642 lines
19 KiB
TeX
642 lines
19 KiB
TeX
\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}
|
|
|
|
\DeclareMathOperator{\sen}{sen}
|
|
\DeclareMathOperator{\tg}{tg}
|
|
\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{Motivation}
|
|
|
|
\begin{frame}{Light polarization and non-commutative probability}
|
|
\begin{figure}
|
|
\includegraphics[width=0.6\textwidth]{../latex/images/Filter_figure.png}
|
|
\end{figure}
|
|
\begin{itemize}
|
|
\item Light passing through a polarizer becomes polarized in the direction of that filter.
|
|
\item If two filters are placed with relative angle $\alpha$, the transmitted intensity decreases as $\alpha$ increases.
|
|
\item In particular, the transmitted intensity vanishes when $\alpha=\pi/2$.
|
|
\end{itemize}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Polarization experiment}
|
|
|
|
\vspace{0.5em}
|
|
Now consider three filters $F_1,F_2,F_3$ with directions
|
|
$$
|
|
\alpha_1,\alpha_2,\alpha_3.
|
|
$$
|
|
Testing them pairwise suggests introducing three $0$--$1$ random variables
|
|
$$
|
|
P_1,P_2,P_3,
|
|
$$
|
|
where $P_i=1$ means that the photon passes filter $F_i$.
|
|
|
|
\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 $P_1,P_2,P_3\in\{0,1\}$,
|
|
$$
|
|
\operatorname{Prob}(P_1=1,P_3=0)
|
|
\leq
|
|
\operatorname{Prob}(P_1=1,P_2=0)
|
|
+
|
|
\operatorname{Prob}(P_2=1,P_3=0).
|
|
$$
|
|
\end{block}
|
|
|
|
\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}{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,
|
|
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 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}
|
|
|
|
\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}
|
|
$$
|
|
\|H(\psi_A)\|_{\mathrm{Lip}}
|
|
\leq
|
|
\sqrt{8}\,\log_2(d_A),
|
|
\qquad d_A\geq 3.
|
|
$$
|
|
\end{block}
|
|
|
|
Levy concentration plus these two estimates produces the exponential entropy tail bound.
|
|
\end{frame}
|
|
|
|
\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=\textwidth]{../latex/images/entropy_vs_dim.png}
|
|
\end{figure}
|
|
\centering
|
|
Entropy vs.\ ambient dimension
|
|
|
|
\column{0.5\textwidth}
|
|
\begin{figure}
|
|
\includegraphics[width=\textwidth]{../latex/images/entropy_vs_dA.png}
|
|
\end{figure}
|
|
\centering
|
|
Entropy vs.\ subsystem dimension
|
|
\end{columns}
|
|
|
|
\vspace{0.6em}
|
|
As dimension increases, the entropy distribution concentrates near the maximal value.
|
|
\end{frame}
|
|
|
|
\section{Conclusion}
|
|
|
|
\begin{frame}{Conclusion and Outlook}
|
|
\begin{itemize}
|
|
\item Concentration of measure explains generic high entanglement in large bipartite systems.
|
|
\item Complex projective space provides the natural geometric setting for pure quantum states.
|
|
\item Observable diameter gives a way to phrase concentration geometrically.
|
|
\item Ongoing directions:
|
|
\begin{itemize}
|
|
\item 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{*}
|
|
\bibliographystyle{apalike}
|
|
\bibliography{references}
|
|
\end{frame}
|
|
|
|
\begin{frame}
|
|
\begin{center}
|
|
Q\&A
|
|
\end{center}
|
|
\end{frame}
|
|
|
|
\end{document}
|