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\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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\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 $0<\kappa<1$,
        $$
        \obdiam(\mathbb{C}P^n(1);-\kappa)\leq O(\sqrt{n}).
        $$
    \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}


    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 $O(\sqrt{n})$.
\end{frame}

\begin{frame}{A conjecture}
    \begin{block}{Wu's conjecture}

        For $0<\kappa<1$,
        $$
        \obdiam(\mathbb{C}P^n(1);-\kappa)= O(\sqrt{n}).
        $$
    
    \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}
