Trance-0/HonorThesis
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

updates

Browse Source
This commit is contained in:
Zheyuan Wu
2026-03-25 01:30:56 -05:00
parent e6e5104de2
commit b3502576eb
5 changed files with 453 additions and 321 deletions
Download Patch File Download Diff File Expand all files Collapse all files

BIN
latex/chapters/chap0.pdf
View File

Binary file not shown.

5
latex/chapters/chap0.tex
View File

@@ -309,7 +309,6 @@ $$
\operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i \operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i
$$ $$
\end{defn} \end{defn}
Or we can define the map $L_v: \mathscr{A}\to \mathscr{A}\otimes \mathscr{B}$ by Or we can define the map $L_v: \mathscr{A}\to \mathscr{A}\otimes \mathscr{B}$ by
@@ -509,7 +508,7 @@ Recall from classical probability theory, we call the initial probability distri
Given a non-commutative probability space $(\mathscr{B}(\mathscr{H}),\mathscr{P})$, Given a non-commutative probability space $(\mathscr{B}(\mathscr{H}),\mathscr{P})$,
A state is a unit vector $\bra{\psi}$ in the Hilbert space $\mathscr{H}$, such that $\bra{\psi}\ket{\psi}=1$. 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: 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} \begin{itemize}
@@ -518,7 +517,7 @@ Recall from classical probability theory, we call the initial probability distri
\end{itemize} \end{itemize}
\end{defn} \end{defn}
Note that the pure states are the density operators that can be represented by a unit vector $\bra{\psi}$ in the Hilbert space $\mathscr{H}$, whereas mixed states are the density operators that cannot be represented by a unit vector in the Hilbert space $\mathscr{H}$. Note that the pure states are the density operators that can be represented by a unit vector $\ket{\psi}$ in the Hilbert space $\mathscr{H}$, whereas mixed states are the density operators that cannot be represented by a unit vector in the Hilbert space $\mathscr{H}$.
If $(|\psi_1\rangle,|\psi_2\rangle,\cdots,|\psi_n\rangle)$ is an orthonormal basis of $\mathscr{H}$ consisting of eigenvectors of $\rho$, for the eigenvalues $p_1,p_2,\cdots,p_n$, then $p_j\geq 0$ and $\sum_{j=1}^n p_j=1$. If $(|\psi_1\rangle,|\psi_2\rangle,\cdots,|\psi_n\rangle)$ is an orthonormal basis of $\mathscr{H}$ consisting of eigenvectors of $\rho$, for the eigenvalues $p_1,p_2,\cdots,p_n$, then $p_j\geq 0$ and $\sum_{j=1}^n p_j=1$.

BIN
latex/chapters/chap2.pdf
View File

Binary file not shown.

BIN
presentation/ZheyuanWu_HonorThesis_Presentation.pdf
View File

Binary file not shown.

727
presentation/ZheyuanWu_HonorThesis_Presentation.tex
View File

@@ -16,6 +16,7 @@
\usepackage{tabularx} \usepackage{tabularx}
\usepackage{colortbl} \usepackage{colortbl}
\usepackage{tikz} \usepackage{tikz}
\usepackage{braket}
\DeclareMathOperator{\sen}{sen} \DeclareMathOperator{\sen}{sen}
\DeclareMathOperator{\tg}{tg} \DeclareMathOperator{\tg}{tg}
@@ -67,243 +68,305 @@
\hypersetup{linkcolor=black} \hypersetup{linkcolor=black}
\tableofcontents \tableofcontents
\end{frame} \end{frame}
\section{Motivation} \section{Formulation of Quantum Entangement}
\begin{frame}{Why I'm here?}
\begin{frame}{Light polarization and non-commutative probability} \centering
\begin{figure} \Large\itshape
\includegraphics[width=0.6\textwidth]{../latex/images/Filter_figure.png} ``I think I can safely say that nobody understands quantum mechanics.''
\end{figure}
\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} \begin{itemize}
\item Light passing through a polarizer becomes polarized in the direction of that filter. \item $\rho(O)=0$, where $O$ is the zero projection, and $\rho(I)=1$, where $I$ is the identity projection.
\item If two filters are placed with relative angle $\alpha$, the transmitted intensity decreases as $\alpha$ increases. \item If $P_1,P_2,\ldots,P_n$ are pairwise disjoint orthogonal projections, then $\rho(P_1 + P_2 + \cdots + P_n) = \sum_{i=1}^n \rho(P_i)$.
\item In particular, the transmitted intensity vanishes when $\alpha=\pi/2$.
\end{itemize} \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} \end{frame}
\begin{frame}{Polarization experiment} \begin{frame}{Quantum measurements}
\vspace{0.5em} \begin{block}{Definition of Quantum Measurement}
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} 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 $\mathscr{H}$.
If these were classical random variables on one probability space, they would satisfy a Bell-type inequality.
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} \end{frame}
\begin{frame}{A classical Bell-type inequality} \begin{frame}{Backgrounds: Motivation of Tensor product}
\begin{block}{Bell-type inequality}
For any classical random variables $P_1,P_2,P_3\in\{0,1\}$, 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
$$ $$
\operatorname{Prob}(P_1=1,P_3=0) (v\otimes w)(\psi,\phi)=\psi(v)\phi(w)
\leq $$
\operatorname{Prob}(P_1=1,P_2=0) \end{block}
+ \end{frame}
\operatorname{Prob}(P_2=1,P_3=0).
\begin{frame}{Backgrounds: Tensor product of vector spaces}
\begin{block}{Definition of Tensor product of vector spaces}
The tensor product of two vector spaces $V$ and $W$ is the vector space $\mathcal{B}(V',W')$
Notice that the basis of such vector space is the linear combination of the basis of $V'$ and $W'$, that is, if $\{e_i\}$ is the basis of $V'$ and $\{f_j\}$ is the basis of $W'$, then $\{e_i\otimes f_j\}$ is the basis of $\mathcal{B}(V', W')$.
Since $\{e_i\}$ and $\{f_j\}$ are bases of $V'$ and $W'$, respectively, then we can always find a set of linear functionals $\{\phi_i\}$ and $\{\psi_j\}$ such that $\phi_i(e_j)=\delta_{ij}$ and $\psi_j(f_i)=\delta_{ij}$. (Here $\delta_{ij}=1$ if $i=j$ and $0$ otherwise.)
$$
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} \end{block}
\vspace{0.5em} 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 $\mathbb{F}$.
\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} \end{frame}
\begin{frame}{Experimental law} \begin{frame}{Backgrounds: Trace}
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, \label{defn:trace}
$$
\operatorname{Prob}(P_i=1)=\frac12,
\qquad
\operatorname{Prob}(P_i=1,P_j=1)=\frac12\cos^2(\alpha_i-\alpha_j),
$$
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} \begin{block}{Trace}
So the experimentally observed probabilities depend only on the angle difference $\alpha_i-\alpha_j$. 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} \end{frame}
\begin{frame}{Violation of the classical inequality} \begin{frame}{Backgrounds: Partial trace}
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 \begin{block}{Definition of Partial trace}
$$
\alpha_1=0,\qquad
\alpha_2=\frac{\pi}{6},\qquad
\alpha_3=\frac{\pi}{3}.
$$
Then 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.
$$
\begin{aligned}
\frac12\sin^2\!\left(-\frac{\pi}{3}\right)
&\leq
\frac12\sin^2\!\left(-\frac{\pi}{6}\right)
+
\frac12\sin^2\!\left(-\frac{\pi}{6}\right) \\
\frac38 &\leq \frac18+\frac18 \\
\frac38 &\leq \frac14,
\end{aligned}
$$
which is false.
\vspace{0.5em} An operator $T$ on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$ can be written as
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} $$
The correct model uses a Hilbert space rather than classical events. T=\sum_{i=1}^n a_i A_i\otimes B_i
$$
\begin{itemize} where $A_i$ is a linear operator on $\mathscr{A}$ and $B_i$ is a linear operator on $\mathscr{B}$.
\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 $\mathscr{B}$-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
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} $$
Assume the incoming light is unpolarized, so its state is the density matrix \operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i
$$ $$
\rho=\frac12 I.
$$
The probability of passing the first filter $P_{\alpha_i}$ is \end{block}
$$
\operatorname{Prob}(P_i=1)
=
\operatorname{tr}(\rho P_{\alpha_i})
=
\frac12\operatorname{tr}(P_{\alpha_i})
=
\frac12.
$$
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} \end{frame}
\begin{frame}{Recovering the observed law from the operator model (cont.)}
\begin{frame}{Information theory in classical systems}
$$ In probability theory, an important measurement of uncertainty is entropy.
\begin{aligned}
\operatorname{Prob}(P_i=1,P_j=0) It characterizes the information content of a random variable.
&=
\operatorname{Prob}(P_i=1) \begin{block}{Shannon entropy}
\bigl(1-\operatorname{Prob}(P_j=1\mid P_i=1)\bigr) \\ Given a classical probability vector $p=(p_1,\dots,p_n)$ with $\sum_i p_i=1$,
&= $$
\frac12\bigl(1-\cos^2(\alpha_i-\alpha_j)\bigr) \\ H(p)=-\sum_{i=1}^n p_i \log_2 p_i.
&= $$
\frac12\sin^2(\alpha_i-\alpha_j). This measures uncertainty of a \emph{chosen measurement outcome}.
\end{aligned} \end{block}
$$
This matches the experiment exactly.
\end{frame} \end{frame}
\begin{frame}{Conclusion} \begin{frame}{Information theory in quantum systems}
\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. \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} \end{frame}
\section{Concentration on Spheres and quantum states} \section{Concentration on Spheres and quantum states}
\begin{frame}{Quantum states: pure vs.\ mixed} \begin{frame}{Quantum states: pure vs.\ mixed}
\begin{itemize} \begin{itemize}
\item A finite-dimensional quantum system is modeled by a complex Hilbert space (a complete inner product space) \item A finite-dimensional quantum system is modeled by a complex Hilbert space (a complete inner product space)
@@ -316,7 +379,7 @@
$$ $$
\item A \textbf{mixed state} is represented by a density matrix \item A \textbf{mixed state} is represented by a density matrix
$$ $$
\rho \geq 0, \qquad \operatorname{tr}(\rho)=1. \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. \item Pure states describe maximal information; mixed states describe probabilistic mixtures or partial information.
\end{itemize} \end{itemize}
@@ -327,7 +390,7 @@
\end{block} \end{block}
\end{frame} \end{frame}
\begin{frame}{Why pure states are not vectors} \begin{frame}{Pure states live in the complex projective space}
\begin{itemize} \begin{itemize}
\item Two nonzero vectors that differ by a nonzero complex scalar represent the same physical state: \item Two nonzero vectors that differ by a nonzero complex scalar represent the same physical state:
$$ $$
@@ -338,16 +401,16 @@
\end{itemize} \end{itemize}
\vspace{0.4em} \vspace{0.4em}
Hence the space of pure states is Hence the space of pure states (denoted by $\mathcal{P}(\mathcal H)$) is
$$ $$
\mathbb P(\mathcal H) \mathcal{P}(\mathcal H)
= =
(\mathcal H \setminus \{0\})/\mathbb C^\times. (\mathcal H \setminus \{0\})/\mathbb C^\times.
$$ $$
After choosing a basis $\mathcal H \cong \mathbb C^{n+1}$, this becomes After choosing a basis $\mathcal H \cong \mathbb C^{n+1}$, this becomes
$$ $$
\mathbb P(\mathcal H) \cong \mathbb C P^n. \mathcal{P}(\mathcal H) \cong \mathbb C P^n.
$$ $$
\end{frame} \end{frame}
@@ -375,83 +438,86 @@
is the \textbf{Hopf fibration}. is the \textbf{Hopf fibration}.
\end{frame} \end{frame}
\begin{frame}{How the metric descends to $\mathbb C P^n$} \begin{frame}{The induced riemmanian metric: Fubini--Study metric}
\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{block}{Definition of Riemannian metric}
\begin{itemize}
\item The metric on $\mathbb C P^n$ obtained from the Hopf quotient is the 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 $p$).
\textbf{Fubini--Study metric}.
\item So the geometric picture is: $g_p(v,v)\geq 0$ 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} S^{2n+1}
\xrightarrow{\text{Hopf fibration}} \xrightarrow{\text{Hopf fibration}}
\mathbb C P^n \mathbb C P^n,
$$ \qquad
$$
\text{round metric} \text{round metric}
\rightsquigarrow \rightsquigarrow
\text{Fubini--Study metric}. \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} \begin{columns}[T]
To prove this carefully, one usually shows: \column{0.5\textwidth}
\begin{enumerate} The sphere $S^{2n+1}\subset \mathbb C^{n+1}$ has the \textbf{round metric}
\item $p:S^{2n+1}\to \mathbb C P^n$ is a smooth surjective submersion, $$
\item the vertical space is the tangent space to the $S^1$-orbit, g_{\mathrm{round}}=\sum_{j=0}^n (dx_j^2+dy_j^2)\big|_{S^{2n+1}},
\item horizontal lifts are well defined, $$
\item the quotient metric is exactly the Fubini--Study metric. induced from the Euclidean metric on $\mathbb R^{2n+2}$.
\end{enumerate}
\end{block} \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} \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{frame}{Maxwell-Boltzmann Distribution Law}
\begin{columns}[T] \begin{figure}[H]
\column{0.58\textwidth} \includegraphics[width=0.7\textwidth]{../latex/images/maxwell.png}
Consider the orthogonal projection \end{figure}
Consider the orthogonal projection $0\leq k< n$
$$ $$
\pi_{n,k}:S^n(\sqrt{n})\to \mathbb{R}^k. \pi_{n,k}:S^n(\sqrt{n})\to \mathbb{R}^k.
$$ $$
Its push-forward measure converges to the standard Gaussian: Its push-forward measure converges to the standard Gaussian as dimensions increase $n\to \infty$.
$$ $$
(\pi_{n,k})_*\sigma^n\to \gamma^k. (\pi_{n,k})_*\sigma^n\to \gamma^k.
$$ $$
\vspace{0.5em} Another familiar name when $k=1$ is the central limit theorem.
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} \end{frame}
\begin{frame}{Levy Concentration} \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 \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\} \mu\{x\in S^n:|f(x)-a_0|\geq \epsilon\}
\leq \leq
@@ -461,33 +527,12 @@
\begin{itemize} \begin{itemize}
\item In high dimension, most Lipschitz observables are almost constant. \item In high dimension, most Lipschitz observables are almost constant.
\item This is the geometric mechanism behind generic entanglement. \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{itemize}
\end{frame} \end{frame}
\section{Main Result} \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{frame}{How the Entropy Observable Fits In}
\begin{figure} \begin{figure}
\centering \centering
@@ -510,8 +555,8 @@
\end{figure} \end{figure}
\begin{itemize} \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. \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{itemize}
\end{frame} \end{frame}
@@ -536,11 +581,40 @@
Levy concentration plus these two estimates produces the exponential entropy tail bound. Levy concentration plus these two estimates produces the exponential entropy tail bound.
\end{frame} \end{frame}
\section{Geometry of State Space}
\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{frame}{Observable diameter: the inner definition}
\begin{block}{Partial diameter on $\mathbb{R}$} \begin{block}{Partial diameter on $\mathbb{R}$}
Let $\nu$ be a Borel probability measure on $\mathbb{R}$ and let $\alpha \in (0,1]$. 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 The \textbf{partial diameter} of $\nu$ at mass level $\alpha$ is
$$ $$
\diameter(\nu;\alpha):= \diameter(\nu;\alpha):=
@@ -586,9 +660,8 @@
\end{frame} \end{frame}
\begin{frame}{A Geometric Consequence} \begin{frame}{A Geometric Consequence}
In this thesis, entropy functions are used as concrete observables to estimate observable diameter, and the Hopf fibration helps transfer information between $S^{2n+1}$ and $\mathbb{C}P^n$.
\vspace{0.4em} \vspace{0.4em}
\begin{block}{Projective-space estimate} \begin{block}{Projective-space estimate from Gromov}
For $0<\kappa<1$, For $0<\kappa<1$,
$$ $$
\obdiam(\mathbb{C}P^n(1);-\kappa)\leq O(\sqrt{n}). \obdiam(\mathbb{C}P^n(1);-\kappa)\leq O(\sqrt{n}).
@@ -600,6 +673,30 @@
\item Then use the Hopf map $S^{2n+1}(1)\to \mathbb{C}P^n$. \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. \item This gives a geometric explanation for why many projective-space observables concentrate.
\end{itemize} \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} \end{frame}
\section{Numerical Section} \section{Numerical Section}
@@ -612,8 +709,7 @@
\item Compare concentration across: \item Compare concentration across:
\begin{itemize} \begin{itemize}
\item real spheres, \item real spheres,
\item complex projective spaces, \item complex projective spaces
\item symmetric states via Majorana stellar representation.
\end{itemize} \end{itemize}
\end{itemize} \end{itemize}
\end{frame} \end{frame}
@@ -639,28 +735,65 @@
As dimension increases, the entropy distribution concentrates near the maximal value. As dimension increases, the entropy distribution concentrates near the maximal value.
\end{frame} \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} \section{Conclusion}
\begin{frame}{Conclusion and Outlook} \begin{frame}{Conclusion and Outlook}
\begin{itemize} \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 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 Observable diameter gives a way to phrase concentration geometrically.
\item Ongoing directions: \item Ongoing directions:
\begin{itemize} \begin{itemize}
\item sharper estimates for $\mathbb{C}P^n$, \item sharper estimates for $\mathbb{C}P^n$
\item deeper use of Fubini--Study geometry, \item deeper use of Fubini--Study geometry
\item Majorana stellar representation for symmetric states. \item recursive learning on new theorems and mathematical tools
\end{itemize} \end{itemize}
\end{itemize} \end{itemize}
\end{frame} \end{frame}
\section{References} % \section{References}
\begin{frame}[allowframebreaks]{References} % \begin{frame}[allowframebreaks]{References}
\nocite{*} % \nocite{*}
\bibliographystyle{apalike} % \bibliographystyle{apalike}
\bibliography{references} % \bibliography{references}
\end{frame} % \end{frame}
\begin{frame} \begin{frame}
\begin{center} \begin{center}