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Zheyuan Wu
2026-03-29 18:08:48 -05:00
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latex/preface.tex
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% 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}
\documentclass[main.tex]{subfiles}
\ifSubfilesClassLoaded{
\addbibresource{main.bib}
}
\begin{document}
\chapter*{Preface}
\addcontentsline{toc}{chapter}{Preface}
This thesis studies how concentration of measure enters quantum information through the geometry of pure states. The basic question is probabilistic but the answer is geometric: once the state space is viewed as a high-dimensional metric-measure space, many physically relevant observables become sharply concentrated. In particular, the entropy of a random bipartite pure state is typically close to its maximal value, which explains why high-dimensional random states are generically highly entangled.
The exposition is organized in layers.
Chapter 0 collects the algebraic, probabilistic, geometric, and quantum-mechanical background needed later.
Chapter 1 develops the motivation from polarization experiments, recalls concentration on spheres, and then states the Hayden--Leung--Winter entanglement bound.
Chapter 2 reformulates the same phenomenon in the language of observable diameter and uses entropy-based simulations as a concrete probe on spheres and complex projective spaces.
Chapter 3 records an exploratory direction through Majorana stellar representation and related holomorphic models. It branches out from our main narrative but is included here for completeness and future reference as our journey into the geometry of quantum states continues.
\section*{How to use the dependency graph}
The graph below is meant to be read from top to bottom. The first row lists undergraduate courses that supply the basic language. The middle rows list the sections and subsection-level themes that are used in the body of the thesis. The bottom node records the concentration-of-measure theorem that motivates the main narrative. Not every reader needs every path: a reader interested mainly in the entanglement theorem can follow the linear algebra, probability, quantum, and concentration branches first, while a reader interested in the geometric reformulation can continue through manifolds, Riemannian geometry, Hopf fibration, and observable diameter.
\bigskip
\begin{figure}[ht]
\centering
\begin{tikzpicture}[
x=1cm,
y=1cm,
scale=0.74,
transform shape,
font=\small,
course/.style={draw, rounded corners, fill=gray!12, align=center, minimum width=2.6cm, minimum height=0.95cm},
topic/.style={draw, rounded corners, fill=blue!5, align=center, minimum width=2.9cm, minimum height=0.95cm},
advanced/.style={draw, rounded corners, fill=green!6, align=center, minimum width=3.1cm, minimum height=0.95cm},
final/.style={draw, rounded corners, fill=red!8, align=center, minimum width=4.0cm, minimum height=1.0cm},
edge/.style={-Latex, semithick}
]
\node[course] (lin) at (-8,0) {Linear algebra};
\node[course] (real) at (-4,0) {Real analysis};
\node[course] (measure) at (0,0) {Measure theory};
\node[course] (prob) at (4,0) {Probability theory};
\node[course] (top) at (8,0) {Topology};
\node[topic] (cvec) at (-8,-2.3) {Complex vector\\ spaces};
\node[topic] (ncprob) at (-3.6,-2.3) {Non-commutative\\ probability theory};
\node[topic] (man) at (0.8,-2.3) {Manifolds};
\node[topic] (quant) at (5.6,-2.3) {Quantum physics\\ and terminologies};
\node[topic] (mot) at (10.0,-2.3) {Motivation};
\node[advanced] (smooth) at (-6.0,-4.8) {Smooth manifolds\\ and Lie groups};
\node[advanced] (riem) at (-1.7,-4.8) {Riemannian\\ manifolds};
\node[advanced] (hopf) at (2.6,-4.8) {Hopf fibration};
\node[advanced] (rand) at (6.9,-4.8) {Random quantum\\ states};
\node[advanced] (conc) at (11.0,-4.8) {Concentration of\\ measure phenomenon};
\node[advanced] (app) at (-4.4,-7.3) {Application in\\ non-commutative probability};
\node[advanced] (obs) at (0.2,-7.3) {Observable\\ diameters};
\node[advanced] (entropy) at (4.9,-7.3) {Entropy estimator\\ on $\mathbb{C}P^n$};
\node[advanced] (majorana) at (9.6,-7.3) {Majorana stellar\\ representation};
\node[final] (final) at (2.6,-9.9) {Concentration of measure};
\draw[edge] (lin) -- (cvec);
\draw[edge] (lin) -- (quant);
\draw[edge] (real) -- (ncprob);
\draw[edge] (measure) -- (ncprob);
\draw[edge] (measure) -- (obs);
\draw[edge] (prob) -- (mot);
\draw[edge] (prob) -- (conc);
\draw[edge] (top) -- (man);
\draw[edge] (cvec) -- (ncprob);
\draw[edge] (cvec) to[out=-25,in=165] (rand);
\draw[edge] (ncprob) -- (app);
\draw[edge] (man) -- (smooth);
\draw[edge] (smooth) -- (riem);
\draw[edge] (riem) -- (hopf);
\draw[edge] (quant) -- (rand);
\draw[edge] (quant) -- (mot);
\draw[edge] (mot) -- (conc);
\draw[edge] (rand) -- (app);
\draw[edge] (conc) -- (app);
\draw[edge] (conc) -- (obs);
\draw[edge] (hopf) -- (obs);
\draw[edge] (hopf) -- (entropy);
\draw[edge] (obs) -- (entropy);
\draw[edge] (rand) -- (entropy);
\draw[edge] (quant) to[out=-70,in=110] (majorana);
\draw[edge] (app) -- (final);
\draw[edge] (obs) -- (final);
\draw[edge] (entropy) -- (final);
\draw[edge] (conc) to[out=-90,in=30] (final);
\end{tikzpicture}
\caption{Dependency graph for the thesis. The central path runs from undergraduate background through geometry and quantum theory to concentration of measure, while the Majorana branch records a later exploratory direction.}
\label{fig:dependency-tree}
\end{figure}
\ifSubfilesClassLoaded{
\printbibliography[title={References}]
}
\end{document}