805 lines
35 KiB
TeX
805 lines
35 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}
|
|
\usepackage{braket}
|
|
|
|
\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{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}
|
|
|
|
\begin{frame}{Backgrounds: Motivation of Tensor product}
|
|
|
|
Recall from the traditional notation of product space of two vector spaces $V$ and $W$, that is, $V\times W$, is the set of all ordered pairs $(\ket{v},\ket{w})$ where $\ket{v}\in V$ and $\ket{w}\in W$.
|
|
|
|
The space has dimension $\dim V+\dim W$.
|
|
|
|
We want to define a vector space with the notation of multiplication of two vectors from different vector spaces.
|
|
|
|
That is
|
|
|
|
$$
|
|
(\ket{v_1}+\ket{v_2})\otimes \ket{w}=(\ket{v_1}\otimes \ket{w})+(\ket{v_2}\otimes \ket{w})
|
|
$$
|
|
$$
|
|
\ket{v}\otimes (\ket{w_1}+\ket{w_2})=(\ket{v}\otimes \ket{w_1})+(\ket{v}\otimes \ket{w_2})
|
|
$$
|
|
|
|
and enables scalar multiplication by
|
|
|
|
$$
|
|
\lambda (\ket{v}\otimes \ket{w})=(\lambda \ket{v})\otimes \ket{w}=\ket{v}\otimes (\lambda \ket{w})
|
|
$$
|
|
|
|
And we wish to build a way to associate the basis of $V$ and $W$ with the basis of $V\otimes W$. That makes the tensor product a vector space with dimension $\dim V\times \dim W$.
|
|
|
|
\end{frame}
|
|
|
|
\begin{frame}{Backgrounds: Tensor product of vectors}
|
|
|
|
\begin{block}{Definition of Bilinear functional}
|
|
A bilinear functional is a bilinear function $\beta:V\times W\to \mathbb{F}$ satisfying the condition that $\ket{v}\to \beta(\ket{v},\ket{w})$ is a linear functional for all $\ket{w}\in W$ and $\ket{w}\to \beta(\ket{v},\ket{w})$ is a linear functional for all $\ket{v}\in V$.
|
|
|
|
\end{block}
|
|
|
|
The vector space of all bilinear functionals is denoted by $\mathcal{B}(V, W)$.
|
|
\begin{block}{Definition of Tensor product of vectors}
|
|
Let $V, W$ be two vector spaces.
|
|
|
|
Let $V'$ and $W'$ be the dual spaces of $V$ and $W$, respectively, that is $V'=\{\psi:V\to \mathbb{F}\}$ and $W'=\{\phi:W\to \mathbb{F}\}$, $\psi, \phi$ are linear functionals.
|
|
|
|
The \textbf{tensor product of vectors} $v\in V$ and $w\in W$ is the bilinear functional defined by $\forall (\psi,\phi)\in V'\times W'$ given by the notation
|
|
|
|
$$
|
|
(v\otimes w)(\psi,\phi)=\psi(v)\phi(w)
|
|
$$
|
|
\end{block}
|
|
\end{frame}
|
|
|
|
\begin{frame}{Backgrounds: Tensor product of vector spaces}
|
|
\begin{block}{Definition of Tensor product of vector spaces}
|
|
The tensor product of two vector spaces $V$ and $W$ is the vector space $\mathcal{B}(V',W')$
|
|
|
|
Notice that the basis of such vector space is the linear combination of the basis of $V'$ and $W'$, that is, if $\{e_i\}$ is the basis of $V'$ and $\{f_j\}$ is the basis of $W'$, then $\{e_i\otimes f_j\}$ is the basis of $\mathcal{B}(V', W')$.
|
|
|
|
Since $\{e_i\}$ and $\{f_j\}$ are bases of $V'$ and $W'$, respectively, then we can always find a set of linear functionals $\{\phi_i\}$ and $\{\psi_j\}$ such that $\phi_i(e_j)=\delta_{ij}$ and $\psi_j(f_i)=\delta_{ij}$. (Here $\delta_{ij}=1$ if $i=j$ and $0$ otherwise.)
|
|
|
|
$$
|
|
V\otimes W=\left\{\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w): \phi_i\in V', \psi_j\in W'\right\}
|
|
$$
|
|
\end{block}
|
|
|
|
Note that $\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w)$ is a bilinear functional that maps $V'\times W'$ to $\mathbb{F}$.
|
|
\end{frame}
|
|
|
|
\begin{frame}{Backgrounds: Trace}
|
|
|
|
\label{defn:trace}
|
|
|
|
\begin{block}{Trace}
|
|
Let $T$ be a linear operator on $\mathscr{H}$, $(e_1,e_2,\cdots,e_n)$ be a basis of $\mathscr{H}$ and $(\epsilon_1,\epsilon_2,\cdots,\epsilon_n)$ be a basis of dual space $\mathscr{H}^*$. Then the trace of $T$ is defined by
|
|
|
|
$$
|
|
\operatorname{Tr}(T)=\sum_{i=1}^n \epsilon_i(T(e_i))=\sum_{i=1}^n \langle e_i,T(e_i)\rangle
|
|
$$
|
|
|
|
\end{block}
|
|
|
|
This is equivalent to the sum of the diagonal elements of $T$.
|
|
|
|
\vspace{1em}
|
|
Q: How we generalize the trace to a subsystem of a larger, entangled quantum system $A\otimes B$?
|
|
\end{frame}
|
|
|
|
\begin{frame}{Backgrounds: Partial trace}
|
|
|
|
\begin{block}{Definition of Partial trace}
|
|
|
|
Let $T$ be a linear operator on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$, where $\mathscr{A}$ and $\mathscr{B}$ are finite-dimensional Hilbert spaces.
|
|
|
|
An operator $T$ on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$ can be written as
|
|
|
|
$$
|
|
T=\sum_{i=1}^n a_i A_i\otimes B_i
|
|
$$
|
|
|
|
where $A_i$ is a linear operator on $\mathscr{A}$ and $B_i$ is a linear operator on $\mathscr{B}$.
|
|
|
|
The $\mathscr{B}$-partial trace of $T$ ($\operatorname{Tr}_{\mathscr{B}}(T):\mathcal{L}(\mathscr{A}\otimes \mathscr{B})\to \mathcal{L}(\mathscr{A})$) is the linear operator on $\mathscr{A}$ defined by
|
|
|
|
$$
|
|
\operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i
|
|
$$
|
|
|
|
\end{block}
|
|
|
|
\end{frame}
|
|
|
|
\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 mixedness 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
|
|
$$
|
|
E(|\Psi\rangle)=S(\rho_A).
|
|
$$
|
|
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
|
|
$$
|
|
\rho=\sum_{j=1}^n p_j|\psi_j\rangle\langle\psi_j|
|
|
$$
|
|
\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}
|
|
$$
|
|
\|H(\psi_A)\|_{\mathrm{Lip}}
|
|
\leq
|
|
\sqrt{8}\,\log_2(d_A),
|
|
\qquad d_A\geq 3.
|
|
$$
|
|
\end{block}
|
|
|
|
Levy concentration plus these two estimates produces the exponential entropy tail bound.
|
|
\end{frame}
|
|
|
|
|
|
\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 natual 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 $\nu$ (nu) be a Borel probability measure on $\mathbb{R}$ and let $\alpha \in (0,1]$.
|
|
The \textbf{partial diameter} of $\nu$ at mass level $\alpha$ is
|
|
$$
|
|
\diameter(\nu;\alpha):=
|
|
\{\diameter(A):A \subseteq \mathcal{B}(\mathbb{R}),
|
|
\nu(A)\ge \alpha
|
|
\},
|
|
$$
|
|
where
|
|
$$
|
|
\diameter(A):=\sup_{x,y\in A}|x-y|.
|
|
$$
|
|
\end{block}
|
|
|
|
\vspace{0.4em}
|
|
\begin{itemize}
|
|
\item This asks for the shortest interval-like region containing at least $\alpha$ of the total mass.
|
|
\item So $\diameter(\nu;1-\kappa)$ 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}
|