diff --git a/latex/chapters/chap0.pdf b/latex/chapters/chap0.pdf index d980ecc..c058eb6 100644 Binary files a/latex/chapters/chap0.pdf and b/latex/chapters/chap0.pdf differ diff --git a/latex/chapters/chap0.tex b/latex/chapters/chap0.tex index 8bbe8c5..00b97b8 100644 --- a/latex/chapters/chap0.tex +++ b/latex/chapters/chap0.tex @@ -309,7 +309,6 @@ $$ \operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i $$ - \end{defn} 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 $(\mathscr{B}(\mathscr{H}),\mathscr{P})$, - 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: \begin{itemize} @@ -518,7 +517,7 @@ Recall from classical probability theory, we call the initial probability distri \end{itemize} \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 $\sum_{j=1}^n p_j=1$. diff --git a/latex/chapters/chap2.pdf b/latex/chapters/chap2.pdf index 7bb4a17..b5ffef0 100644 Binary files a/latex/chapters/chap2.pdf and b/latex/chapters/chap2.pdf differ diff --git a/presentation/ZheyuanWu_HonorThesis_Presentation.pdf b/presentation/ZheyuanWu_HonorThesis_Presentation.pdf index d8b1fb8..f11149e 100644 Binary files a/presentation/ZheyuanWu_HonorThesis_Presentation.pdf and b/presentation/ZheyuanWu_HonorThesis_Presentation.pdf differ diff --git a/presentation/ZheyuanWu_HonorThesis_Presentation.tex b/presentation/ZheyuanWu_HonorThesis_Presentation.tex index fac8934..281b47c 100644 --- a/presentation/ZheyuanWu_HonorThesis_Presentation.tex +++ b/presentation/ZheyuanWu_HonorThesis_Presentation.tex @@ -16,6 +16,7 @@ \usepackage{tabularx} \usepackage{colortbl} \usepackage{tikz} +\usepackage{braket} \DeclareMathOperator{\sen}{sen} \DeclareMathOperator{\tg}{tg} @@ -67,243 +68,305 @@ \hypersetup{linkcolor=black} \tableofcontents \end{frame} -\section{Motivation} +\section{Formulation of Quantum Entangement} +\begin{frame}{Why I'm here?} -\begin{frame}{Light polarization and non-commutative probability} - \begin{figure} - \includegraphics[width=0.6\textwidth]{../latex/images/Filter_figure.png} - \end{figure} + \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 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 $\alpha=\pi/2$. + \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}{Polarization experiment} +\begin{frame}{Quantum measurements} - \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 $F_i$. +\begin{block}{Definition of Quantum Measurement} - \vspace{0.5em} - If these were classical random variables on one probability space, they would satisfy a Bell-type inequality. + 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}{A classical Bell-type inequality} - \begin{block}{Bell-type inequality} - For any classical random variables $P_1,P_2,P_3\in\{0,1\}$, +\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.) + $$ - \operatorname{Prob}(P_1=1,P_3=0) - \leq - \operatorname{Prob}(P_1=1,P_2=0) - + - \operatorname{Prob}(P_2=1,P_3=0). + V\otimes W=\left\{\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w): \phi_i\in V', \psi_j\in W'\right\} $$ \end{block} - \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} +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}{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{frame}{Backgrounds: Trace} +\label{defn:trace} - $$ - \begin{aligned} - \operatorname{Prob}(P_i=1,P_j=0) - &= - \operatorname{Prob}(P_i=1) - \bigl(1-\operatorname{Prob}(P_j=1\mid P_i=1)\bigr) \\ - &= - \frac12\bigl(1-\cos^2(\alpha_i-\alpha_j)\bigr) \\ - &= - \frac12\sin^2(\alpha_i-\alpha_j). - \end{aligned} - $$ +\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 - This matches the experiment exactly. +$$ +\operatorname{Tr}(T)=\sum_{i=1}^n \epsilon_i(T(e_i))=\sum_{i=1}^n \langle e_i,T(e_i)\rangle +$$ + +\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}{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} +\begin{frame}{Backgrounds: Partial trace} - \vspace{0.6em} - This is one of the basic motivations for passing from classical probability to non-commutative probability. +\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) @@ -315,8 +378,8 @@ \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. + $$ + \rho=\sum_{j=1}^n p_j|\psi_j\rangle\langle\psi_j| $$ \item Pure states describe maximal information; mixed states describe probabilistic mixtures or partial information. \end{itemize} @@ -327,7 +390,7 @@ \end{block} \end{frame} -\begin{frame}{Why pure states are not vectors} +\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: $$ @@ -338,16 +401,16 @@ \end{itemize} \vspace{0.4em} - Hence the space of pure states is + Hence the space of pure states (denoted by $\mathcal{P}(\mathcal H)$) is $$ - \mathbb P(\mathcal H) + \mathcal{P}(\mathcal H) = (\mathcal H \setminus \{0\})/\mathbb C^\times. $$ After choosing a basis $\mathcal H \cong \mathbb C^{n+1}$, this becomes $$ - \mathbb P(\mathcal H) \cong \mathbb C P^n. + \mathcal{P}(\mathcal H) \cong \mathbb C P^n. $$ \end{frame} @@ -375,83 +438,86 @@ is the \textbf{Hopf fibration}. \end{frame} -\begin{frame}{How the metric descends to $\mathbb C P^n$} - \begin{itemize} - \item The sphere $S^{2n+1}$ inherits the round metric from the Euclidean metric on - $$ - \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\}. - $$ - \item Tangent vectors split into: - \begin{itemize} - \item \textbf{vertical directions}: tangent to the $S^1$-fiber, - \item \textbf{horizontal directions}: orthogonal complement to the fiber. - \end{itemize} - \item The differential $dp$ identifies horizontal vectors on the sphere with tangent vectors on $\mathbb C P^n$. - \end{itemize} +\begin{frame}{The induced riemmanian metric: Fubini--Study metric} - \vspace{0.4em} - This allows the round metric on $S^{2n+1}$ to define a metric on $\mathbb C P^n$. + +\begin{block}{Definition of Riemannian metric} + + Let $M$ be a smooth manifold. A \textit{\textbf{Riemannian metric}} on $M$ is a smooth covariant tensor field $g\in \mathcal{T}^2(M)$ such that for each $p\in M$, $g_p$ is an inner product on $T_pM$ (Vector space formed by the tangent vectors relative to the manifold $M$ at $p$). + + $g_p(v,v)\geq 0$ for each $p\in M$ and each $v\in T_pM$. equality holds if and only if $v=0$. + +\end{block} +\begin{itemize} + \item + The geometric picture is + $$ + 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}, + $$ + \end{columns} + +\end{itemize} \end{frame} -\begin{frame}{The induced metric: Fubini--Study metric} - \begin{itemize} - \item The metric on $\mathbb C P^n$ obtained from the Hopf quotient is the - \textbf{Fubini--Study metric}. - \item So the geometric picture is: - $$ - S^{2n+1} - \xrightarrow{\text{Hopf fibration}} - \mathbb C P^n - $$ - $$ - \text{round metric} - \rightsquigarrow - \text{Fubini--Study metric}. - $$ - \item The normalized Riemannian volume measure induced by this metric gives the natural probability measure on pure states. - \end{itemize} +\begin{frame}{So what?} - \vspace{0.5em} - \begin{block}{Proof roadmap} - To prove this carefully, one usually shows: - \begin{enumerate} - \item $p:S^{2n+1}\to \mathbb C P^n$ is a smooth surjective submersion, - \item the vertical space is the tangent space to the $S^1$-orbit, - \item horizontal lifts are well defined, - \item the quotient metric is exactly the Fubini--Study metric. - \end{enumerate} - \end{block} + With everything we have here, we are ready to answer the question: + + \vspace{2em} + + \begin{center} + \textbf{How a random bipartite pure state $\mathcal{P}(A\otimes B)$ is distributed on the complex projective space? And how entangled $H(\psi_A)$ it is?} + \end{center} + \end{frame} +\section{Volume Distribution in High Dimensional Spaces} + \begin{frame}{Maxwell-Boltzmann Distribution Law} - \begin{columns}[T] - \column{0.58\textwidth} - Consider the orthogonal projection + \begin{figure}[H] + \includegraphics[width=0.7\textwidth]{../latex/images/maxwell.png} + \end{figure} + Consider the orthogonal projection $0\leq k< n$ $$ \pi_{n,k}:S^n(\sqrt{n})\to \mathbb{R}^k. $$ - Its push-forward measure converges to the standard Gaussian: + Its push-forward measure converges to the standard Gaussian as dimensions increase $n\to \infty$. $$ (\pi_{n,k})_*\sigma^n\to \gamma^k. $$ - \vspace{0.5em} - This explains why Gaussian behavior emerges from high-dimensional spheres and supports the proof strategy for Levy concentration. - - \column{0.42\textwidth} - \begin{figure} - \includegraphics[width=\textwidth]{../latex/images/maxwell.png} - \end{figure} - \end{columns} + Another familiar name when $k=1$ is the central limit theorem. \end{frame} \begin{frame}{Levy Concentration} - \begin{block}{Levy's theorem} - If $f:S^n\to \mathbb{R}$ is $1$-Lipschitz, then there exists a median $a_0$ such that + + \begin{block}{Definition of Lipschitz function} + A function $f:X\to Y$, where $X,Y$ are metric spaces, is $L$-Lipschitz if there exists a constant $L$ such that $|f(x)-f(y)|\leq L|y-x|$ for all $x,y\in S^n$. + \end{block} + \begin{block}{Levy's lemma} + If $f:S^n\to \mathbb{R}$ is $1$-Lipschitz, then there exists a $a_0$ such that for $\epsilon>0$, $$ \mu\{x\in S^n:|f(x)-a_0|\geq \epsilon\} \leq @@ -461,33 +527,12 @@ \begin{itemize} \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{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 @@ -510,8 +555,8 @@ \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. - \item The projective description is natural because global phase does not change the physical state. \end{itemize} \end{frame} @@ -536,11 +581,40 @@ Levy concentration plus these two estimates produces the exponential entropy tail bound. \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{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 $$ \diameter(\nu;\alpha):= @@ -586,9 +660,8 @@ \end{frame} \begin{frame}{A Geometric Consequence} - In this thesis, entropy functions are used as concrete observables to estimate observable diameter, and the Hopf fibration helps transfer information between $S^{2n+1}$ and $\mathbb{C}P^n$. \vspace{0.4em} - \begin{block}{Projective-space estimate} + \begin{block}{Projective-space estimate from Gromov} For $0<\kappa<1$, $$ \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 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} @@ -612,8 +709,7 @@ \item Compare concentration across: \begin{itemize} \item real spheres, - \item complex projective spaces, - \item symmetric states via Majorana stellar representation. + \item complex projective spaces \end{itemize} \end{itemize} \end{frame} @@ -639,28 +735,65 @@ 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 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 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 Majorana stellar representation for symmetric states. + \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} +% \section{References} +% \begin{frame}[allowframebreaks]{References} +% \nocite{*} +% \bibliographystyle{apalike} +% \bibliography{references} +% \end{frame} \begin{frame} \begin{center}