diff --git a/presentation/images/stereographic.png b/latex/images/stereographic.png similarity index 100% rename from presentation/images/stereographic.png rename to latex/images/stereographic.png diff --git a/presentation/images/strengthvisuals.jpg b/latex/images/strengthvisuals.jpg similarity index 100% rename from presentation/images/strengthvisuals.jpg rename to latex/images/strengthvisuals.jpg diff --git a/presentation/ZheyuanWu_HonorThesis_Presentation.pdf b/presentation/ZheyuanWu_HonorThesis_Presentation.pdf index bf8b144..2066138 100644 Binary files a/presentation/ZheyuanWu_HonorThesis_Presentation.pdf and b/presentation/ZheyuanWu_HonorThesis_Presentation.pdf differ diff --git a/presentation/ZheyuanWu_HonorThesis_Presentation.tex b/presentation/ZheyuanWu_HonorThesis_Presentation.tex index acc9516..3b1f33f 100644 --- a/presentation/ZheyuanWu_HonorThesis_Presentation.tex +++ b/presentation/ZheyuanWu_HonorThesis_Presentation.tex @@ -13,27 +13,23 @@ \usepackage{mathrsfs} \usepackage{mathtools} \usepackage{graphicx} -\usepackage{tabularx} +\usepackage{tabularx} \usepackage{colortbl} -% for drawing the graph \usepackage{tikz} -% declare some math operators here \DeclareMathOperator{\sen}{sen} \DeclareMathOperator{\tg}{tg} +\DeclareMathOperator{\obdiam}{ObsDiam} +\DeclareMathOperator{\diameter}{diam} \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}{} +\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% @@ -42,7 +38,7 @@ \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 + \usebeamerfont{title in head/foot}\insertsectionhead \end{beamercolorbox}% \begin{beamercolorbox}[wd=.15\paperwidth,ht=2.25ex,dp=1ex,center]{date in head/foot}% \usebeamerfont{author in head/foot}\insertshortdate @@ -53,42 +49,13 @@ \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}% -% --------------------------------------------------------- +\institute[]{Washington University in St. Louis} +\date{\today} \begin{document} @@ -97,43 +64,431 @@ \end{frame} \begin{frame}{Table of Contents} - \hypersetup{linkcolor=black} -\tableofcontents + \tableofcontents +\end{frame} +\section{Motivation} + +\begin{frame}{Light polarization and non-commutative probability} + \begin{figure} + \includegraphics[width=0.6\textwidth]{../latex/images/Filter_figure.png} + \end{figure} + \begin{itemize} + \item Light passing through a polarizer becomes polarized in the direction of that filter. + \item If two filters are placed with relative angle $\alpha$, the transmitted intensity decreases as $\alpha$ increases. + \item In particular, the transmitted intensity vanishes when $\alpha=\pi/2$. + \end{itemize} \end{frame} -\section{Memes} +\begin{frame}{Polarization experiment} -\begin{frame}{Memes} + \vspace{0.5em} + Now consider three filters $F_1,F_2,F_3$ with directions + $$ + \alpha_1,\alpha_2,\alpha_3. + $$ + Testing them pairwise suggests introducing three $0$--$1$ random variables + $$ + P_1,P_2,P_3, + $$ + where $P_i=1$ means that the photon passes filter $F_i$. - \begin{figure} - \includegraphics[width=0.5\textwidth]{./images/strengthvisuals.jpg} - \end{figure} - - Note that the count of the beams is actually less than before. + \vspace{0.5em} + If these were classical random variables on one probability space, they would satisfy a Bell-type inequality. \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 $d_B\geq d_A\geq 3$. +\begin{frame}{A classical Bell-type inequality} + \begin{block}{Bell-type inequality} + For any classical random variables $P_1,P_2,P_3\in\{0,1\}$, + $$ + \operatorname{Prob}(P_1=1,P_3=0) + \leq + \operatorname{Prob}(P_1=1,P_2=0) + + + \operatorname{Prob}(P_2=1,P_3=0). + $$ \end{block} - \cite{Hayden_2006} - Recall that the von Neumann entropy is defined as $H(\psi_A)=-\operatorname{Tr}(\psi_A\log_2(\psi_A))$. + \vspace{0.5em} + \begin{proof} + The event $\{P_1=1,P_3=0\}$ splits into two disjoint cases according to whether $P_2=0$ or $P_2=1$: + $$ + \{P_1=1,P_3=0\} + = + \{P_1=1,P_2=0,P_3=0\} + \sqcup + \{P_1=1,P_2=1,P_3=0\}. + $$ + Therefore, + $$ + \begin{aligned} + \operatorname{Prob}(P_1=1,P_3=0) + &= + \operatorname{Prob}(P_1=1,P_2=0,P_3=0) \\ + &\quad+ + \operatorname{Prob}(P_1=1,P_2=1,P_3=0) \\ + &\leq + \operatorname{Prob}(P_1=1,P_2=0) + + + \operatorname{Prob}(P_2=1,P_3=0). + \end{aligned} + $$ + \end{proof} \end{frame} -\begin{frame}{What the system actually looks like} +\begin{frame}{Experimental law} + For unpolarized incoming light, the \textbf{observed transition law} for a pair of filters is + $$ + \operatorname{Prob}(P_i=1,P_j=0) + = + \operatorname{Prob}(P_i=1)-\operatorname{Prob}(P_i=1,P_j=1). + $$ + + Using the polarization law, + $$ + \operatorname{Prob}(P_i=1)=\frac12, + \qquad + \operatorname{Prob}(P_i=1,P_j=1)=\frac12\cos^2(\alpha_i-\alpha_j), + $$ + hence + $$ + \operatorname{Prob}(P_i=1,P_j=0) + = + \frac12-\frac12\cos^2(\alpha_i-\alpha_j) + = + \frac12\sin^2(\alpha_i-\alpha_j). + $$ + + \vspace{0.5em} + So the experimentally observed probabilities depend only on the angle difference $\alpha_i-\alpha_j$. +\end{frame} + +\begin{frame}{Violation of the classical inequality} + Substituting the experimental law into the classical inequality gives + $$ + \frac12\sin^2(\alpha_1-\alpha_3) + \leq + \frac12\sin^2(\alpha_1-\alpha_2) + + + \frac12\sin^2(\alpha_2-\alpha_3). + $$ + + Choose + $$ + \alpha_1=0,\qquad + \alpha_2=\frac{\pi}{6},\qquad + \alpha_3=\frac{\pi}{3}. + $$ + + Then + $$ + \begin{aligned} + \frac12\sin^2\!\left(-\frac{\pi}{3}\right) + &\leq + \frac12\sin^2\!\left(-\frac{\pi}{6}\right) + + + \frac12\sin^2\!\left(-\frac{\pi}{6}\right) \\ + \frac38 &\leq \frac18+\frac18 \\ + \frac38 &\leq \frac14, + \end{aligned} + $$ + which is false. + + \vspace{0.5em} + Therefore the pairwise polarization data cannot come from one classical probability model with random variables $P_1,P_2,P_3$. +\end{frame} + +\begin{frame}{The quantum model of polarization} + The correct model uses a Hilbert space rather than classical events. + + \begin{itemize} + \item A pure polarization state is a vector + $$ + \psi=\alpha|0\rangle+\beta|1\rangle \in \mathbb{C}^2. + $$ + \item A filter at angle $\alpha$ is represented by the orthogonal projection + $$ + P_\alpha= + \begin{pmatrix} + \cos^2\alpha & \cos\alpha\sin\alpha \\ + \cos\alpha\sin\alpha & \sin^2\alpha + \end{pmatrix}. + $$ + \item For a pure state $\psi$, the probability of passing the filter is + $$ + \langle P_\alpha\psi,\psi\rangle. + $$ + \end{itemize} + + \vspace{0.4em} + The key point is that sequential measurements are described by \emph{ordered products} of projections, and these need not commute. +\end{frame} + +\begin{frame}{Recovering the observed law from the operator model} + Assume the incoming light is unpolarized, so its state is the density matrix + $$ + \rho=\frac12 I. + $$ + + The probability of passing the first filter $P_{\alpha_i}$ is + $$ + \operatorname{Prob}(P_i=1) + = + \operatorname{tr}(\rho P_{\alpha_i}) + = + \frac12\operatorname{tr}(P_{\alpha_i}) + = + \frac12. + $$ + + If the photon passes the first filter, the post-measurement state is + $$ + \rho_i + = + \frac{P_{\alpha_i}\rho P_{\alpha_i}}{\operatorname{tr}(\rho P_{\alpha_i})} + = + P_{\alpha_i}. + $$ + + + $$ + P_\alpha= + \begin{pmatrix} + \cos^2\alpha & \cos\alpha\sin\alpha \\ + \cos\alpha\sin\alpha & \sin^2\alpha + \end{pmatrix}. + $$ + + + Therefore + $$ + \operatorname{Prob}(P_j=1\mid P_i=1) + = + \operatorname{tr}(\rho_i P_{\alpha_j}) + = + \operatorname{tr}(P_{\alpha_i}P_{\alpha_j}) + = + \cos^2(\alpha_i-\alpha_j). + $$ + +\end{frame} +\begin{frame}{Recovering the observed law from the operator model (cont.)} + + + $$ + \begin{aligned} + \operatorname{Prob}(P_i=1,P_j=0) + &= + \operatorname{Prob}(P_i=1) + \bigl(1-\operatorname{Prob}(P_j=1\mid P_i=1)\bigr) \\ + &= + \frac12\bigl(1-\cos^2(\alpha_i-\alpha_j)\bigr) \\ + &= + \frac12\sin^2(\alpha_i-\alpha_j). + \end{aligned} + $$ + + This matches the experiment exactly. +\end{frame} + +\begin{frame}{Conclusion} + \begin{itemize} + \item The classical model predicts a Bell-type inequality for three $0$--$1$ random variables. + \item The polarization experiment violates that inequality. + \item The resolution is that the quantities measured are \emph{sequential probabilities}, not joint probabilities of classical random variables. + \item In quantum probability, events are modeled by projections on a Hilbert space, and measurement order matters. + \end{itemize} + + \vspace{0.6em} + This is one of the basic motivations for passing from classical probability to non-commutative probability. +\end{frame} + +\section{Concentration on Spheres and quantum states} +\begin{frame}{Quantum states: pure vs.\ mixed} + \begin{itemize} + \item A finite-dimensional quantum system is modeled by a complex Hilbert space (a complete inner product space) + $$ + \mathcal H \cong \mathbb C^{n+1}. + $$ + \item A \textbf{pure state} is represented by a unit vector + $$ + \psi \in \mathcal H, \qquad \|\psi\|=1. + $$ + \item A \textbf{mixed state} is represented by a density matrix + $$ + \rho \geq 0, \qquad \operatorname{tr}(\rho)=1. + $$ + \item Pure states describe maximal information; mixed states describe probabilistic mixtures or partial information. + \end{itemize} + + \vspace{0.4em} + \begin{block}{Key distinction} + Pure states form a curved geometric space; mixed states form a convex set inside the space of matrices. + \end{block} +\end{frame} + +\begin{frame}{Why pure states are not vectors} + \begin{itemize} + \item Two nonzero vectors that differ by a nonzero complex scalar represent the same physical state: + $$ + \psi \sim \lambda \psi, \qquad \lambda \in \mathbb C^\times. + $$ + \item In particular, multiplying by a phase $e^{i\theta}$ does not change any physical predictions. + \item Therefore the physical pure state is not a single vector, but the \emph{complex line} spanned by that vector. + \end{itemize} + + \vspace{0.4em} + Hence the space of pure states is + $$ + \mathbb P(\mathcal H) + = + (\mathcal H \setminus \{0\})/\mathbb C^\times. + $$ + + After choosing a basis $\mathcal H \cong \mathbb C^{n+1}$, this becomes + $$ + \mathbb P(\mathcal H) \cong \mathbb C P^n. + $$ +\end{frame} + +\begin{frame}{Relation with the sphere} + \begin{itemize} + \item Every nonzero vector can be normalized, so each pure state has a representative on the unit sphere + $$ + S^{2n+1} \subset \mathbb C^{n+1}. + $$ + \item Two unit vectors represent the same pure state exactly when they differ by a phase: + $$ + z \sim e^{i\theta} z. + $$ + \item Therefore + $$ + \mathbb C P^n = S^{2n+1}/S^1. + $$ + \end{itemize} + + \vspace{0.4em} + The quotient map + $$ + p:S^{2n+1}\to \mathbb C P^n, \qquad p(z)=[z]=\{\lambda z : \lambda \in \mathbb C^\times\}, + $$ + is the \textbf{Hopf fibration}. +\end{frame} + +\begin{frame}{How the metric descends to $\mathbb C P^n$} + \begin{itemize} + \item The sphere $S^{2n+1}$ inherits the round metric from the Euclidean metric on + $$ + \mathbb C^{n+1} \cong \mathbb R^{2n+2}. + $$ + \item The fibers of the Hopf map are circles + $$ + p^{-1}([z]) = \{e^{i\theta}z : \theta \in \mathbb R\}. + $$ + \item Tangent vectors split into: + \begin{itemize} + \item \textbf{vertical directions}: tangent to the $S^1$-fiber, + \item \textbf{horizontal directions}: orthogonal complement to the fiber. + \end{itemize} + \item The differential $dp$ identifies horizontal vectors on the sphere with tangent vectors on $\mathbb C P^n$. + \end{itemize} + + \vspace{0.4em} + This allows the round metric on $S^{2n+1}$ to define a metric on $\mathbb C P^n$. +\end{frame} + +\begin{frame}{The induced metric: Fubini--Study metric} + \begin{itemize} + \item The metric on $\mathbb C P^n$ obtained from the Hopf quotient is the + \textbf{Fubini--Study metric}. + \item So the geometric picture is: + $$ + S^{2n+1} + \xrightarrow{\text{Hopf fibration}} + \mathbb C P^n + $$ + $$ + \text{round metric} + \rightsquigarrow + \text{Fubini--Study metric}. + $$ + \item The normalized Riemannian volume measure induced by this metric gives the natural probability measure on pure states. + \end{itemize} + + \vspace{0.5em} + \begin{block}{Proof roadmap} + To prove this carefully, one usually shows: + \begin{enumerate} + \item $p:S^{2n+1}\to \mathbb C P^n$ is a smooth surjective submersion, + \item the vertical space is the tangent space to the $S^1$-orbit, + \item horizontal lifts are well defined, + \item the quotient metric is exactly the Fubini--Study metric. + \end{enumerate} + \end{block} +\end{frame} + +\begin{frame}{Maxwell-Boltzmann Distribution Law} + \begin{columns}[T] + \column{0.58\textwidth} + Consider the orthogonal projection + $$ + \pi_{n,k}:S^n(\sqrt{n})\to \mathbb{R}^k. + $$ + Its push-forward measure converges to the standard Gaussian: + $$ + (\pi_{n,k})_*\sigma^n\to \gamma^k. + $$ + + \vspace{0.5em} + This explains why Gaussian behavior emerges from high-dimensional spheres and supports the proof strategy for Levy concentration. + + \column{0.42\textwidth} + \begin{figure} + \includegraphics[width=\textwidth]{../latex/images/maxwell.png} + \end{figure} + \end{columns} +\end{frame} + +\begin{frame}{Levy Concentration} + \begin{block}{Levy's theorem} + If $f:S^n\to \mathbb{R}$ is $1$-Lipschitz, then there exists a median $a_0$ such that + $$ + \mu\{x\in S^n:|f(x)-a_0|\geq \epsilon\} + \leq + 2\exp\left(-\frac{(n-1)\epsilon^2}{2}\right). + $$ + \end{block} + + \begin{itemize} + \item In high dimension, most Lipschitz observables are almost constant. + \item This is the geometric mechanism behind generic entanglement. + \end{itemize} +\end{frame} + + +\section{Main Result} + +\begin{frame}{Generic Entanglement Theorem} + \begin{block}{Hayden--Leung--Winter} + Let $\psi\in \mathcal{P}(A\otimes B)$ be Haar-random and define + $$ + \beta=\frac{1}{\ln(2)}\frac{d_A}{d_B}. + $$ + For $d_B\geq d_A\geq 3$, + $$ + \operatorname{Pr}[H(\psi_A)<\log_2(d_A)-\alpha-\beta] + \leq + \exp\left( + -\frac{1}{8\pi^2\ln(2)} + \frac{(d_Ad_B-1)\alpha^2}{(\log_2 d_A)^2} + \right). + $$ + \end{block} + + With overwhelming probability, a random pure state is almost maximally entangled. +\end{frame} + +\begin{frame}{How the Entropy Observable Fits In} \begin{figure} \centering \begin{tikzpicture}[node distance=30mm, thick, @@ -142,224 +497,145 @@ 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] {$[0,\infty)$}; + \node[main] (sa) [below of=pa] {$\mathcal{S}(A)$}; + \node[main] (rng) [right of=sa] {$[0,\log_2 d_A]$}; - % 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); + \draw[towards_imp] (pa) -- node[above right] {$\psi\mapsto H(\psi_A)$} (rng); + \draw[towards] (sa) -- node[above] {$H$} (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 $A$ + \item The red arrow is the observable to which concentration is applied. + \item The projective description is natural because global phase does not change the physical state. \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 $\gamma^k$. - +\begin{frame}{Ingredients Behind the Tail Bound} + \begin{block}{Page-type lower bound} $$ - d\gamma^k(x)\coloneqq\frac{1}{\sqrt{2\pi}^k}\exp(-\frac{1}{2}\|x\|^2)dx - $$ - - $x\in \mathbb{R}^k$, $\|x\|^2=\sum_{i=1}^k x_i^2$ 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, - - $$ - (\pi_{n,k})_*\sigma^n\to \gamma^k\text{ weakly as }n\to \infty + \mathbb{E}[H(\psi_A)] + \geq + \log_2(d_A)-\frac{1}{2\ln(2)}\frac{d_A}{d_B}. $$ \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. + \begin{block}{Lipschitz estimate} $$ - \operatorname{Pr}[|f(x)-M_f|>\epsilon]\leq \exp(-\frac{n\epsilon^2}{2\eta^2}) + \|H(\psi_A)\|_{\mathrm{Lip}} + \leq + \sqrt{8}\,\log_2(d_A), + \qquad d_A\geq 3. $$ - $$ - \operatorname{Pr}[|f(x)-\langle f\rangle|>\epsilon]\leq 2\exp(-\frac{(n-1)\epsilon^2}{2\eta^2}) - $$ - \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 $f_n:S^{n-1}(\sqrt{n})\to \mathbb{R}$. - + Levy concentration plus these two estimates produces the exponential entropy tail bound. \end{frame} -\begin{frame}{Majorana stellar representation of the quantum state} +\section{Geometry of State Space} - \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{frame}{Observable Diameter} + \begin{block}{Definition} + For a metric-measure space $X$ and $\kappa>0$, + $$ + \obdiam_{\mathbb{R}}(X;-\kappa) + = + \sup_{f\in \operatorname{Lip}_1(X,\mathbb{R})} + \diameter(f_*\mu_X;1-\kappa). + $$ + \end{block} \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. + \item It asks how concentrated every $1$-Lipschitz real observable must be. + \item In the thesis, entropy is used as a concrete observable-diameter proxy. + \item Hopf fibration lets us compare $\mathbb{C}P^n$ with spheres. + \end{itemize} +\end{frame} + +\begin{frame}{A Geometric Consequence} + \begin{block}{Projective-space estimate} + For $0<\kappa<1$, + $$ + \obdiam(\mathbb{C}P^n(1);-\kappa)\leq O(\sqrt{n}). + $$ + \end{block} + + \begin{itemize} + \item First estimate observable diameter on spheres via Gaussian limits. + \item Then use the Hopf map $S^{2n+1}(1)\to \mathbb{C}P^n$. + \item This gives a geometric explanation for why many projective-space observables concentrate. + \end{itemize} +\end{frame} + +\section{Numerical Section} + +\begin{frame}{Entropy-Based Simulations} + \begin{itemize} + \item Sample Haar-random pure states in $\mathbb{C}^{d_A}\otimes\mathbb{C}^{d_B}$. + \item Compute reduced density matrices and entanglement entropy. + \item Measure shortest intervals containing mass $1-\kappa$ in the entropy distribution. + \item Compare concentration across: \begin{itemize} - \item Fiber bundles - \item Fubini-Study metric - \item Space of entangled states + \item real spheres, + \item complex projective spaces, + \item symmetric states via Majorana stellar representation. \end{itemize} - \item Riemannian geometry of $\mathbb{C}P^n$. + \end{itemize} +\end{frame} + +\begin{frame}{What the Data Suggests} + \begin{columns}[T] + \column{0.5\textwidth} + \begin{figure} + \includegraphics[width=\textwidth]{../latex/images/entropy_vs_dim.png} + \end{figure} + \centering + Entropy vs.\ ambient dimension + + \column{0.5\textwidth} + \begin{figure} + \includegraphics[width=\textwidth]{../latex/images/entropy_vs_dA.png} + \end{figure} + \centering + Entropy vs.\ subsystem dimension + \end{columns} + + \vspace{0.6em} + As dimension increases, the entropy distribution concentrates near the maximal value. +\end{frame} + +\section{Conclusion} + +\begin{frame}{Conclusion and Outlook} + \begin{itemize} + \item Concentration of measure explains generic high entanglement in large bipartite systems. + \item Complex projective space provides the natural geometric setting for pure quantum states. + \item Observable diameter gives a way to phrase concentration geometrically. + \item Ongoing directions: \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 + \item sharper estimates for $\mathbb{C}P^n$, + \item deeper use of Fubini--Study geometry, + \item Majorana stellar representation for symmetric states. \end{itemize} \end{itemize} \end{frame} \section{References} \begin{frame}[allowframebreaks]{References} - \nocite{*} % This will include all entries from the bibliography file + \nocite{*} \bibliographystyle{apalike} \bibliography{references} \end{frame} \begin{frame} - \begin{center} Q\&A \end{center} - \end{frame} -\end{document} \ No newline at end of file +\end{document} diff --git a/presentation/images/maxwell.png b/presentation/images/maxwell.png deleted file mode 100644 index d02910f..0000000 Binary files a/presentation/images/maxwell.png and /dev/null differ