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presentation/ZheyuanWu_HonorThesis_Presentation.tex

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+\DeclareMathOperator{\diameter}{diam}
 
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-% 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}
 
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-%\logo{} 
 \institute[]{Washington University in St. Louis}
 \date{\today}
-%\subject{}
-
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 \begin{document}
 
@@ -97,43 +64,431 @@
 \end{frame}
 
 \begin{frame}{Table of Contents}
-
   \hypersetup{linkcolor=black}
-\tableofcontents 
+  \tableofcontents
 \end{frame}
+\section{Motivation}
 
-\section{Memes}
-
-\begin{frame}{Memes}
-
+\begin{frame}{Light polarization and non-commutative probability}
         \begin{figure}
-        \includegraphics[width=0.5\textwidth]{./images/strengthvisuals.jpg}
+            \includegraphics[width=0.6\textwidth]{../latex/images/Filter_figure.png}
         \end{figure}
-
-    Note that the count of the beams is actually less than before.
+    \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{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
+\begin{frame}{Polarization experiment}
 
+    \vspace{0.5em}
+    Now consider three filters $F_1,F_2,F_3$ with directions
     $$
-        \operatorname{Pr}[H(\psi_A) < \log_2(d_A)-\alpha-\beta] \leq \exp\left(-\frac{1}{8\pi^2\ln(2)}\frac{(d_Ad_B-1)\alpha^2}{(\log_2(d_A))^2}\right)
+    \alpha_1,\alpha_2,\alpha_3.
+    $$
+    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 $F_i$.
+
+    \vspace{0.5em}
+    If these were classical random variables on one probability space, they would satisfy a Bell-type inequality.
+\end{frame}
+
+\begin{frame}{A classical Bell-type inequality}
+    \begin{block}{Bell-type inequality}
+    For any classical random variables $P_1,P_2,P_3\in\{0,1\}$,
+    $$
+    \operatorname{Prob}(P_1=1,P_3=0)
+    \leq
+    \operatorname{Prob}(P_1=1,P_2=0)
+    +
+    \operatorname{Prob}(P_2=1,P_3=0).
     $$
-        where $d_B\geq d_A\geq 3$.
     \end{block}
-    \cite{Hayden_2006}
-    Recall that the von Neumann entropy is defined as $H(\psi_A)=-\operatorname{Tr}(\psi_A\log_2(\psi_A))$.
 
+    \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}{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{frame}{Wait, but what is $\mathbb{C}P^n$ and where they are coming from?}
+    \begin{block}{Lipschitz estimate}
+        $$
+        \|H(\psi_A)\|_{\mathrm{Lip}}
+        \leq
+        \sqrt{8}\,\log_2(d_A),
+        \qquad d_A\geq 3.
+        $$
+    \end{block}
 
-    $\mathbb{C}P^n$ 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} 
+    Levy concentration plus these two estimates produces the exponential entropy tail bound.
+\end{frame}
 
-    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.
+\section{Geometry of State Space}
 
+
+\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 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 real spheres,
+            \item complex projective spaces,
+            \item symmetric states via Majorana stellar representation.
+        \end{itemize}
+    \end{itemize}
+\end{frame}
+
+\begin{frame}{What the Data Suggests}
+    \begin{columns}[T]
+        \column{0.5\textwidth}
         \begin{figure}
-        \includegraphics[width=0.5\textwidth]{./images/stereographic.png}
+            \includegraphics[width=\textwidth]{../latex/images/entropy_vs_dim.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}
+        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}
 
-    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 $\mathbb{C}P^n=S^2\times S^2\times \cdots \times S^2/S_n$.
-
+    \vspace{0.6em}
+    As dimension increases, the entropy distribution concentrates near the maximal value.
 \end{frame}
 
-\section{Future Plans}
-
-\begin{frame}{Future Plans}
+\section{Conclusion}
 
+\begin{frame}{Conclusion and Outlook}
     \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 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 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
+            \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}