diff --git a/presentation/ZheyuanWu_HonorThesis_Presentation.pdf b/presentation/ZheyuanWu_HonorThesis_Presentation.pdf new file mode 100644 index 0000000..bf8b144 Binary files /dev/null and b/presentation/ZheyuanWu_HonorThesis_Presentation.pdf differ diff --git a/presentation/ZheyuanWu_HonorThesis_Presentation.tex b/presentation/ZheyuanWu_HonorThesis_Presentation.tex new file mode 100644 index 0000000..acc9516 --- /dev/null +++ b/presentation/ZheyuanWu_HonorThesis_Presentation.tex @@ -0,0 +1,365 @@ +\documentclass[11pt]{beamer} +\usetheme{Madrid} +\usecolortheme{beaver} +\usefonttheme{serif} + +\usepackage[utf8]{inputenc} +\usepackage[english]{babel} +\usepackage[T1]{fontenc} + +\usepackage{amsmath} +\usepackage{amsfonts} +\usepackage{amssymb} +\usepackage{mathrsfs} +\usepackage{mathtools} +\usepackage{graphicx} +\usepackage{tabularx} +\usepackage{colortbl} +% for drawing the graph +\usepackage{tikz} + +% declare some math operators here +\DeclareMathOperator{\sen}{sen} +\DeclareMathOperator{\tg}{tg} + +\setbeamertemplate{caption}[numbered] + +% set the author, title, and email +\author[Zheyuan Wu]{Zheyuan Wu} +\title{Measure concentration in complex projective space and quantum entanglement} +\newcommand{\email}{w.zheyuan@wustl.edu} + +% \setbeamercovered{transparent} +\setbeamertemplate{navigation symbols}{} + +% the code below is from http://tex.stackexchange.com/questions/170394/modify-beamer-footer-portions +% do not change unless you know what you are doing +\setbeamertemplate{footline} +{ + \leavevmode% + \hbox{% + \begin{beamercolorbox}[wd=.15\paperwidth,ht=2.25ex,dp=1ex,center]{author in head/foot}% + \usebeamerfont{author in head/foot}\insertshortauthor + \end{beamercolorbox}% + \begin{beamercolorbox}[wd=.6\paperwidth,ht=2.25ex,dp=1ex,center]{title in head/foot}% + \usebeamerfont{title in head/foot}\insertshorttitle + \end{beamercolorbox}% + \begin{beamercolorbox}[wd=.15\paperwidth,ht=2.25ex,dp=1ex,center]{date in head/foot}% + \usebeamerfont{author in head/foot}\insertshortdate + \end{beamercolorbox}% + \begin{beamercolorbox}[wd=.1\paperwidth,ht=2.25ex,dp=1ex,center]{institute in head/foot}% + \insertframenumber{} / \inserttotalframenumber\hspace*{1ex} + \end{beamercolorbox}}% + \vskip0pt% +} + +% set definition color +% the code below is from https://tex.stackexchange.com/questions/647650/modifiying-environments-color-theorem-definition-etc-in-beamer +% do not change unless you know what you are doing +\setbeamercolor{block title}{fg=white, bg=red!50!black!60} +\setbeamercolor{block body}{fg=black, bg=red!5} +\setbeamercolor{item}{fg=red!60!black} +\setbeamercolor{section number projected}{fg=white, bg=red!70!black} + +%\logo{} +\institute[]{Washington University in St. Louis} +\date{\today} +%\subject{} + +% --------------------------------------------------------- +% Selecione um estilo de referência +% \bibliographystyle{apalike} + +%\bibliographystyle{abbrv} +%\setbeamertemplate{bibliography item}{\insertbiblabel} +% --------------------------------------------------------- + +% --------------------------------------------------------- +% Incluir os slides nos quais as referências foram citadas +%\usepackage[brazilian,hyperpageref]{backref} + +%\renewcommand{\backrefpagesname}{Citado na(s) página(s):~} +%\renewcommand{\backref}{} +%\renewcommand*{\backrefalt}[4]{ +% \ifcase #1 % +% Nenhuma citação no texto.% +% \or +% Citado na página #2.% +% \else +% Citado #1 vezes nas páginas #2.% +% \fi}% +% --------------------------------------------------------- + +\begin{document} + +\begin{frame} +\titlepage +\end{frame} + +\begin{frame}{Table of Contents} + + \hypersetup{linkcolor=black} +\tableofcontents +\end{frame} + +\section{Memes} + +\begin{frame}{Memes} + + \begin{figure} + \includegraphics[width=0.5\textwidth]{./images/strengthvisuals.jpg} + \end{figure} + + Note that the count of the beams is actually less than before. +\end{frame} + +\section{Decomposing the statements} + +\begin{frame}{Decomposing the statements} + + \begin{block}{Concentration of measure effect} + + 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 $d_B\geq d_A\geq 3$. + \end{block} + \cite{Hayden_2006} + Recall that the von Neumann entropy is defined as $H(\psi_A)=-\operatorname{Tr}(\psi_A\log_2(\psi_A))$. + +\end{frame} + +\begin{frame}{What the system actually looks like} + \begin{figure} + \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] {$[0,\infty)$}; + + % 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} + \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} +\end{frame} + +\section{Geometry of Quantum States} + +\begin{frame}{Wait, but what is $\mathbb{C}P^n$ and where they are coming from?} + + $\mathbb{C}P^n$ 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} + + 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. + + \begin{figure} + \includegraphics[width=0.5\textwidth]{./images/stereographic.png} + \end{figure} + + Detailed proof involves the Hopf fibration structures. + It's a natural projective Hilbert space. +\end{frame} + +\begin{frame}{Some interesting claims about $\mathbb{C}P^n$} + + ..... 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} +\end{frame} + +\begin{frame}{Initial attempts for Levy's concentration lemma} + Consider the orthogonal projection from $\mathbb{R}^{n+1}$, the space where $S^n$ is embedded, to $\mathbb{R}^k$, we denote the restriction of the projection as $\pi_{n,k}:S^n(\sqrt{n})\to \mathbb{R}^k$. Note that $\pi_{n,k}$ is a 1-Lipschitz function (projection will never increase the distance between two points). + + We denote the normalized Riemannian volume measure on $S^n(\sqrt{n})$ as $\sigma^n(\cdot)$, and $\sigma^n(S^n(\sqrt{n}))=1$. + + \begin{block}{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{block} + Basically, you can consider the Gaussian measure as the normalized Lebesgue measure on $\mathbb{R}^k$ with standard deviation $1$. + +\end{frame} + +\begin{frame}{Maxwell-Boltzmann distribution law} + + \begin{block}{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{block} +\end{frame} + +\begin{frame}{Maxwell-Boltzmann distribution law} + + 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} + \includegraphics[width=0.8\textwidth]{./images/maxwell.png} + \end{figure} + +\end{frame} + +\begin{frame}{Proof of Maxwell-Boltzmann distribution law I} + + This part is from \cite{shioya2014metricmeasuregeometry}. + + 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 $a>0$. +\end{frame} + +\begin{frame}{Proof of Maxwell-Boltzmann distribution law II} + + $(n-\|x\|^2)^{\frac{n-k}{2}}=\left(n(1-\frac{\|x\|^2}{n})\right)^{\frac{n-k}{2}}\to n^{\frac{n-k}{2}}\exp(-\frac{\|x\|^2}{2})$ + + 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{frame} + +\begin{frame}{Levy's concentration lemma} + \begin{block}{Levy's concentration lemma} + + 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).) + + Select a random point $x\in S^{n-1}$ with $n>2$ according to the uniform measure (Haar measure). Then the probability of observing a value of $f$ much different from the reference value is exponentially small. + + $$ + \operatorname{Pr}[|f(x)-M_f|>\epsilon]\leq \exp(-\frac{n\epsilon^2}{2\eta^2}) + $$ + $$ + \operatorname{Pr}[|f(x)-\langle f\rangle|>\epsilon]\leq 2\exp(-\frac{(n-1)\epsilon^2}{2\eta^2}) + $$ + + \end{block} + + The Maxwell-Boltzmann distribution law will help us find the limit of measures on hemisphere $S^{n-1}$ under the series of functions $f_n:S^{n-1}(\sqrt{n})\to \mathbb{R}$. + +\end{frame} + +\begin{frame}{Majorana stellar representation of the quantum state} + + \begin{figure} + \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} + \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 permutation of $n$ complex numbers (or $S^2$) to represent the $\mathbb{C}P^n$, that is $\mathbb{C}P^n=S^2\times S^2\times \cdots \times S^2/S_n$. + +\end{frame} + +\section{Future Plans} + +\begin{frame}{Future Plans} + + \begin{itemize} + \item The physical meaning of the mathematical structures, the correspondence, and the relationship between the measures, quantum states, and the geometry of topological spaces. + \begin{itemize} + \item Fiber bundles + \item Fubini-Study metric + \item Space of entangled states + \end{itemize} + \item Riemannian geometry of $\mathbb{C}P^n$. + \begin{itemize} + \item Ricci curvature + \item Levy's Isoperimetric inequality + \item Lipschitz constants and Levi-Civita connection + \item Local operations and classical communication (LOCC) + \end{itemize} + \item The proof of the Page's formula. + \item Majorana stellar representation of the quantum state. And possibly the concentration of measure effect on that. + \item Relations to Gromov's works \cite{MGomolovs} + \begin{itemize} + \item Levy families + \item Observable diameters + \end{itemize} + \end{itemize} +\end{frame} + +\section{References} +\begin{frame}[allowframebreaks]{References} + \nocite{*} % This will include all entries from the bibliography file + \bibliographystyle{apalike} + \bibliography{references} +\end{frame} + +\begin{frame} + +\begin{center} + Q\&A +\end{center} + +\end{frame} + +\end{document} \ No newline at end of file diff --git a/presentation/images/maxwell.png b/presentation/images/maxwell.png new file mode 100644 index 0000000..d02910f Binary files /dev/null and b/presentation/images/maxwell.png differ diff --git a/presentation/images/stereographic.png b/presentation/images/stereographic.png new file mode 100644 index 0000000..becd43c Binary files /dev/null and b/presentation/images/stereographic.png differ diff --git a/presentation/images/strengthvisuals.jpg b/presentation/images/strengthvisuals.jpg new file mode 100644 index 0000000..3cc5af5 Binary files /dev/null and b/presentation/images/strengthvisuals.jpg differ diff --git a/presentation/references.bib b/presentation/references.bib new file mode 100644 index 0000000..cb18060 --- /dev/null +++ b/presentation/references.bib @@ -0,0 +1,59 @@ +@book{MGomolovs, + title = {Metric structures for Riemannian and non-Riemannian spaces}, + author = {M. Gromov}, + year = {1981}, + publisher = {Birkhäuser} +} + +@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{Bengtsson_Życzkowski_2017, + place={Cambridge}, + edition={2}, + title={Geometry of Quantum States: An Introduction to Quantum Entanglement}, + publisher={Cambridge University Press}, + author={Bengtsson, Ingemar and Życzkowski, Karol}, + year={2017} +} + +@book{romanvershyni, + title = {High-dimensional probability: an introduction with applications in data science}, + author = {Roman Vershynin}, + year = {2018}, + publisher = {Cambridge University Press} +} + +@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}, +} \ No newline at end of file