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This commit is contained in:
Zheyuan Wu
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presentation/ZheyuanWu_HonorThesis_Presentation.tex
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presentation/ZheyuanWu_HonorThesis_Presentation.tex
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\documentclass[11pt]{beamer}
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\usetheme{Madrid}
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\usecolortheme{beaver}
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\usefonttheme{serif}
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\usepackage[utf8]{inputenc}
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\usepackage[english]{babel}
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\usepackage[T1]{fontenc}
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\usepackage{amsmath}
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\usepackage{amsfonts}
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\usepackage{amssymb}
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\usepackage{mathrsfs}
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\usepackage{mathtools}
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\usepackage{graphicx}
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\usepackage{tabularx}
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\usepackage{colortbl}
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% for drawing the graph
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\usepackage{tikz}
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% declare some math operators here
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\DeclareMathOperator{\sen}{sen}
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\DeclareMathOperator{\tg}{tg}
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\setbeamertemplate{caption}[numbered]
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% set the author, title, and email
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\author[Zheyuan Wu]{Zheyuan Wu}
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\title{Measure concentration in complex projective space and quantum entanglement}
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\newcommand{\email}{w.zheyuan@wustl.edu}
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% \setbeamercovered{transparent}
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\setbeamertemplate{navigation symbols}{}
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\setbeamertemplate{footline}
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{
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\leavevmode%
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\hbox{%
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\begin{beamercolorbox}[wd=.15\paperwidth,ht=2.25ex,dp=1ex,center]{author in head/foot}%
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\usebeamerfont{author in head/foot}\insertshortauthor
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\begin{beamercolorbox}[wd=.6\paperwidth,ht=2.25ex,dp=1ex,center]{title in head/foot}%
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\end{beamercolorbox}%
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\begin{beamercolorbox}[wd=.15\paperwidth,ht=2.25ex,dp=1ex,center]{date in head/foot}%
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\begin{beamercolorbox}[wd=.1\paperwidth,ht=2.25ex,dp=1ex,center]{institute in head/foot}%
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\insertframenumber{} / \inserttotalframenumber\hspace*{1ex}
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\end{beamercolorbox}}%
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\vskip0pt%
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}
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% set definition color
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% the code below is from https://tex.stackexchange.com/questions/647650/modifiying-environments-color-theorem-definition-etc-in-beamer
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% do not change unless you know what you are doing
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\setbeamercolor{block title}{fg=white, bg=red!50!black!60}
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\setbeamercolor{block body}{fg=black, bg=red!5}
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\setbeamercolor{item}{fg=red!60!black}
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\setbeamercolor{section number projected}{fg=white, bg=red!70!black}
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%\logo{}
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\institute[]{Washington University in St. Louis}
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\date{\today}
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%\subject{}
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% ---------------------------------------------------------
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% Selecione um estilo de referência
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% \bibliographystyle{apalike}
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%\bibliographystyle{abbrv}
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%\setbeamertemplate{bibliography item}{\insertbiblabel}
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% ---------------------------------------------------------
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% ---------------------------------------------------------
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% Incluir os slides nos quais as referências foram citadas
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%\usepackage[brazilian,hyperpageref]{backref}
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%\renewcommand{\backrefpagesname}{Citado na(s) página(s):~}
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%\renewcommand{\backref}{}
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%\renewcommand*{\backrefalt}[4]{
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% \ifcase #1 %
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% Nenhuma citação no texto.%
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% \or
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% Citado na página #2.%
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% \else
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% Citado #1 vezes nas páginas #2.%
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% \fi}%
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% ---------------------------------------------------------
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\begin{document}
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\begin{frame}
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\titlepage
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\end{frame}
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\begin{frame}{Table of Contents}
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\hypersetup{linkcolor=black}
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\tableofcontents
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\end{frame}
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\section{Memes}
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\begin{frame}{Memes}
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\begin{figure}
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\includegraphics[width=0.5\textwidth]{./images/strengthvisuals.jpg}
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\end{figure}
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Note that the count of the beams is actually less than before.
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\end{frame}
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\section{Decomposing the statements}
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\begin{frame}{Decomposing the statements}
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\begin{block}{Concentration of measure effect}
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Let $\psi\in \mathcal{P}(A\otimes B)$ be a random pure state on $A\otimes B$.
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If we define $\beta=\frac{1}{\ln(2)}\frac{d_A}{d_B}$, then we have
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$$
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\operatorname{Pr}[H(\psi_A) < \log_2(d_A)-\alpha-\beta] \leq \exp\left(-\frac{1}{8\pi^2\ln(2)}\frac{(d_Ad_B-1)\alpha^2}{(\log_2(d_A))^2}\right)
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$$
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where $d_B\geq d_A\geq 3$.
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\end{block}
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\cite{Hayden_2006}
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Recall that the von Neumann entropy is defined as $H(\psi_A)=-\operatorname{Tr}(\psi_A\log_2(\psi_A))$.
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\end{frame}
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\begin{frame}{What the system actually looks like}
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\begin{figure}
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\centering
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\begin{tikzpicture}[node distance=30mm, thick,
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main/.style={draw, draw=white},
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towards/.style={->},
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towards_imp/.style={->,red},
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mutual/.style={<->}
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]
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% define nodes
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\node[main] (cp) {$\mathbb{C}P^{d_A d_B-1}$};
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\node[main] (pa) [left of=cp] {$\mathcal{P}(A\otimes B)$};
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\node[main] (sa) [below of=pa] {$S_A$};
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\node[main] (rng) [right of=sa] {$[0,\infty)$};
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% draw edges
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\draw[mutual] (cp) -- (pa);
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\draw[towards] (pa) -- node[left] {$\operatorname{Tr}_B$} (sa);
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\draw[towards_imp] (pa) -- node[above right] {$f$} (rng);
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\draw[towards] (sa) -- node[above] {$H(\psi_A)$} (rng);
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\end{tikzpicture}
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\end{figure}
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\begin{itemize}
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\item The red arrow is the concentration of measure effect. $f=H(\operatorname{Tr}_B(\psi))$.
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\item $S_A$ denotes the mixed states on $A$
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\end{itemize}
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\end{frame}
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\section{Geometry of Quantum States}
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\begin{frame}{Wait, but what is $\mathbb{C}P^n$ and where they are coming from?}
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$\mathbb{C}P^n$ is the set of all complex lines in $\mathbb{C}^{n+1}$, or equivalently the space of equivalence classes of $n+1$ complex numbers up to a scalar multiple. \cite{Bengtsson_Życzkowski_2017}
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One can find that every odd dimensional sphere $S^{2n+1}$ under the group action of $S^1$, denoted by $S^{2n+1}/S^1$, is a complex projective space $\mathbb{C}P^n$ (complex-dimensional). Recall Math 416.
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\begin{figure}
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\includegraphics[width=0.5\textwidth]{./images/stereographic.png}
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\end{figure}
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Detailed proof involves the Hopf fibration structures.
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It's a natural projective Hilbert space.
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\end{frame}
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\begin{frame}{Some interesting claims about $\mathbb{C}P^n$}
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..... The claim is that every physical system can be modelled by $\mathbb{C}P^n$ for some (possibly infinite) value of $n$, provided taht a definite correspondence between the system and the point of $\mathbb{C}P^n$ is set up. \cite{Bengtsson_Życzkowski_2017}
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\end{frame}
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\begin{frame}{Initial attempts for Levy's concentration lemma}
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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).
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We denote the normalized Riemannian volume measure on $S^n(\sqrt{n})$ as $\sigma^n(\cdot)$, and $\sigma^n(S^n(\sqrt{n}))=1$.
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\begin{block}{Gaussian measure}
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We denote the Gaussian measure on $\mathbb{R}^k$ as $\gamma^k$.
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$$
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d\gamma^k(x)\coloneqq\frac{1}{\sqrt{2\pi}^k}\exp(-\frac{1}{2}\|x\|^2)dx
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$$
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$x\in \mathbb{R}^k$, $\|x\|^2=\sum_{i=1}^k x_i^2$ is the Euclidean norm, and $dx$ is the Lebesgue measure on $\mathbb{R}^k$.
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\end{block}
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Basically, you can consider the Gaussian measure as the normalized Lebesgue measure on $\mathbb{R}^k$ with standard deviation $1$.
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\end{frame}
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\begin{frame}{Maxwell-Boltzmann distribution law}
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\begin{block}{Maxwell-Boltzmann distribution law}
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For any natural number $k$,
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$$
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\frac{d(\pi_{n,k})_*\sigma^n(x)}{dx}\to \frac{d\gamma^k(x)}{dx}
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$$
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where $(\pi_{n,k})_*\sigma^n$ is the push-forward measure of $\sigma^n$ by $\pi_{n,k}$.
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In other words,
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$$
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(\pi_{n,k})_*\sigma^n\to \gamma^k\text{ weakly as }n\to \infty
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$$
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\end{block}
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\end{frame}
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\begin{frame}{Maxwell-Boltzmann distribution law}
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It also has another name, the Projective limit theorem. \cite{romanvershyni}
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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 $n\to \infty$.
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\begin{figure}
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\includegraphics[width=0.8\textwidth]{./images/maxwell.png}
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\end{figure}
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\end{frame}
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\begin{frame}{Proof of Maxwell-Boltzmann distribution law I}
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This part is from \cite{shioya2014metricmeasuregeometry}.
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We denote the $n$ dimensional volume measure on $\mathbb{R}^k$ as $\operatorname{vol}_k$.
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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 $\pi_{n,k}$).
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So,
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$$
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\begin{aligned}
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\frac{d(\pi_{n,k})_*\sigma^n(x)}{dx}&=\frac{\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(x))}{\operatorname{vol}_k(S^n(\sqrt{n}))}\\
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&=\frac{(n-\|x\|^2)^{\frac{n-k}{2}}}{\int_{\|x\|\leq \sqrt{n}}(n-\|x\|^2)^{\frac{n-k}{2}}dx}\\
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\end{aligned}
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$$
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as $n\to \infty$.
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note that $\lim_{n\to \infty}{(1-\frac{a}{n})^n}=e^{-a}$ for any $a>0$.
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\end{frame}
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\begin{frame}{Proof of Maxwell-Boltzmann distribution law II}
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$(n-\|x\|^2)^{\frac{n-k}{2}}=\left(n(1-\frac{\|x\|^2}{n})\right)^{\frac{n-k}{2}}\to n^{\frac{n-k}{2}}\exp(-\frac{\|x\|^2}{2})$
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So
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$$
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\begin{aligned}
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\frac{(n-\|x\|^2)^{\frac{n-k}{2}}}{\int_{\|x\|\leq \sqrt{n}}(n-\|x\|^2)^{\frac{n-k}{2}}dx}&=\frac{e^{-\frac{\|x\|^2}{2}}}{\int_{x\in \mathbb{R}^k}e^{-\frac{\|x\|^2}{2}}dx}\\
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&=\frac{1}{(2\pi)^{\frac{k}{2}}}e^{-\frac{\|x\|^2}{2}}\\
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&=\frac{d\gamma^k(x)}{dx}
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\end{aligned}
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$$
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\end{frame}
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\begin{frame}{Levy's concentration lemma}
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\begin{block}{Levy's concentration lemma}
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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 (spaces that locally resembles Euclidean space).)
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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.
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$$
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\operatorname{Pr}[|f(x)-M_f|>\epsilon]\leq \exp(-\frac{n\epsilon^2}{2\eta^2})
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$$
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$$
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\operatorname{Pr}[|f(x)-\langle f\rangle|>\epsilon]\leq 2\exp(-\frac{(n-1)\epsilon^2}{2\eta^2})
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$$
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\end{block}
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The Maxwell-Boltzmann distribution law will help us find the limit of measures on hemisphere $S^{n-1}$ under the series of functions $f_n:S^{n-1}(\sqrt{n})\to \mathbb{R}$.
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\end{frame}
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\begin{frame}{Majorana stellar representation of the quantum state}
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\begin{figure}
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\centering
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\begin{tikzpicture}[node distance=40mm, thick,
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main/.style={draw, draw=white},
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towards/.style={->},
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towards_imp/.style={<->,red},
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mutual/.style={<->}
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]
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\node[main] (cp) {$\mathbb{C}P^{n}$};
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\node[main] (c) [below of=cp] {$\mathbb{C}^{n+1}$};
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\node[main] (p) [right of=cp] {$\mathbb{P}^n$};
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\node[main] (sym) [below of=p] {$\operatorname{Sym}_n(\mathbb{C}P^1)$};
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% draw edges
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\draw[towards] (c) -- (cp) node[midway, left] {$z\sim \lambda z$};
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\draw[towards] (c) -- (p) node[midway, fill=white] {$w(z)=\sum_{i=0}^n Z_i z^i$};
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\draw[towards_imp] (cp) -- (p) node[midway, above] {$w(z)\sim w(\lambda z)$};
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\draw[mutual] (p) -- (sym) node[midway, right] {root of $w(z)$};
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\end{tikzpicture}
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\end{figure}
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Basically, there is a bijection between the complex projective space $\mathbb{C}P^n$ and the set of roots of a polynomial of degree $n$.
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We can use a symmetric group of permutation of $n$ complex numbers (or $S^2$) to represent the $\mathbb{C}P^n$, that is $\mathbb{C}P^n=S^2\times S^2\times \cdots \times S^2/S_n$.
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\end{frame}
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\section{Future Plans}
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\begin{frame}{Future Plans}
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\begin{itemize}
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\item The physical meaning of the mathematical structures, the correspondence, and the relationship between the measures, quantum states, and the geometry of topological spaces.
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\begin{itemize}
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\item Fiber bundles
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\item Fubini-Study metric
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\item Space of entangled states
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\end{itemize}
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\item Riemannian geometry of $\mathbb{C}P^n$.
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\begin{itemize}
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\item Ricci curvature
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\item Levy's Isoperimetric inequality
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\item Lipschitz constants and Levi-Civita connection
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\item Local operations and classical communication (LOCC)
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\end{itemize}
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\item The proof of the Page's formula.
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\item Majorana stellar representation of the quantum state. And possibly the concentration of measure effect on that.
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\item Relations to Gromov's works \cite{MGomolovs}
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\begin{itemize}
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\item Levy families
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\item Observable diameters
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\end{itemize}
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\end{itemize}
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\end{frame}
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\section{References}
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\begin{frame}[allowframebreaks]{References}
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\nocite{*} % This will include all entries from the bibliography file
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\bibliographystyle{apalike}
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\bibliography{references}
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\end{frame}
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\begin{frame}
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\begin{center}
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Q\&A
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\end{center}
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\end{frame}
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\end{document}
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