\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} \usepackage{tikz} \usepackage{braket} \DeclareMathOperator{\obdiam}{ObsDiam} \DeclareMathOperator{\diameter}{diam} \setbeamertemplate{caption}[numbered] \author[Zheyuan Wu]{Zheyuan Wu} \title{Measure concentration in complex projective space and quantum entanglement} \newcommand{\email}{w.zheyuan@wustl.edu} \setbeamertemplate{navigation symbols}{} \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}\insertsectionhead \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% } \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} \institute[]{Washington University in St. Louis} \date{\today} \begin{document} \begin{frame} \titlepage \end{frame} \begin{frame}{Table of Contents} \hypersetup{linkcolor=black} \tableofcontents \end{frame} \section{Formulation of Quantum Entangement} \begin{frame}{Why I'm here?} \centering \Large\itshape ``I think I can safely say that nobody understands quantum mechanics.'' \vspace{1em} \normalsize --- Richard Feynman \end{frame} \begin{frame}{Non-commutative probability space} We begin our discussion on a general type of probability space. \begin{block}{Non-commutative probability space} \label{defn:non-commutative_probability_space} A non-commutative probability space is a pair $(\mathscr{B}(\mathscr{H}),\mathscr{P})$, where $\mathscr{B}(\mathscr{H})$ is the set of all \textbf{bounded} linear operators on $\mathscr{H}$. $\mathscr{P}$ is the set of all orthogonal projections on $\mathscr{B}(\mathscr{H})$. The set $\mathscr{P}=\{P\in\mathscr{B}(\mathscr{H}):P^*=P=P^2\}$ is the set of all orthogonal projections on $\mathscr{B}(\mathscr{H})$. \end{block} \begin{table}[H] \centering \renewcommand{\arraystretch}{1} \label{tab:analog_of_classical_probability_theory_and_non_commutative_probability_theory} {\tiny \begin{tabular}{|p{0.45\linewidth}|p{0.45\linewidth}|} \hline \textbf{Classical probability} & \textbf{Non-commutative probability} \\ \hline Sample space $\Omega$, cardinality $\vert\Omega\vert=n$, example: $\Omega=\{0,1\}$ & Complex Hilbert space $\mathscr{H}$, dimension $\dim\mathscr{H}=n$, example: $\mathscr{H}=\mathbb{C}^2$ \\ \hline Common algebra of $\mathbb{C}$ valued functions & Algebra of bounded operators $\mathcal{B}(\mathscr{H})$ \\ \hline Events: indicator functions of sets & Projections: space of orthogonal projections $\mathscr{P}\subseteq\mathscr{B}(\mathscr{H})$ \\ \hline functions $f$ such that $f^2=f=\overline{f}$ & orthogonal projections $P$ such that $P^*=P=P^2$ \\ \hline $\mathbb{I}_{f^{-1}(\{\lambda\})}$ is the indicator function of the set $f^{-1}(\{\lambda\})$ & $P(\lambda)$ is the orthogonal projection to eigenspace \\ \hline $f=\sum_{\lambda\in \operatorname{Range}(f)}\lambda \mathbb{I}_{f^{-1}(\{\lambda\})}$ & $A=\sum_{\lambda\in \operatorname{sp}(A)}\lambda P(\lambda)$ \\ \hline Probability measure $\mu$ on $\Omega$ & Density operator $\rho$ on $\mathscr{H}$ \\ \hline \end{tabular} } \end{table} \end{frame} \begin{frame}{Quantum states} Given a non-commutative probability space $(\mathscr{B}(\mathscr{H}),\mathscr{P})$, \begin{block}{Definition of (Quantum) State} A state is a unit vector $\ket{\psi}$ in the Hilbert space $\mathscr{H}$, such that $\bra{\psi}\ket{\psi}=1$. Every state uniquely defines a map $\rho:\mathscr{P}\to[0,1]$, $\rho(P)=\bra{\psi}P\ket{\psi}$ (commonly named as density operator) such that: \begin{itemize} \item $\rho(O)=0$, where $O$ is the zero projection, and $\rho(I)=1$, where $I$ is the identity projection. \item If $P_1,P_2,\ldots,P_n$ are pairwise disjoint orthogonal projections, then $\rho(P_1 + P_2 + \cdots + P_n) = \sum_{i=1}^n \rho(P_i)$. \end{itemize} \end{block} Here $\psi$ is just a label for the vector. $\ket{\cdot}$ is called the ket (column vector), where the counterpart $\bra{\psi}$ is called the bra, used to denote the vector dual to $\psi$ (row vector/linear functional of $\ket{\psi})$. \end{frame} \begin{frame}{Quantum measurements} \begin{block}{Definition of Quantum Measurement} A measurement (observation) of a system prepared in a given state produces an outcome $x$, $x$ is a physical event that is a subset of the set of all possible outcomes. For each $x$, we associate a measurement operator $M_x$ on $\mathscr{H}$. Given the initial state (pure state, unit vector) $u$, the probability of measurement outcome $x$ is given by: $$ p(x)=\|M_xu\|^2 $$ Note that to make sense of this definition, the collection of measurement operators $\{M_x\}$ must satisfy the completeness requirement: $$ 1=\sum_{x\in X} p(x)=\sum_{x\in X}\|M_xu\|^2=\sum_{x\in X}\langle M_xu,M_xu\rangle=\langle u,(\sum_{x\in X}M_x^*M_x)u\rangle $$ So $\sum_{x\in X}M_x^*M_x=I$ (Law of total probability). \end{block} \end{frame} % \input{./backgrounds.tex} \begin{frame}{Information theory in classical systems} In probability theory, an important measurement of uncertainty is entropy. It characterizes the information content of a random variable. \begin{block}{Shannon entropy} Given a classical probability vector $p=(p_1,\dots,p_n)$ with $\sum_i p_i=1$, $$ H(p)=-\sum_{i=1}^n p_i \log_2 p_i. $$ This measures uncertainty of a \emph{chosen measurement outcome}. \end{block} \end{frame} \begin{frame}{Information theory in quantum systems} \begin{block}{von Neumann entropy} For a density matrix $\rho$, $$ S(\rho)=-\operatorname{Tr}(\rho\log_2\rho). $$ This measures the intrinsic uncertainty of the quantum state and is basis-independent. \end{block} \begin{block}{Entanglement entropy} For a bipartite pure state $|\Psi\rangle\in \mathcal{H}_A\otimes\mathcal{H}_B$, define the reduced state $\rho_A=\operatorname{Tr}_B\bigl(|\Psi\rangle\langle\Psi|\bigr).$ Its entanglement entropy is $$ E(|\Psi\rangle)=H(\rho_A). $$ Thus entanglement entropy is the von Neumann entropy of a subsystem, and it measures how entangled the bipartite pure state is. \end{block} \end{frame} \begin{frame}{Conclusion of Non-commutative probability space} \begin{table}[H] \centering {\tiny \begin{tabular}{|p{0.45\linewidth}|p{0.45\linewidth}|} \hline \textbf{Classical probability} & \textbf{Non-commutative probability} \\ \hline Sample space $\Omega$, cardinality $\vert\Omega\vert=n$, example: $\Omega=\{0,1\}$ & Complex Hilbert space $\mathscr{H}$, dimension $\dim\mathscr{H}=n$, example: $\mathscr{H}=\mathbb{C}^2$ \\ \hline Common algebra of $\mathbb{C}$ valued functions & Algebra of bounded operators $\mathcal{B}(\mathscr{H})$ \\ \hline $f\mapsto \bar{f}$ complex conjugation & $P\mapsto P^*$ adjoint \\ \hline Events: indicator functions of sets & Projections: space of orthogonal projections $\mathscr{P}\subseteq\mathscr{B}(\mathscr{H})$ \\ \hline functions $f$ such that $f^2=f=\overline{f}$ & orthogonal projections $P$ such that $P^*=P=P^2$ \\ \hline $\mathbb{R}$-valued functions $f=\overline{f}$ & self-adjoint operators $A=A^*$ \\ \hline $\mathbb{I}_{f^{-1}(\{\lambda\})}$ is the indicator function of the set $f^{-1}(\{\lambda\})$ & $P(\lambda)$ is the orthogonal projection to eigenspace \\ \hline $f=\sum_{\lambda\in \operatorname{Range}(f)}\lambda \mathbb{I}_{f^{-1}(\{\lambda\})}$ & $A=\sum_{\lambda\in \operatorname{sp}(A)}\lambda P(\lambda)$ \\ \hline Probability measure $\mu$ on $\Omega$ & Density operator $\rho$ on $\mathscr{H}$ \\ \hline Delta measure $\delta_\omega$ & Pure state $\rho=\vert\psi\rangle\langle\psi\vert$ \\ \hline $\mu$ is non-negative measure and $\sum_{i=1}^n\mu(\{i\})=1$ & $\rho$ is positive semi-definite and $\operatorname{Tr}(\rho)=1$ \\ \hline Expected value of random variable $f$ is $\mathbb{E}_{\mu}(f)=\sum_{i=1}^n f(i)\mu(\{i\})$ & Expected value of operator $A$ is $\mathbb{E}_\rho(A)=\operatorname{Tr}(\rho A)$ \\ \hline Variance of random variable $f$ is $\operatorname{Var}_\mu(f)=\sum_{i=1}^n (f(i)-\mathbb{E}_\mu(f))^2\mu(\{i\})$ & Variance of operator $A$ is $\operatorname{Var}_\rho(A)=\operatorname{Tr}(\rho A^2)-\operatorname{Tr}(\rho A)^2$ \\ \hline Covariance of random variables $f$ and $g$ is $\operatorname{Cov}_\mu(f,g)=\sum_{i=1}^n (f(i)-\mathbb{E}_\mu(f))(g(i)-\mathbb{E}_\mu(g))\mu(\{i\})$ & Covariance of operators $A$ and $B$ is $\operatorname{Cov}_\rho(A,B)=\operatorname{Tr}(\rho A\circ B)-\operatorname{Tr}(\rho A)\operatorname{Tr}(\rho B)$ \\ \hline Composite system is given by Cartesian product of the sample spaces $\Omega_1\times\Omega_2$ & Composite system is given by tensor product of the Hilbert spaces $\mathscr{H}_1\otimes\mathscr{H}_2$ \\ \hline Product measure $\mu_1\times\mu_2$ on $\Omega_1\times\Omega_2$ & Tensor product of space $\rho_1\otimes\rho_2$ on $\mathscr{H}_1\otimes\mathscr{H}_2$ \\ \hline Marginal distribution $\pi_*v$ & Partial trace $\operatorname{Tr}_2(\rho)$ \\ \hline \end{tabular} } \vspace{0.5cm} \end{table} \end{frame} \begin{frame}{So what?} \begin{block}{Lemma: That's all we need.} All quantum operations can be constructed by composing four kinds of transformations: \begin{enumerate} \item Unitary operations. $U(\cdot)$ for any quantum state. $A^* A=AA^*=I$, $A$ is the matrix of $U$. (It is possible to apply a non-unitary operation for an open quantum system, but usually leads to non-recoverable loss of information) \item Extend the system. Given a quantum state $\rho\in\mathcal{H}^N$, we can extend it to a larger quantum system by "entangle" it with some new states $\sigma\in \mathcal{H}^K$ and get $\rho'=\rho\otimes\sigma\in \mathcal{H}^N\otimes \mathcal{H}^K$. \item Partial trace. Given a quantum state $\rho\in\mathcal{H}^N$ and some reference state $\sigma\in\mathcal {H}^K$, we can trace out some subsystems and get a new state $\rho'\in\mathcal{H}^{N-K}$. \item Selective measurement. Given a quantum state, we measure it and get a classical bit. \end{enumerate} \end{block} \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 $$ \rho=\sum_{j=1}^n p_j|\psi_j\rangle\langle\psi_j|, \qquad \sum_{j=1}^n p_j=1, \qquad p_j\geq 0. $$ \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}{Pure states live in the complex projective space} \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 (denoted by $\mathcal{P}(\mathcal H)$) is $$ \mathcal{P}(\mathcal H) = (\mathcal H \setminus \{0\})/\mathbb C^\times. $$ After choosing a basis $\mathcal H \cong \mathbb C^{n+1}$, this becomes $$ \mathcal{P}(\mathcal H) \cong \mathbb C P^n. $$ \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}{The induced riemmanian metric: Fubini--Study metric} \begin{block}{Definition of Riemannian metric} Let $M$ be a smooth manifold. A \textit{\textbf{Riemannian metric}} on $M$ is a smooth covariant tensor field $g\in \mathcal{T}^2(M)$ such that for each $p\in M$, $g_p$ is an inner product on $T_pM$ (Vector space formed by the tangent vectors relative to the manifold $M$ at $p$). $g_p(v,v)\geq 0$ for each $p\in M$ and each $v\in T_pM$. equality holds if and only if $v=0$. \end{block} \begin{itemize} \item The geometric picture is $$ S^{2n+1} \xrightarrow{\text{Hopf fibration}} \mathbb C P^n, \qquad \text{round metric} \rightsquigarrow \text{Fubini--Study metric}. $$ \begin{columns}[T] \column{0.5\textwidth} The sphere $S^{2n+1}\subset \mathbb C^{n+1}$ has the \textbf{round metric} $$ g_{\mathrm{round}}=\sum_{j=0}^n (dx_j^2+dy_j^2)\big|_{S^{2n+1}}, $$ induced from the Euclidean metric on $\mathbb R^{2n+2}$. \column{0.5\textwidth} In homogeneous coordinates $[z]\in\mathbb C P^n$, the \textbf{Fubini--Study metric} is $$ g_{FS} = \frac{\langle dz,dz\rangle \langle z,z\rangle-|\langle z,dz\rangle|^2}{\langle z,z\rangle^2}, $$ \end{columns} \end{itemize} \end{frame} \begin{frame}{So what?} With everything we have here, we are ready to answer the question: \vspace{2em} \begin{center} \textbf{How a random bipartite pure state $\mathcal{P}(A\otimes B)$ is distributed on the complex projective space? And how entangled $H(\psi_A)$ it is?} \end{center} \end{frame} \section{Volume Distribution in High Dimensional Spaces} \begin{frame}{Maxwell-Boltzmann Distribution Law} \begin{figure}[H] \includegraphics[width=0.7\textwidth]{../latex/images/maxwell.png} \end{figure} Consider the orthogonal projection $0\leq k< n$ $$ \pi_{n,k}:S^n(\sqrt{n})\to \mathbb{R}^k. $$ Its push-forward measure converges to the standard Gaussian as dimensions increase $n\to \infty$. $$ (\pi_{n,k})_*\sigma^n\to \gamma^k. $$ Another familiar name when $k=1$ is the central limit theorem. \end{frame} \begin{frame}{Levy Concentration} % \begin{block}{Definition of Lipschitz function} % A function $f:X\to Y$, where $X,Y$ are metric spaces, is $L$-Lipschitz if there exists a constant $L$ such that $|f(x)-f(y)|\leq L|y-x|$ for all $x,y\in S^n$. % \end{block} \begin{block}{Levy's lemma} If $f:S^n\to \mathbb{R}$ is $1$-Lipschitz, then there exists a $a_0$ such that for $\epsilon>0$, $$ \mu\{x\in S^n:|f(x)-a_0|\geq \epsilon\} \leq 2\exp\left(-\frac{(n-1)\epsilon^2}{2}\right). $$ \end{block} \begin{itemize} \item In high dimension, most Lipschitz observables are almost constant. \item Here $a_0$ resembles the "median" of the set $f(S^n)$, that is half of the measure of the observations is bounded below/above by $a_0$. \end{itemize} \end{frame} \section{Main Result} \begin{frame}{How the Entropy Observable Fits In} \begin{figure} \centering \begin{tikzpicture}[node distance=30mm, thick, main/.style={draw, draw=white}, towards/.style={->}, towards_imp/.style={->,red}, mutual/.style={<->} ] \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] {$\mathcal{S}(A)$}; \node[main] (rng) [right of=sa] {$[0,\log_2 d_A]$}; \draw[mutual] (cp) -- (pa); \draw[towards] (pa) -- node[left] {$\operatorname{Tr}_B$} (sa); \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 Recall that $\mathcal{P}(A\otimes B)$ is the set of pure states on $A\otimes B$. $\operatorname{Tr}_B$ is the partial trace over $B$. $\mathcal{S}(A)$ is the set of mixed states on $A$. $H$ is the shannon entropy function, $H(\psi_A)$ is the entanglement entropy function. \item The red arrow is the observable to which concentration is applied. \end{itemize} \end{frame} \begin{frame}{Ingredients Behind the Tail Bound} \begin{block}{Page-type lower bound} $$ \mathbb{E}[H(\psi_A)] \geq \log_2(d_A)-\frac{1}{2\ln(2)}\frac{d_A}{d_B}. $$ \end{block} \begin{block}{Lipschitz estimate for $H(\psi_A)$} The Lipschitz constant for the function $ H(\psi_A)$ should be upper bounded by $ \sqrt{8}\,\log_2(d_A)$, for $d_A\geq 3$. \end{block} Levy concentration plus these two estimates produces the exponential entropy tail bound. \end{frame} \begin{frame}{Generic Entanglement Theorem} \begin{block}{Hayden--Leung--Winter} Let $\psi\in \mathcal{P}(A\otimes B)$ be a random pure state on $A\otimes B$ and define $$ \beta=\frac{1}{\ln(2)}\frac{d_A}{d_B}. $$ For $d_B\geq d_A\geq 3$, with $\alpha\geq 0$ by our choice, $$ \operatorname{Pr}[H(\psi_A)<\log_2(d_A)-\alpha-\beta] \leq \exp\left( -\frac{1}{8\pi^2\ln(2)} \frac{(d_Ad_B-1)\alpha^2}{(\log_2 d_A)^2} \right). $$ \end{block} As $d_B\to \infty$, with overwhelming probability $1-\exp\left( -\frac{1}{8\pi^2\ln(2)} \frac{(d_Ad_B-1)\alpha^2}{(\log_2 d_A)^2} \right)=1-\Theta(e^{-c d_B})$, a random pure state is almost maximally entangled $\log_2(d_A)-\frac{1}{2\ln(2)}\frac{d_A}{d_B}=\log_2(d_A)-\Theta(\frac{1}{d_B})$. \end{frame} \begin{frame}{A natural question from the observables} \textbf{What does the hayden--leung--winter theorem generalize the behavior of the lipschitz function $S^n\to \mathbb{R}$ and the lipschitz function $\mathbb{C}P^n\to \mathbb{R}$ as $n\to \infty$?} \end{frame} \section{Metric-Measure space after Gromov} \begin{frame}{Observable diameter: the inner definition} \begin{block}{Partial diameter on $\mathbb{R}$} Let $\mu$ be a Borel probability measure on $\mathbb{R}$ and let $\alpha \in (0,1]$. The \textbf{partial diameter} of $\mu$ at mass level $\alpha$ is $$ \diameter(\mu;\alpha):=\inf_{A\subset \mathbb{R}}\left\{ \diameter(A): \mu(A)\geq \alpha \right\}. $$ where $$ \diameter(A):=\sup_{x,y\in A}|x-y|. $$ \end{block} \vspace{0.4em} \begin{itemize} \item The partial diameter asks for: what is the shortest interval I need to capture at least $\alpha$ of the mass (measure)? \item If $1$ is the total measure of the space, $\diameter(\mu;1-\kappa)$, measures how tightly we can capture \emph{most} of the distribution, allowing us to discard a set of mass at most $\kappa$. \end{itemize} \end{frame} \begin{frame}{Observable diameter of a metric-measure space} \begin{block}{Definition} Let $X=(X,d_X,\mu_X)$ be a metric-measure space and let $\kappa>0$. The \textbf{observable diameter} of $X$ is $$ \obdiam_{\mathbb{R}}(X;-\kappa) := \sup_{f\in \operatorname{Lip}_1(X,\mathbb{R})} \diameter(f_*\mu_X;1-\kappa), $$ where $\operatorname{Lip}_1(X,\mathbb{R})$ is the set of all $1$-Lipschitz functions $f:X\to\mathbb{R}$, and $f_*\mu_X$ is the pushforward measure on $\mathbb{R}$. \end{block} \vspace{0.4em} \begin{itemize} \item Each $1$-Lipschitz function $f$ is viewed as an \textbf{observable} on $X$. \item The pushforward measure $f_*\mu_X$ is the distribution of the values of that observable. \item If $\obdiam_{\mathbb{R}}(X;-\kappa)$ is small, then \emph{every} $1$-Lipschitz observable is strongly concentrated. \end{itemize} \end{frame} \begin{frame}{A Geometric Consequence} \vspace{0.4em} \begin{block}{Projective-space estimate from Gromov} For $0<\kappa<1$, $$ \obdiam(\mathbb{C}P^n(1);-\kappa)\leq O\left(\frac{1}{\sqrt{n}}\right). $$ \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} Hayden's work suggests that if the entropy function is a ``good'' proxy for the observable diameter, the difference of \textbf{the order of the growth} between $\obdiam(\mathbb{C}P^n(1);-\kappa)$ and $\obdiam(S^{2n+1}(1);-\kappa)$ should be smaller than $O\left(\frac{1}{\sqrt{n}}\right)$. \end{frame} \begin{frame}{A conjecture} \begin{block}{Wu's conjecture} For $0<\kappa<1$, $$ \obdiam(\mathbb{C}P^n(1);-\kappa)= O\left(\frac{1}{\sqrt{n}}\right). $$ \end{block} Additional works need to be done to verify this conjecture. \begin{itemize} \item Entropy function is not globally lipschitz, so we need to bound the deficit of the entropy function. \item Normalize by the Lipschitz constant of $f$ to obtain a weak lower bound for the observable diameter with the algebraic varieties on $\mathbb{C}P^n(1)$. \item Continue to study and interpret the overall concentration mechanism geometrically through the positive Ricci curvature of the Fubini--Study metric and Lévy--Gromov type inequalities. \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 real spheres, \item complex projective spaces \end{itemize} \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} \begin{frame}{Results for concentration of random states in lower dimensional spaces} \begin{columns}[T] \column{0.5\textwidth} \begin{figure} \includegraphics[width=\textwidth]{../results/exp-20260311-154003/hist_sphere_16.png} \end{figure} \centering Entropy distribution for $S^{15}$ \column{0.5\textwidth} \begin{figure} \includegraphics[width=\textwidth]{../results/exp-20260311-154003/hist_cp_16x16.png} \end{figure} \centering Entropy distribution for $\mathbb{C}P^{16}\otimes\mathbb{C}P^{16}$ \end{columns} \end{frame} \begin{frame}{Results for concentration of random states in higher dimensional spaces} \begin{columns}[T] \column{0.5\textwidth} \begin{figure} \includegraphics[width=\textwidth]{../results/exp-20260311-154003/hist_sphere_256.png} \end{figure} \centering Entropy distribution for $S^{255}$ \column{0.5\textwidth} \begin{figure} \includegraphics[width=\textwidth]{../results/exp-20260311-154003/hist_cp_256x256.png} \end{figure} \centering Entropy distribution for $\mathbb{C}P^{256}\otimes\mathbb{C}P^{256}$ \end{columns} \end{frame} \section{Conclusion} \begin{frame}{Conclusion and Outlook} \begin{itemize} \item Complex projective space provides the natural geometric setting for pure quantum states. \item Concentration of measure explains generic high entanglement in large bipartite systems. \item Observable diameter gives a way to phrase concentration geometrically. \item Ongoing directions: \begin{itemize} \item sharper estimates for $\mathbb{C}P^n$ \item deeper use of Fubini--Study geometry \item recursive learning on new theorems and mathematical tools \end{itemize} \end{itemize} \end{frame} % \section{References} % \begin{frame}[allowframebreaks]{References} % \nocite{*} % \bibliographystyle{apalike} % \bibliography{references} % \end{frame} \begin{frame} \begin{center} Q\&A \end{center} \end{frame} \end{document}