangled qubit to a system of one of four orthogonal Bell states.
+
+$B$ performs a measurement on the combined state of the one qubit and the entangled qubits he holds.
+
+$B$ decodes the result and obtains the 2 classical bits sent by $A$.
+
+\begin{figure}[h]
+	\centering
+	\includegraphics[width=0.8\textwidth]{superdense_coding.png}
+	\caption{Superdense coding, image from \cite{Hayden}}
+	\label{fig:superdense_coding}
+\end{figure}
+
+Note that superdense coding is a way to send 2 classical bits of information by sending 1 qubit with 1 entangled qubit. \textbf{The role of the entangled qubit} is to help them to distinguish the 4 possible states of the total 3 qubits system where 2 of them (the pair of entangled qubits) are mathematically the same.
+
+Additionally, no information can be gained by measuring a pair of entangled qubits. To send information from  $A$ to $B$, we need to physically send the qubits from $A$ to $B$. That means, we cannot send information faster than the speed of light.
+
+% TODO: FILL the description of the superdense coding here.
+
+\subsection{Hayden's concentration of measure phenomenon}
+
+The application of the concentration of measure phenomenon in the superdense coding can be realized in random sampling the entangled qubits~\cite{Hayden}:
+
+It is a theorem connecting the following mathematical structure:
+
+\begin{figure}[h]
+    \centering
+    \begin{tikzpicture}[node distance=30mm, thick,
+        main/.style={draw, draw=white},
+        towards/.style={->},
+        towards_imp/.style={->,red},
+        mutual/.style={<->}
+        ]
+        % define nodes
+        \node[main] (cp) {$\mathbb{C}P^{d_A d_B-1}$};
+        \node[main] (pa) [left of=cp] {$\mathcal{P}(A\otimes B)$};
+        \node[main] (sa) [below of=pa] {$S_A$};
+        \node[main] (rng) [right of=sa] {$[0,\infty)\subset \mathbb{R}$};
+
+        % draw edges
+        \draw[mutual] (cp) -- (pa);
+        \draw[towards] (pa) -- node[left] {$\operatorname{Tr}_B$} (sa);
+        \draw[towards_imp] (pa) -- node[above right] {$f$} (rng);
+        \draw[towards] (sa) -- node[above] {$H(\psi_A)$} (rng);
+    \end{tikzpicture}
+    \caption{Mathematical structure for Hayden's concentration of measure phenomenon}
+    \label{fig:Hayden_concentration_of_measure_phenomenon}
+\end{figure}
+
+\begin{itemize}
+    \item The red arrow is the concentration of measure effect. $f=H(\operatorname{Tr}_B(\psi))$.
+    \item $S_A$ denotes the mixed states on $A$.
+\end{itemize}
+
+To prove the concentration of measure phenomenon, we need to analyze the following elements involved in figure~\ref{fig:Hayden_concentration_of_measure_phenomenon}:
+
+    
+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.
+
+Due to time constrains of the projects, the following lemma is demonstrated but not investigated thoroughly through the research:
+
+
+\begin{lemma}
+    \label{pages_lemma}
+
+    Page's lemma for expected entropy of mixed states
+
+    Choose a random pure state $\sigma=|\psi\rangle\langle\psi|$ from $A'\otimes A$.
+
+    The expected value of the entropy of entanglement is known and satisfies a concentration inequality known as Page's formula~\cite{Pages_conjecture,Pages_conjecture_simple_proof,Bengtsson_Zyczkowski_2017}[15.72].
+
+    $$
+    \mathbb{E}[H(\psi_A)]=\frac{1}{\ln(2)}\left(\sum_{j=d_B+1}^{d_Ad_B}\frac{1}{j}-\frac{d_A-1}{2d_B}\right) \geq \log_2(d_A)-\frac{1}{2\ln(2)}\frac{d_A}{d_B}
+    $$
+
+\end{lemma}
+
+It basically provides a lower bound for the expected entropy of entanglement. Experimentally, we can have the following result (see Figure~\ref{fig:entropy_vs_dim}):
+
+\begin{figure}[h]
+	\centering
+	\includegraphics[width=0.8\textwidth]{entropy_vs_dim.png}
+	\caption{Entropy vs dimension}
+	\label{fig:entropy_vs_dim}
+\end{figure}
+
+Then we have bound for Lipschitz constant $\eta$ of the map $S(\varphi_A): \mathcal{P}(A\otimes B)\to \R$
+
+\begin{lemma}
+    The Lipschitz constant $\eta$ of $S(\varphi_A)$ is upper bounded by $\sqrt{8}\log_2(d_A)$ for $d_A\geq 3$.
+\end{lemma}
+
+\begin{proof}
+    Consider the Lipschitz constant of the function $g:A\otimes B\to \R$ defined as $g(\varphi)=H(M(\varphi_A))$, where $M:A\otimes B\to \mathcal{P}(A)$ is any fixed complete von Neumann measurement and $H: \mathcal{P}(A)\otimes \mathcal{P}(B)\to \R$ is the Shannon entropy.
+
+    Let $\{\ket{e_j}_A\}$ be the orthonormal basis for $A$ and $\{\ket{f_k}_B\}$ be the orthonormal basis for $B$. Then we decompose the state as spectral form $\ket{\varphi}=\sum_{j=1}^{d_A}\sum_{k=1}^{d_B}\varphi_{jk}\ket{e_j}_A\ket{f_k}_B$.
+
+    By unitary invariance, suppose $M_j=\ket{e_j}\bra{e_j}_A$, and define
+
+    $$
+    p_j(\varphi)=\bra{e_j}\varphi_A \ket{e_j}=\sum_{k=1}^{d_B}|\varphi_{jk}|^2
+    $$
+
+    Then 
+    
+    $$
+    g(\varphi)=H(M(\varphi_A))=-\sum_{j=1}^{d_A}p_j(\varphi)\log_2(p_j(\varphi))
+    $$
+
+    Let $h(p)=-p\log_2(p)$, $h(p)=-\frac{p\ln p}{\ln 2}$, and $h'(p)=-\frac{\ln p+1}{\ln 2}$. Let $\varphi_{jk}=x_{jk}+i y_{jk}$, then $p_j(\varphi)=\sum_{k=1}^{d_B}(x_{jk}^2+y_{jk}^2)$, $\frac{\partial p_j}{\partial x_{jk}}=2x_{jk}$, $\frac{\partial p_j}{\partial y_{jk}}=2y_{jk}$.
+
+    Therefore 
+    
+    $$
+    \frac{\partial g}{\partial x_{jk}}=\frac{\partial g}{\partial p_j}\frac{\partial p_j}{x_{jk}}=-\frac{1+\ln p_j}{\ln 2}\cdot 2x_{jk}
+    \qquad 
+    \frac{\partial g}{\partial y_{jk}}=-\frac{1+\ln p_j}{\ln 2}\cdot 2y_{jk}
+    $$
+    
+    Then the lipschitz constant of $g$ is
+
+    $$
+    \begin{aligned}
+    \eta^2&=\sup_{\langle \varphi|\varphi\rangle \leq 1}\nabla g\cdot \nabla g\\
+    &=\sum_{j=1}^{d_A}\sum_{k=1}^{d_B}\left(\frac{\partial g}{\partial x_{jk}}\right)^2+\left(\frac{\partial g}{\partial y_{jk}}\right)^2\\
+    &=\sum_{j=1}^{d_A}\sum_{k=1}^{d_B}\frac{4(x_{jk}^2+y_{jk}^2)}{(\ln 2)^2}[1+\ln p_j(\varphi)]^2\\
+    &=\sum_{j=1}^{d_A}\sum_{k=1}^{d_B}\frac{4|\varphi_{jk}|^2}{(\ln 2)^2}[1+\ln p_j(\varphi)]^2\\
+    \end{aligned}
+    $$
+
+    Note that $\sum_{k=1}^{d_B}|\varphi_{jk}|^2=p_j(\varphi)$, $\nabla g\cdot \nabla g=\frac{4}{(\ln 2)^2}\sum_{j=1}^{d_A}p_j(\varphi)(1+\ln p_j(\varphi))^2$.
+
+    Since $0\leq p_j\leq 1$, we have $\ln p_j(\varphi)\leq 0$, hence $\sum_{j=0}^{d_A}p_j(\varphi)\ln p_j(\varphi)\leq 0$.
+    
+    $$
+    \begin{aligned}
+        \sum_{j=1}^{d_A}p_j(\varphi)(1+\ln p_j(\varphi))^2&=\sum_{j=1}^{d_A}p_j(\varphi)(1+2\ln p_j(\varphi)+(\ln p_j(\varphi))^2)\\
+        &=1+2\sum_{j=1}^{d_A} p_j(\varphi)\ln p_j(\varphi)+\sum_{j=1}^{d_A}p_j(\varphi)(\ln p_j(\varphi))^2\\
+        &\leq 1+\sum_{j=1}^{d_A}p_j(\varphi)(\ln p_j(\varphi))^2\\
+    \end{aligned}
+    $$
+
+    Thus,
+    $$
+    \begin{aligned}
+    \nabla g\cdot \nabla g&\leq \frac{4}{(\ln 2)^2}[1+\sum_{j=1}^{d_A}p_j(\varphi)(\ln p_j(\varphi))^2]\\
+    &\leq \frac{4}{(\ln 2)^2}[1+(\ln d_A)^2]\\
+    &\leq 8(\log_2 d_A)^2
+    \end{aligned}
+    $$
+
+    Proving $\sum_j^{d_A} p_j(\varphi)\ln p_j(\varphi)\leq (\ln d_A)^2$ for $d_A\geq 3$ takes some efforts and we will continue that later.
+
+    Consider any two unit vectors $\ket{\varphi}$ and $\ket{\psi}$, assume $S(\varphi_A)\leq S(\psi_A)$. If we choose the measurement $M$ to be along the eigenbasis of $\varphi_A$, $H(M(\varphi_A))=S(\varphi_A)$ and we have
+
+    $$
+    S(\psi_A)-S(\varphi_A)\leq H(M(\psi_A))-H(M(\varphi_A))\leq \eta\|\ket{\psi}-\ket{\varphi}\|
+    $$
+
+    Thus the lipschitz constant of $S(\varphi_A)$ is upper bounded by $\sqrt{8}\log_2(d_A)$.
+\end{proof}
+
+From Levy's lemma, we have
+
+If we define $\beta=\frac{1}{\ln(2)}\frac{d_A}{d_B}$, then we have
+
+$$
+\operatorname{Pr}[H(\psi_A) < \log_2(d_A)-\alpha-\beta] \leq \exp\left(-\frac{1}{8\pi^2\ln(2)}\frac{(d_Ad_B-1)\alpha^2}{(\log_2(d_A))^2}\right)
+$$
+
+where $d_B\geq d_A\geq 3$~\cite{Hayden_2006}.
+
+Experimentally, we can have the following result:
+
+As the dimension of the Hilbert space increases, the chance of getting an almost maximally entangled state increases (see Figure~\ref{fig:entropy_vs_dA}).
+
+\begin{figure}[h]
+	\centering
+	\includegraphics[width=0.8\textwidth]{entropy_vs_dA.png}
+	\caption{Entropy vs $d_A$}
+	\label{fig:entropy_vs_dA}
+\end{figure}
+
+% When compiled standalone, print this chapter's references at the end.
+\ifSubfilesClassLoaded{
+  \printbibliography[title={References for Chapter 1}]
+}
+
+\end{document}

latex/chapters/chap2.tex

@@ -0,0 +1,244 @@
+% chapters/chap2.tex
+\documentclass[../main.tex]{subfiles}
+
+\ifSubfilesClassLoaded{
+  \addbibresource{../main.bib}
+}
+
+\begin{document}
+
+\chapter{Levy's family and observable diameters}
+
+In this section, we will explore how the results from Hayden's concentration of measure theorem can be understood in terms of observable diameters from Gromov's perspective and what properties it reveals for entropy functions.
+
+We will try to use the results from previous sections to estimate the observable diameter for complex projective spaces.
+
+\section{Observable diameters}
+
+Recall from previous sections, 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.
+
+\begin{defn}
+  \label{defn:mm-space}
+
+  Let $X$ be a topological space with the following:
+
+  \begin{enumerate}
+    \item $X$ is a complete (every Cauchy sequence converges)
+    \item $X$ is a metric space with metric $d_X$
+    \item $X$ has a Borel probability measure $\mu_X$
+  \end{enumerate}
+
+  Then $(X,d_X,\mu_X)$ is a \textbf{metric measure space}.
+\end{defn}
+
+\begin{defn}
+  \label{defn:diameter}
+
+  Let $(X,d_X)$ be a metric space. The \textbf{diameter} of a set $A\subset X$ is defined as
+  $$
+  \diam(A)=\sup_{x,y\in A}d_X(x,y).
+  $$
+\end{defn}
+
+\begin{defn}
+  \label{defn:partial-diameter}
+
+  Let $(X,d_X,\mu_X)$ be a metric measure space, For any real number $\alpha\leq 1$, the \textbf{partial diameter} of $X$ is defined as
+  $$
+  \diam(A;\alpha)=\inf_{A\subseteq X|\mu_X(A)\geq \alpha}\diam(A).
+  $$
+\end{defn}
+
+This definition generalize the relation between the measure and metric in the metric-measure space. Intuitively, the space with smaller partial diameter can take more volume with the same diameter constrains.
+
+However, in higher dimensions, the volume may tend to concentrates more around a small neighborhood of the set, as we see in previous chapters with high dimensional sphere as example. We can safely cut $\kappa>0$ volume to significantly reduce the diameter, this yields better measure for concentration for shapes in spaces with high dimension.
+
+
+\begin{defn}
+  \label{defn:observable-diameter}
+  Let $X$ be a metric-measure space, $Y$ be a metric space, and $f:X\to Y$ be a 1-Lipschitz function. Then $f_*\mu_X=\mu_Y$ is a push forward measure on $Y$.
+  
+  For any real number $\kappa>0$, the \textbf{$\kappa$-observable diameter with screen $Y$} is defined as
+
+  $$
+  \obdiam_Y(X;\kappa)=\sup\{\diam(f_*\mu_X;1-\kappa)\}
+  $$
+
+  And the \textbf{obbservable diameter with screen $Y$} is defined as
+
+  $$
+  \obdiam_Y(X)=\inf_{\kappa>0}\max\{\obdiam_Y(X;\kappa)\}
+  $$
+
+  If $Y=\R$, we call it the \textbf{observable diameter}.
+
+\end{defn}
+
+If we collapse it naively via
+$$
+  \inf_{\kappa>0}\obdiam_Y(X;\kappa),
+$$
+we typically get something degenerate: as $\kappa\to 1$, the condition ``mass $\ge 1-\kappa$'' becomes almost empty space, so $\diam(\nu;1-\kappa)$ can be forced to be $0$ (take a tiny set of positive mass), and hence the infimum tends to $0$ for essentially any non-atomic space.
+
+This is why one either:
+\begin{enumerate}
+  \item keeps $\obdiam_Y(X;\kappa)$ as a \emph{function of $\kappa$} (picking $\kappa$ to be small but not $0$), or
+  \item if one insists on a single number, balances ``spread'' against ``exceptional mass'' by defining $\obdiam_Y(X)=\inf_{\kappa>0}\max\{\obdiam_Y(X;\kappa),\kappa\}$ as above.
+\end{enumerate}
+The point of the $\max\{\cdot,\kappa\}$ is that it prevents cheating by taking $\kappa$ close to $1$: if $\kappa$ is large then the maximum is large regardless of how small $\obdiam_Y(X;\kappa)$ is, so the infimum is forced to occur where the exceptional mass and the observable spread are small.
+
+Few additional proposition in \cite{shioya2014metricmeasuregeometry} will help us to estimate the observable diameter for complex projective spaces.
+
+\begin{prop}
+  \label{prop:observable-diameter-domination}
+  Let $X$ and $Y$ be two metric-measure spaces and $\kappa>0$, and let $f:Y\to X$ be a 1-Lipschitz function ($Y$ dominates $X$, denoted as $X\prec Y$) then:
+
+  \begin{enumerate}
+    \item
+    $
+    \diam(X,1-\kappa)\leq \diam(Y,1-\kappa)
+    $
+    \item $\obdiam(X;-\kappa)\leq \diam(X;1-\kappa)$, and $\obdiam(X)$ is finite.
+    \item
+    $
+    \obdiam(X;-\kappa)\leq \obdiam(Y;-\kappa)
+    $
+  \end{enumerate}
+\end{prop}
+
+\begin{proof}
+  Since $f$ is 1-Lipschitz, we have $f_*\mu Y=\mu_X$. Let $A$ be any Borel set of $Y$ with $\mu_Y(A)\geq 1-\kappa$ and $\overline{f(A)}$ be the closure of $f(A)$ in $X$. We have $\mu_X(\overline{f(A)})=\mu_Y(f^{-1}(\overline{f(A)}))\geq \mu_Y(A)\geq 1-\kappa$ and by the 1-lipschitz property, $\diam(\overline{f(A)})\leq \diam(A)$, so $\diam(X;1-\kappa)\leq \diam(A)\leq \diam(Y;1-\kappa)$.
+
+  Let $g:X\to \R$ be any 1-lipschitz function, since $(\R,|\cdot|,g_*\mu_X)$ is dominated by $X$, $\diam(\R;1-\kappa)\leq \diam(X;1-\kappa)$. Therefore, $\obdiam(X;-\kappa)\leq \diam(X;1-\kappa)$.
+ 
+  and
+  $$
+  \diam(g_*\mu_X;1-\kappa)\leq \diam((f\circ g)_*\mu_Y;1-\kappa)\leq \obdiam(Y;1-\kappa)
+  $$
+\end{proof}
+
+\begin{prop}
+  \label{prop:observable-diameter-scale}
+  Let $X$ be an metric-measure space. Then for any real number $t>0$, we have
+  
+  $$
+  \obdiam(tX;-\kappa)=t\obdiam(X;-\kappa)
+  $$
+
+  Where $tX=(X,tdX,\mu X)$.
+\end{prop}
+
+\begin{proof}
+  $$
+  \begin{aligned}
+    \obdiam(tX;-\kappa)&=\sup\{\diam(f_*\mu_X;1-\kappa)|f:tX\to \R \text{ is 1-Lipschitz}\}\\
+    &=\sup\{\diam(f_*\mu_X;1-\kappa)|t^{-1}f:X\to \R \text{ is 1-Lipschitz}\}\\
+    &=\sup\{\diam((tg)_*\mu_X;1-\kappa)|g:X\to \R \text{ is 1-Lipschitz}\}\\
+    &=t\sup\{\diam(g_*\mu_X;1-\kappa)|g:X\to \R \text{ is 1-Lipschitz}\}\\
+    &=t\obdiam(X;-\kappa)
+  \end{aligned}
+  $$
+\end{proof}
+
+\subsection{Observable diameter for class of spheres}
+
+In this section, we will try to use the results from previous sections to estimate the observable diameter for class of spheres.
+
+\begin{theorem}
+  \label{thm:observable-diameter-sphere}
+  For any real number $\kappa$ with $0<\kappa<1$, we have
+  $$
+  \obdiam(S^n(1);-\kappa)=O(\sqrt{n})
+  $$
+\end{theorem}
+
+\begin{proof}
+  First, recall that by maxwell boltzmann distribution, we have that for any $n>0$, let $I(r)$ denote the measure of standard gaussian measure on the interval $[0,r]$. Then we have
+  
+  $$
+  \begin{aligned}
+    \lim_{n\to \infty} \obdiam(S^n(\sqrt{n});-\kappa)&=\lim_{n\to \infty} \sup\{\diam((\pi_{n,k})_*\sigma^n;1-\kappa)|\pi_{n,k} \text{ is 1-Lipschitz}\}\\
+    &=\lim_{n\to \infty} \sup\{\diam(\gamma^1;1-\kappa)|\gamma^1 \text{ is the standard gaussian measure}\}\\
+    &=\diam(\gamma^1;1-\kappa)\\
+    &=2I^{-1}(\frac{1-\kappa}{2})\text { cutting the extremum for normal distribution}\\
+  \end{aligned}
+  $$
+
+  By proposition \ref{prop:observable-diameter-scale}, we have
+
+  $$
+  \obdiam(S^n(\sqrt{n});-\kappa)=\sqrt{n}\obdiam(S^n(1);-\kappa)
+  $$
+
+  So $\obdiam(S^n(1);-\kappa)=\sqrt{n}(2I^{-1}(\frac{1-\kappa}{2}))=O(\sqrt{n})$.
+\end{proof}
+
+From the previous discussion, we see that the only remaining for finding observable diameter of $\C P^n$ is to find the lipchitz function that is isometric with consistent push-forward measure.
+
+To find such metric, we need some additional results from previous sections.
+
+A natural measure for $\C P^n$ is the normalized volume measure on $\C P^n$ induced from the Fubini-Study metric. \cite{lee_introduction_2018} Example 2.30
+
+\begin{defn}
+  \label{defn:fubini-study-metric}
+
+  Let $n$ be a positive integer, and consider the complex projective space $\C P^n$ defined as the quotient space of $\C^{n+1}$ by the equivalence relation $z\sim z'$ if there exists $\lambda \in \C$ such that $z=\lambda z'$. The map $\pi:\C^{n+1}\setminus\{0\}\to \C P^n$ sending each point in $\C^{n+1}\setminus\{0\}$ to its span is surjective smooth submersion.
+
+  Identifying $\C^{n+1}$ with $\R^{2n+2}$ with its Euclidean metric, we can view the unit sphere $S^{2n+1}$ with its roud metric $\mathring{g}$ as an embedded Riemannian submanifold of $\C^{n+1}\setminus\{0\}$. Let $p:S^{2n+1}\to \C P^n$ denote the restriction of the map $\pi$. Then $p$ is smooth, and its is surjective, because every 1-dimensional complex subspace contains elements of unit norm.
+
+\end{defn}
+
+There are many additional properties for such construction, we will check them just for curiosity.
+
+We need to show that it is a submersion.
+
+\begin{proof}
+   Let $z_0\in S^{2n+1}$ and set $\zeta_0=p(z_0)\in \C P^n$. Since $\pi$ is a smooth submersion, it has a smooth local section $\sigma: U\to \C^{n+1}$ defined on a neighborhood $U$ of $\zeta_0$ and satisfying $\sigma(\zeta_0)=z_0$ by the local section theorem (Theorem \ref{theorem:local_section_theorem}). Let $v:\C^{n+1}\setminus\{0\}\to S^{2n+1}$ be the radial projection on to the sphere:
+
+   $$
+   v(z)=\frac{z}{|z|}
+   $$
+
+   Since dividing an element of $\C^{n+1}$ by a nonzero scalar does not change its span, it follows that $p\circ v=\pi$. Therefore, if we set $\tilde{\sigma}=v\circ \sigma$, then $\tilde{\sigma}$ is a smooth local section of $p$. Apply the local section theorem (Theorem \ref{theorem:local_section_theorem}) again, this shows that $p$ is a submersion.
+
+   Define an action of $S^1$ on $S^{2n+1}$ by complex multiplication:
+
+   $$
+   \lambda (z^1,z^2,\ldots,z^{n+1})=(\lambda z^1,\lambda z^2,\ldots,\lambda z^{n+1})
+   $$
+
+   for $\lambda\in S^1$ (viewed as complex number of norm 1) and $z=(z^1,z^2,\ldots,z^{n+1})\in S^{2n+1}$. This is easily seen to be isometric, vertical, and transitive on fibers of $p$.
+
+   By (Theorem \ref{theorem:riemannian-submersion}). Therefore, there is a unique metric on $\C P^n$ such that the map $p:S^{2n+1}\to \C P^n$ is a Riemannian submersion. This metric is called the Fubini-study metric.
+\end{proof}
+
+\subsection{Observable diameter for complex projective spaces}
+
+Using the projection map and Hopf's fibration, we can estimate the observable diameter for complex projective spaces from the observable diameter of spheres.
+
+\begin{theorem}
+  \label{thm:observable-diameter-complex-projective-space}
+  For any real number $\kappa$ with $0<\kappa<1$, we have
+  $$
+  \obdiam(\mathbb{C}P^n(1);-\kappa)\leq O(\sqrt{n})
+  $$
+\end{theorem}
+
+\begin{proof}
+  Recall from Example 2.30 in \cite{lee_introduction_2018}, the Hopf fibration $f_n:S^{2n+1}(1)\to \C P^n$ is 1-Lipschitz continuous with respect to the Fubini-Study metric on $\C P^n$. and the push-forward $(f_n)_*\sigma^{2n+1}$ coincides with the normalized volume measure on $\C P^n$ induced from the Fubini-Study metric.
+
+  By proposition \ref{prop:observable-diameter-domination}, we have $\obdiam(\mathbb{C}P^n(1);-\kappa)\leq \obdiam(S^{2n+1}(1);-\kappa)\leq O(\sqrt{n})$.
+
+\end{proof}
+
+\section{Use entropy function as estimator of observable diameter for complex projective spaces}
+
+In this section, we wish to use observable diameter to estimate the statics of thermal dynamics of some classical systems.
+
+
+
+\ifSubfilesClassLoaded{
+  \printbibliography[title={References}]
+}
+
+\end{document}

latex/chapters/chap3.tex

@@ -0,0 +1,57 @@
+% chapters/chap2.tex
+\documentclass[../main.tex]{subfiles}
+
+\ifSubfilesClassLoaded{
+  \addbibresource{../main.bib}
+}
+
+\begin{document}
+
+\chapter{Seigel-Bargmann Space}
+
+In this chapter, we will collect ideas and other perspective we have understanding the concentration of measure phenomenon. Especially with symmetric product of $\C P^1$ and see how it relates to Riemman surfaces and Seigel-Bargmann spaces.
+
+\begin{figure}[h]
+    \centering
+    \begin{tikzpicture}[node distance=40mm, thick,
+        main/.style={draw, draw=white},
+        towards/.style={->},
+        towards_imp/.style={<->,red},
+        mutual/.style={<->}
+        ]
+        \node[main] (cp) {$\mathbb{C}P^{n}$};
+        \node[main] (c) [below of=cp] {$\mathbb{C}^{n+1}$};
+        \node[main] (p) [right of=cp] {$\mathbb{P}^n$};
+        \node[main] (sym) [below of=p] {$\operatorname{Sym}_n(\mathbb{C}P^1)$};
+        % draw edges
+        \draw[towards] (c) -- (cp) node[midway, left] {$z\sim \lambda z$};
+        \draw[towards] (c) -- (p) node[midway, fill=white] {$w(z)=\sum_{i=0}^n Z_i z^i$};
+        \draw[towards_imp] (cp) -- (p) node[midway, above] {$w(z)\sim w(\lambda z)$};
+        \draw[mutual] (p) -- (sym) node[midway, right] {root of $w(z)$};
+    \end{tikzpicture}
+    \caption{Majorana stellar representation}
+    \label{fig:majorana_stellar_representation}
+\end{figure}
+
+Basically, there is a bijection between the complex projective space $\mathbb{C}P^n$ and the set of roots of a polynomial of degree $n$.
+
+We can use a symmetric group of permutations of $n$ complex numbers (or $S^2$) to represent the $\mathbb{C}P^n$, that is, $\mathbb{C}P^n=S^2\times S^2\times \cdots \times S^2/S_n$.
+
+One might be interested in the random sampling over the $\operatorname{Sym}_n(\mathbb{C}P^1)$ and the concentration of measure phenomenon on that.
+
+\section{Majorana stellar representation of the quantum state}
+
+\begin{defn}
+    Let $n$ be a positive integer. The Majorana stellar representation of the quantum state is the set of all roots of a polynomial of degree $n$ in $\mathbb{C}$.
+
+    
+\end{defn}
+\section{Space of complex valued functions and pure states}
+
+
+
+\ifSubfilesClassLoaded{
+  \printbibliography[title={References for Chapter 2}]
+}
+
+\end{document}

latex/main.bib

@@ -0,0 +1,189 @@
+@book{parthasarathy1992quantum,
+  title     = {An Introduction to Quantum Stochastic Calculus},
+  author    = {Parthasarathy, K. R.},
+  series    = {Monographs in Mathematics},
+  volume    = {85},
+  year      = {1992},
+  publisher = {Birkh{\"a}user Basel},
+  doi       = {10.1007/978-3-0348-8641-3},
+  isbn      = {978-3-0348-9711-2},
+  eisbn     = {978-3-0348-8641-3},
+  pages     = {XI, 292},
+  topics    = {Probability Theory and Stochastic Processes}
+}
+
+@book{Elizabeth_book,
+  title     ={The Random Matrix Theory of the Classical Compact Groups},
+  author    ={Elizabeth Meckes}
+}
+
+@book{parthasarathy2005mathematical,
+  title     = {Mathematical Foundation of Quantum Mechanics},
+  author    = {Parthasarathy, K. R.},
+  series    = {Texts and Readings in Mathematics},
+  volume    = {85},
+  year      = {2005},
+  publisher = {Hindustan Book Agency},
+  doi       = {10.1007/978-93-86279-28-6},
+  isbn      = {978-93-86279-28-6},
+  eisbn     = {978-93-86279-28-6},
+  pages     = {XI, 292},
+  topics    = {Mathematics, general}
+}
+
+@book{Vershynin_book,
+  title     = {High-dimensional probability: an introduction with applications in data science},
+  author    = {Vershynin, Roman},
+  year      = {2018},
+  publisher = {Cambridge University Press},
+  doi       = {10.1017/9781316278289},
+  isbn      = {9781316278289},
+  eisbn     = {9781316278289},
+  pages     = {X, 368}
+}
+
+@inbook{kummer1998elements,
+  author    = {B. Kümmer and H. Maassen},
+  title     = {Elements of quantum probability},
+  booktitle = {Quantum Probability Communications},
+  chapter   = {},
+  pages     = {73-100},
+  doi       = {10.1142/9789812816054_0003},
+  url       = {https://www.worldscientific.com/doi/abs/10.1142/9789812816054_0003},
+  abstract  = { Abstract This is an introductory article presenting some basic ideas of quantum probability. From a discussion of simple experiments with polarized light and a card game we deduce the necessity of extending the body of classical probability theory. For a class of systems, containing classical systems with finitely many states, a probabilistic model is developed. It can describe, in particular, the polarization experiments. Some examples of “quantum coin tosses” are discussed, closely related to V.F.R. Jones’ approach to braid group representations, to spin relaxation, and to nuclear magnetic resonance. In an appendix we indicate the steps which lead to the full mathematical model of quantum probability. }
+}
+
+@misc{Feres,
+  title   = {Math 444 Lecture notes – the mathematics of quantum theory},
+  url     = {https://www.math.wustl.edu/~feres/Math444Spring25/Math444Spring25Syllabus.html},
+  journal = {Math 444 – the mathematics of quantum theory},
+  author  = {Feres, Renato}
+} 
+
+@book{romanvershyni,
+  title     = {High-dimensional probability: an introduction with applications in data science},
+  author    = {Roman Vershynin},
+  year      = {2018},
+  publisher = {Cambridge University Press}
+}
+
+@book{MGomolovs,
+  title     = {Metric structures for Riemannian and non-Riemannian spaces},
+  author    = {M. Gromov},
+  year      = {1981},
+  publisher = {Birkhäuser}
+}
+
+@misc{shioya2014metricmeasuregeometry,
+      title={Metric measure geometry}, 
+      author={Takashi Shioya},
+      year={2014},
+      eprint={1410.0428},
+      archivePrefix={arXiv},
+      primaryClass={math.MG},
+      url={https://arxiv.org/abs/1410.0428}, 
+}
+
+@book{lee_introduction_2018,
+  title = {Introduction to {{Riemannian Manifolds}}},
+  author = {Lee, John M.},
+  year = {2018},
+  series = {Graduate {{Texts}} in {{Mathematics}}},
+  edition = {2nd},
+  publisher = {Springer},
+  address = {Cham, Switzerland},
+  isbn = {978-3-319-91755-9}
+}
+
+@book{lee_introduction_2012,
+	address = {New York},
+	title = {Introduction to {Smooth} {Manifolds}},
+	isbn = {978-1-4899-9475-2 978-1-4419-9982-5},
+	language = {eng},
+	publisher = {Springer},
+	author = {Lee, John M. and Lee, John M.},
+	year = {2012},
+}
+
+@inproceedings{Hayden,
+  title     = {Concentration of measure effects in quantum information},
+  author    = {Hayden, Patrick},
+  booktitle = {Quantum Information Science and Its Contributions to Mathematics},
+  series    = {Proceedings of Symposia in Applied Mathematics},
+  volume    = {68},
+  pages     = {211--260},
+  year      = {2010},
+  publisher = {American Mathematical Society},
+  isbn      = {978-0-8218-4828-9},
+  doi       = {10.1090/psapm/068}
+}
+
+@article{Hayden_2006,
+  title={Aspects of Generic Entanglement},
+  volume={265},
+  ISSN={1432-0916},
+  url={http://dx.doi.org/10.1007/s00220-006-1535-6},
+  DOI={10.1007/s00220-006-1535-6},
+  number={1},
+  journal={Communications in Mathematical Physics},
+  publisher={Springer Science and Business Media LLC},
+  author={Hayden, Patrick and Leung, Debbie W. and Winter, Andreas},
+  year={2006},
+  month=mar, pages={95-117} 
+}
+
+@book{Haar_book,
+  title     = {The random Matrix Theory of the Classical Compact groups},
+  author    = {E. M. S. Meckes},
+  year      = {2013},
+  publisher = {Princeton University Press}
+}
+
+@book{Bengtsson_Zyczkowski_2017,
+  title     = {Geometry of Quantum States: An Introduction to Quantum Entanglement},
+  author    = {Bengtsson, Ingemar and Zyczkowski, Karol},
+  year      = {2017},
+  publisher = {Cambridge University Press}
+}
+
+@article{Pages_conjecture,
+  title   = {Page's conjecture},
+  author  = {Page, Don N.},
+  journal = {Physical Review Letters},
+}
+
+@article{Pages_conjecture_simple_proof,
+  title   = {Page's conjecture simple proof},
+  author  = {Jorge Sanchez-Ruiz},
+  journal = {Departament de Fisica Fonamental, Universitat de Barcelona, Diagonal 6/7, 08028 Barcelona, Spain},
+  year    = {1995},
+  journal = {Physical Review E},
+}
+
+@book{Nielsen_Chuang_2010, 
+  place={Cambridge}, 
+  title={Quantum Computation and Quantum Information: 10th Anniversary Edition}, 
+  publisher={Cambridge University Press}, 
+  author={Nielsen, Michael A. and Chuang, Isaac L.}, 
+  year={2010}
+} 
+
+@misc{gupta2015functionalanalysisquantuminformation,
+      title={The Functional Analysis of Quantum Information Theory}, 
+      author={Ved Prakash Gupta and Prabha Mandayam and V. S. Sunder},
+      year={2015},
+      eprint={1410.7188},
+      archivePrefix={arXiv},
+      primaryClass={quant-ph},
+      url={https://arxiv.org/abs/1410.7188}, 
+}
+
+@book{axler2023linear,
+  title={Linear Algebra Done Right},
+  author={Axler, S.},
+  isbn={9783031410260},
+  series={Undergraduate Texts in Mathematics},
+  url={https://books.google.com/books?id=OdnfEAAAQBAJ},
+  year={2023},
+  publisher={Springer International Publishing}
+}

latex/main.tex

@@ -0,0 +1,112 @@
+% main.tex
+\documentclass[11pt]{book}
+
+% --- Math + structure ---
+\usepackage{amsmath,amssymb,amsthm}
+\usepackage{hyperref}
+\usepackage{subfiles} % allows chapters to compile independently
+
+% --- Formatting ---
+\usepackage{fancyhdr,parskip}
+\usepackage{fullpage}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% add special notation supports
+\usepackage[mathscr]{euscript}
+\usepackage{mathtools}
+\usepackage{braket}
+
+%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
+% add image package and directory
+\usepackage{graphicx}
+\usepackage{tikz}
+\graphicspath{{./images/}}
+% dependency graph
+\usetikzlibrary{trees,positioning,arrows.meta,backgrounds}
+% floating graph
+\usepackage{float}
+
+% --- Bibliography: biblatex + biber ---
+\usepackage[
+  backend=biber,
+  style=alphabetic,
+  sorting=nyt,
+  giveninits=true
+]{biblatex}
+
+% --- Beamer-like blocks (printer-friendly) ---
+\usepackage[most]{tcolorbox}
+\usepackage{xcolor}
+
+% A dedicated "Examples" block (optional convenience wrapper)
+\newtcolorbox{examples}[1][Example]{%
+  enhanced,
+  breakable,
+  colback=white,
+  colframe=black!90,
+  coltitle=white,            % title text color
+  colbacktitle=black!90,      % <<< grey 80 title bar
+  boxrule=0.6pt,
+  arc=1.5mm,
+  left=1.2mm,right=1.2mm,top=1.0mm,bottom=1.0mm,
+  fonttitle=\bfseries,
+  title=#1
+}
+
+
+% In the assembled book, we load *all* chapter bib files here,
+% and print one combined bibliography at the end.
+
+\addbibresource{main.bib}
+
+%%
+% Some convenient commands if you need to use integrals
+\newcommand{\is}{\hspace{2pt}}
+\newcommand{\dx}{\is dx}
+
+
+%%%%%%%%%%%%%%%%%%%%%%
+% These are commands you can use that will generate nice things in TeX.  Feel free to define your own, too.
+\newcommand{\Z}{\mathbb{Z}} % integers
+\newcommand{\Q}{\mathbb{Q}} % rationals
+\newcommand{\R}{\mathbb{R}} % reals
+\newcommand{\C}{\mathbb{C}} % complex numbers
+\newcommand{\ds}{\displaystyle} % invoke "display style", which makes fractions come out big, etc.
+\newcommand{\charac}{\operatorname{char}} % characteristic of a field
+\newcommand{\st}{\ensuremath{\,:\,}} % Makes the colon in set-builder notation space properly
+
+%%%%%%%%%%%%%%%%%%%%%%
+% These commands are for convenient notation for the concentration of measure theorem
+\newcommand{\obdiam}{\operatorname{ObserDiam}}
+\newcommand{\diam}{\operatorname{diam}}
+
+
+%%%%%%%%%%%%%%%%%%%%%%
+% These commands create theorem-like environments.
+\newtheorem{theorem}{Theorem}
+\newtheorem{lemma}[theorem]{Lemma}
+\newtheorem{prop}[theorem]{Proposition}
+\newtheorem{defn}[theorem]{Definition}
+
+\title{Concentration of Measure And Quantum Entanglement}
+\author{Zheyuan Wu}
+\date{\today}
+
+\begin{document}
+\frontmatter
+\maketitle
+\tableofcontents
+\mainmatter
+
+% Each chapter is in its own file and included as a subfile.
+% \subfile{preface}
+\subfile{chapters/chap0}
+\subfile{chapters/chap1}
+\subfile{chapters/chap2}
+% \subfile{chapters/chap3}
+
+\backmatter
+\cleardoublepage
+\printbibliography[title={References}]
+
+\end{document}

latex/preface.tex

@@ -0,0 +1,86 @@
+% preface.tex
+\documentclass[main.tex]{subfiles}
+
+\ifSubfilesClassLoaded{
+  \addbibresource{main.bib}
+}
+
+\begin{document}
+
+\chapter*{Preface}
+\addcontentsline{toc}{chapter}{Preface}
+
+Non-commutative probability theory is a branch of generalized probability theory that studies the probability of events in non-commutative algebras (e.g. the algebra of observables in quantum mechanics). In the 20th century, non-commutative probability theory has been applied to the study of quantum mechanics as the classical probability theory is not enough to describe quantum mechanics~\cite{kummer1998elements}.
+
+Recently, the concentration of measure phenomenon has been applied to the study of non-commutative probability theory. Basically, the non-trivial observation, citing from Gromov's work~\cite{MGomolovs}, states that 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,
+
+$$
+\mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp\left(-\frac{(n-1)\epsilon^2}{2}\right)
+$$
+
+is applied to computing the probability that, given a bipartite system $A\otimes B$, assume $\dim(B)\geq \dim(A)\geq 3$, as the dimension of the smaller system $A$ increases, with very high probability, a random pure state $\sigma=|\psi\rangle\langle\psi|$ selected from $A\otimes B$ is almost as good as the maximally entangled state.
+
+Mathematically, that is:
+
+Let $\psi\in \mathcal{P}(A\otimes B)$ be a random pure state on $A\otimes B$.
+
+If we define $\beta=\frac{1}{\ln(2)}\frac{d_A}{d_B}$, then we have
+
+$$
+\operatorname{Pr}[H(\psi_A) < \log_2(d_A)-\alpha-\beta] \leq \exp\left(-\frac{1}{8\pi^2\ln(2)}\frac{(d_Ad_B-1)\alpha^2}{(\log_2(d_A))^2}\right)
+$$
+
+where $d_B\geq d_A\geq 3$~\cite{Hayden_2006}.
+
+In this report, we will show the process of my exploration of the concentration of measure phenomenon in the context of non-commutative probability theory. We assume the reader is an undergraduate student in mathematics and is familiar with the basic concepts of probability theory, measure theory, linear algebra, and some basic skills of mathematical analysis. To make the report more self-contained, we will add detailed annotated proofs that I understand and references for the original sources.
+
+\section*{How to use the dependency graph}
+
+Since our topic integrates almost everything I've learned during undergraduate study, I will try to make some dependency graph for reader and for me to keep track of what are the necessary knowledge to understand part of the report.
+
+One can imagine the project as a big tree, where the root is in undergrad math and branches out to the topics of the report, including many advanced topics and motivation to study them.
+
+\bigskip
+
+% --- Dependency tree graph (TikZ) ---
+\begin{figure}[ht]
+\centering
+\begin{tikzpicture}[
+  node distance=10mm and 18mm,
+  box/.style={draw, rectangle, fill=white, align=center, inner sep=4pt},
+  arrow/.style={-Latex}
+]
+
+% \node[box] (lin) {Linear Algebra\\(bases, maps, eigenvalues)};
+% \node[box, right=of lin] (real) {Real Analysis\\(limits, continuity, measure-lite)};
+% \node[box, below=of lin] (prob) {Probability\\(expectation, variance, concentration)};
+% \node[box, below=of real] (top) {Topology/Geometry\\(metrics, compactness)};
+
+% \node[box, below=12mm of prob] (func) {Functional Analysis\\($L^p$, Hilbert spaces, operators)};
+% \node[box, below=12mm of top] (quant) {Quantum Formalism\\(states, observables, partial trace)};
+
+% \node[box, below=14mm of func, xshift=25mm] (book) {This Book\\(Chapters 1--n)};
+% % draw arrows behind nodes
+% \begin{scope}[on background layer]
+%   \draw[arrow] (lin) -- (func);
+%   \draw[arrow] (real) -- (func);
+%   \draw[arrow] (prob) -- (func);
+%   \draw[arrow] (func) -- (quant);
+%   \draw[arrow] (lin) -- (quant);
+%   \draw[arrow] (top) -- (quant);
+
+%   \draw[arrow] (func) -- (book);
+%   \draw[arrow] (quant) -- (book);
+%   \draw[arrow] (prob) -- (book);
+% \end{scope}
+
+\end{tikzpicture}
+\caption{Dependency tree: prerequisites and how they feed into the main text.}
+\label{fig:dependency-tree}
+\end{figure}
+
+\ifSubfilesClassLoaded{
+  \printbibliography[title={References}]
+}
+
+\end{document}

latex/snippets/compile.sh

@@ -0,0 +1,50 @@
+#!/bin/bash
+
+set -e
+
+echo "Starting batch processing of .tex files in chapters/ directory"
+echo "==============================================================="
+
+total_files=$(find chapters -name "*.tex" -type f | wc -l)
+processed_files=0
+
+if [ $total_files -eq 0 ]; then
+    echo "No .tex files found in chapters/ directory"
+    exit 0
+fi
+
+echo "Found $total_files .tex file(s) to process"
+echo ""
+
+for texfile in chapters/*.tex; do
+    if [ -f "$texfile" ]; then
+        processed_files=$((processed_files + 1))
+        base="${texfile%.*}"
+        filename=$(basename "$texfile")
+
+        echo "[$processed_files/$total_files] Processing: $filename"
+        echo "  └─ Running biber on $base..."
+        if biber "$base" 2>&1 | tee -a "$base.biber.log"; then
+            echo "  └─ Biber completed successfully"
+        else
+            echo "  └─ ERROR: Biber failed for $filename"
+            echo "     Check $base.biber.log for details"
+            exit 1
+        fi
+
+        echo "  └─ Running pdflatex on $filename..."
+        if pdflatex -interaction=nonstopmode "$texfile" 2>&1 | tee -a "$base.pdflatex.log"; then
+            echo "  └─ pdflatex completed successfully"
+        else
+            echo "  └─ ERROR: pdflatex failed for $filename"
+            echo "     Check $base.pdflatex.log for details"
+            exit 1
+        fi
+
+        echo "  └─ Finished processing $filename"
+        echo ""
+    fi
+done
+
+echo "==============================================================="
+echo "Batch processing complete: Successfully processed $processed_files/$total_files file(s)"