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\author[Zheyuan Wu]{Zheyuan Wu}
\title{Measure concentration in complex projective space and quantum entanglement}
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\begin{document}

\begin{frame}
\titlepage
\end{frame}

\begin{frame}{Table of Contents}

  \hypersetup{linkcolor=black}
\tableofcontents 
\end{frame}

\section{Memes}

\begin{frame}{Memes}

    \begin{figure}
        \includegraphics[width=0.5\textwidth]{./images/strengthvisuals.jpg}
    \end{figure}

    Note that the count of the beams is actually less than before.
\end{frame}

\section{Decomposing the statements}

\begin{frame}{Decomposing the statements}

    \begin{block}{Concentration of measure effect}
        
        Let $\psi\in \mathcal{P}(A\otimes B)$ be a random pure state on $A\otimes B$.

        If we define $\beta=\frac{1}{\ln(2)}\frac{d_A}{d_B}$, then we have

        $$
        \operatorname{Pr}[H(\psi_A) < \log_2(d_A)-\alpha-\beta] \leq \exp\left(-\frac{1}{8\pi^2\ln(2)}\frac{(d_Ad_B-1)\alpha^2}{(\log_2(d_A))^2}\right)
        $$
        where $d_B\geq d_A\geq 3$.
    \end{block}
    \cite{Hayden_2006}
    Recall that the von Neumann entropy is defined as $H(\psi_A)=-\operatorname{Tr}(\psi_A\log_2(\psi_A))$.

\end{frame}

\begin{frame}{What the system actually looks like}
    \begin{figure}
        \centering
        \begin{tikzpicture}[node distance=30mm, thick,
            main/.style={draw, draw=white},
            towards/.style={->},
            towards_imp/.style={->,red},
            mutual/.style={<->}
            ]
            % define nodes
            \node[main] (cp) {$\mathbb{C}P^{d_A d_B-1}$};
            \node[main] (pa) [left of=cp] {$\mathcal{P}(A\otimes B)$};
            \node[main] (sa) [below of=pa] {$S_A$};
            \node[main] (rng) [right of=sa] {$[0,\infty)$};

            % draw edges
            \draw[mutual] (cp) -- (pa);
            \draw[towards] (pa) -- node[left] {$\operatorname{Tr}_B$} (sa);
            \draw[towards_imp] (pa) -- node[above right] {$f$} (rng);
            \draw[towards] (sa) -- node[above] {$H(\psi_A)$} (rng);
        \end{tikzpicture}
    \end{figure}

    \begin{itemize}
        \item The red arrow is the concentration of measure effect. $f=H(\operatorname{Tr}_B(\psi))$.
        \item $S_A$ denotes the mixed states on $A$
    \end{itemize}
\end{frame}

\section{Geometry of Quantum States}

\begin{frame}{Wait, but what is $\mathbb{C}P^n$ and where they are coming from?}

    $\mathbb{C}P^n$ is the set of all complex lines in $\mathbb{C}^{n+1}$, or equivalently the space of equivalence classes of $n+1$ complex numbers up to a scalar multiple. \cite{Bengtsson_Życzkowski_2017} 

    One can find that every odd dimensional sphere $S^{2n+1}$ under the group action of $S^1$, denoted by $S^{2n+1}/S^1$, is a complex projective space $\mathbb{C}P^n$ (complex-dimensional). Recall Math 416.

    \begin{figure}
        \includegraphics[width=0.5\textwidth]{./images/stereographic.png}
    \end{figure}

    Detailed proof involves the Hopf fibration structures.
    It's a natural projective Hilbert space.
\end{frame}

\begin{frame}{Some interesting claims about $\mathbb{C}P^n$}

    ..... The claim is that every physical system can be modelled by $\mathbb{C}P^n$ for some (possibly infinite) value of $n$, provided taht a definite correspondence between the system and the point of $\mathbb{C}P^n$ is set up. \cite{Bengtsson_Życzkowski_2017}
\end{frame}

\begin{frame}{Initial attempts for Levy's concentration lemma}
    Consider the orthogonal projection from $\mathbb{R}^{n+1}$, the space where $S^n$ is embedded, to $\mathbb{R}^k$, we denote the restriction of the projection as $\pi_{n,k}:S^n(\sqrt{n})\to \mathbb{R}^k$. Note that $\pi_{n,k}$ is a 1-Lipschitz function (projection will never increase the distance between two points).

    We denote the normalized Riemannian volume measure on $S^n(\sqrt{n})$ as $\sigma^n(\cdot)$, and $\sigma^n(S^n(\sqrt{n}))=1$.

    \begin{block}{Gaussian measure}
        We denote the Gaussian measure on $\mathbb{R}^k$ as $\gamma^k$.

        $$
        d\gamma^k(x)\coloneqq\frac{1}{\sqrt{2\pi}^k}\exp(-\frac{1}{2}\|x\|^2)dx
        $$
        
        $x\in \mathbb{R}^k$, $\|x\|^2=\sum_{i=1}^k x_i^2$ is the Euclidean norm, and $dx$ is the Lebesgue measure on $\mathbb{R}^k$.
        
    \end{block}
    Basically, you can consider the Gaussian measure as the normalized Lebesgue measure on $\mathbb{R}^k$ with standard deviation $1$.

\end{frame}

\begin{frame}{Maxwell-Boltzmann distribution law}
    
    \begin{block}{Maxwell-Boltzmann distribution law}
        For any natural number $k$,
        
        $$
        \frac{d(\pi_{n,k})_*\sigma^n(x)}{dx}\to \frac{d\gamma^k(x)}{dx}
        $$
        
        where $(\pi_{n,k})_*\sigma^n$ is the push-forward measure of $\sigma^n$ by $\pi_{n,k}$.
        
        In other words,
        
        $$
        (\pi_{n,k})_*\sigma^n\to \gamma^k\text{ weakly as }n\to \infty
        $$
    \end{block}
\end{frame}

\begin{frame}{Maxwell-Boltzmann distribution law}

    It also has another name, the Projective limit theorem. \cite{romanvershyni}

    If $X\sim \operatorname{Unif}(S^n(\sqrt{n}))$, then for any fixed unit vector $x$ we have $\langle X,x\rangle\to N(0,1)$ in distribution as $n\to \infty$.

    \begin{figure}
        \includegraphics[width=0.8\textwidth]{./images/maxwell.png}
    \end{figure}

\end{frame}

\begin{frame}{Proof of Maxwell-Boltzmann distribution law I}

    This part is from \cite{shioya2014metricmeasuregeometry}.
    
    We denote the $n$ dimensional volume measure on $\mathbb{R}^k$ as $\operatorname{vol}_k$.

    Observe that $\pi_{n,k}^{-1}(x),x\in \mathbb{R}^k$ is isometric to $S^{n-k}(\sqrt{n-\|x\|^2})$, that is, for any $x\in \mathbb{R}^k$, $\pi_{n,k}^{-1}(x)$ is a sphere with radius $\sqrt{n-\|x\|^2}$ (by the definition of $\pi_{n,k}$).
    
    So,
    
    $$
    \begin{aligned}
    \frac{d(\pi_{n,k})_*\sigma^n(x)}{dx}&=\frac{\operatorname{vol}_{n-k}(\pi_{n,k}^{-1}(x))}{\operatorname{vol}_k(S^n(\sqrt{n}))}\\
    &=\frac{(n-\|x\|^2)^{\frac{n-k}{2}}}{\int_{\|x\|\leq \sqrt{n}}(n-\|x\|^2)^{\frac{n-k}{2}}dx}\\
    \end{aligned}
    $$
    
    as $n\to \infty$.
    
    note that $\lim_{n\to \infty}{(1-\frac{a}{n})^n}=e^{-a}$ for any $a>0$.
\end{frame}

\begin{frame}{Proof of Maxwell-Boltzmann distribution law II}
    
    $(n-\|x\|^2)^{\frac{n-k}{2}}=\left(n(1-\frac{\|x\|^2}{n})\right)^{\frac{n-k}{2}}\to n^{\frac{n-k}{2}}\exp(-\frac{\|x\|^2}{2})$
    
    So
    
    $$
    \begin{aligned}
    \frac{(n-\|x\|^2)^{\frac{n-k}{2}}}{\int_{\|x\|\leq \sqrt{n}}(n-\|x\|^2)^{\frac{n-k}{2}}dx}&=\frac{e^{-\frac{\|x\|^2}{2}}}{\int_{x\in \mathbb{R}^k}e^{-\frac{\|x\|^2}{2}}dx}\\
    &=\frac{1}{(2\pi)^{\frac{k}{2}}}e^{-\frac{\|x\|^2}{2}}\\
    &=\frac{d\gamma^k(x)}{dx}
    \end{aligned}
    $$    

\end{frame}

\begin{frame}{Levy's concentration lemma}
    \begin{block}{Levy's concentration lemma}

        Let $f:S^{n-1}\to \mathbb{R}$ be a $\eta$-Lipschitz function. Let $M_f$ denote the median of $f$ and $\langle f\rangle$ denote the mean of $f$. (Note this can be generalized to many other manifolds (spaces that locally resembles Euclidean space).)

        Select a random point $x\in S^{n-1}$ with $n>2$ according to the uniform measure (Haar measure). Then the probability of observing a value of $f$ much different from the reference value is exponentially small.

        $$
        \operatorname{Pr}[|f(x)-M_f|>\epsilon]\leq \exp(-\frac{n\epsilon^2}{2\eta^2})
        $$
        $$
        \operatorname{Pr}[|f(x)-\langle f\rangle|>\epsilon]\leq 2\exp(-\frac{(n-1)\epsilon^2}{2\eta^2})
        $$

    \end{block}

    The Maxwell-Boltzmann distribution law will help us find the limit of measures on hemisphere $S^{n-1}$ under the series of functions $f_n:S^{n-1}(\sqrt{n})\to \mathbb{R}$.

\end{frame}

\begin{frame}{Majorana stellar representation of the quantum state}

    \begin{figure}
        \centering
        \begin{tikzpicture}[node distance=40mm, thick,
            main/.style={draw, draw=white},
            towards/.style={->},
            towards_imp/.style={<->,red},
            mutual/.style={<->}
            ]
            \node[main] (cp) {$\mathbb{C}P^{n}$};
            \node[main] (c) [below of=cp] {$\mathbb{C}^{n+1}$};
            \node[main] (p) [right of=cp] {$\mathbb{P}^n$};
            \node[main] (sym) [below of=p] {$\operatorname{Sym}_n(\mathbb{C}P^1)$};
            % draw edges
            \draw[towards] (c) -- (cp) node[midway, left] {$z\sim \lambda z$};
            \draw[towards] (c) -- (p) node[midway, fill=white] {$w(z)=\sum_{i=0}^n Z_i z^i$};
            \draw[towards_imp] (cp) -- (p) node[midway, above] {$w(z)\sim w(\lambda z)$};
            \draw[mutual] (p) -- (sym) node[midway, right] {root of $w(z)$};
        \end{tikzpicture}
    \end{figure}

    Basically, there is a bijection between the complex projective space $\mathbb{C}P^n$ and the set of roots of a polynomial of degree $n$.

    We can use a symmetric group of permutation of $n$ complex numbers (or $S^2$) to represent the $\mathbb{C}P^n$, that is $\mathbb{C}P^n=S^2\times S^2\times \cdots \times S^2/S_n$.

\end{frame}

\section{Future Plans}

\begin{frame}{Future Plans}

    \begin{itemize}
        \item The physical meaning of the mathematical structures, the correspondence, and the relationship between the measures, quantum states, and the geometry of topological spaces.
        \begin{itemize}
            \item Fiber bundles
            \item Fubini-Study metric
            \item Space of entangled states
        \end{itemize}
        \item Riemannian geometry of $\mathbb{C}P^n$.
        \begin{itemize}
            \item Ricci curvature
            \item Levy's Isoperimetric inequality
            \item Lipschitz constants and Levi-Civita connection
            \item Local operations and classical communication (LOCC)
        \end{itemize}
        \item The proof of the Page's formula.
        \item Majorana stellar representation of the quantum state. And possibly the concentration of measure effect on that.
        \item Relations to Gromov's works \cite{MGomolovs}
        \begin{itemize}
            \item Levy families
            \item Observable diameters
        \end{itemize}
    \end{itemize}
\end{frame}

\section{References}
\begin{frame}[allowframebreaks]{References}
    \nocite{*} % This will include all entries from the bibliography file
    \bibliographystyle{apalike}
    \bibliography{references}
\end{frame}

\begin{frame}

\begin{center}
    Q\&A
\end{center}

\end{frame}

\end{document}