updates
This commit is contained in:
Zheyuan Wu
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@@ -309,7 +309,6 @@ $$
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\operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i
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$$
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\end{defn}
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Or we can define the map $L_v: \mathscr{A}\to \mathscr{A}\otimes \mathscr{B}$ by
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@@ -509,7 +508,7 @@ Recall from classical probability theory, we call the initial probability distri
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Given a non-commutative probability space $(\mathscr{B}(\mathscr{H}),\mathscr{P})$,
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A state is a unit vector $\bra{\psi}$ in the Hilbert space $\mathscr{H}$, such that $\bra{\psi}\ket{\psi}=1$.
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A state is a unit vector $\ket{\psi}$ in the Hilbert space $\mathscr{H}$, such that $\bra{\psi}\ket{\psi}=1$.
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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:
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\begin{itemize}
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@@ -518,7 +517,7 @@ Recall from classical probability theory, we call the initial probability distri
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\end{itemize}
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\end{defn}
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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 $\mathscr{H}$.
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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 $\mathscr{H}$.
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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 $\sum_{j=1}^n p_j=1$.
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@@ -16,6 +16,7 @@
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\usepackage{tabularx}
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\usepackage{colortbl}
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\usepackage{tikz}
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\usepackage{braket}
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\DeclareMathOperator{\sen}{sen}
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\DeclareMathOperator{\tg}{tg}
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@@ -67,243 +68,305 @@
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\hypersetup{linkcolor=black}
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\tableofcontents
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\end{frame}
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\section{Motivation}
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\section{Formulation of Quantum Entangement}
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\begin{frame}{Why I'm here?}
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\begin{frame}{Light polarization and non-commutative probability}
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\begin{figure}
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\includegraphics[width=0.6\textwidth]{../latex/images/Filter_figure.png}
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\end{figure}
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\centering
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\Large\itshape
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``I think I can safely say that nobody understands quantum mechanics.''
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\vspace{1em}
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\normalsize --- Richard Feynman
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\end{frame}
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\begin{frame}{Non-commutative probability space}
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We begin our discussion on a general type of probability space.
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\begin{block}{Non-commutative probability space}
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\label{defn:non-commutative_probability_space}
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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 $\mathscr{H}$.
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$\mathscr{P}$ is the set of all orthogonal projections on $\mathscr{B}(\mathscr{H})$.
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The set $\mathscr{P}=\{P\in\mathscr{B}(\mathscr{H}):P^*=P=P^2\}$ is the set of all orthogonal projections on $\mathscr{B}(\mathscr{H})$.
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\end{block}
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\begin{table}[H]
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\centering
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\renewcommand{\arraystretch}{1}
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\label{tab:analog_of_classical_probability_theory_and_non_commutative_probability_theory}
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{\tiny
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\begin{tabular}{|p{0.45\linewidth}|p{0.45\linewidth}|}
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\hline
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\textbf{Classical probability} & \textbf{Non-commutative probability} \\
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\hline
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Sample space $\Omega$, cardinality $\vert\Omega\vert=n$, example: $\Omega=\{0,1\}$ & Complex Hilbert space $\mathscr{H}$, dimension $\dim\mathscr{H}=n$, example: $\mathscr{H}=\mathbb{C}^2$ \\
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\hline
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Common algebra of $\mathbb{C}$ valued functions & Algebra of bounded operators $\mathcal{B}(\mathscr{H})$ \\
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\hline
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Events: indicator functions of sets & Projections: space of orthogonal projections $\mathscr{P}\subseteq\mathscr{B}(\mathscr{H})$ \\
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\hline
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functions $f$ such that $f^2=f=\overline{f}$ & orthogonal projections $P$ such that $P^*=P=P^2$ \\
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\hline
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$\mathbb{I}_{f^{-1}(\{\lambda\})}$ is the indicator function of the set $f^{-1}(\{\lambda\})$ & $P(\lambda)$ is the orthogonal projection to eigenspace \\
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\hline
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$f=\sum_{\lambda\in \operatorname{Range}(f)}\lambda \mathbb{I}_{f^{-1}(\{\lambda\})}$ & $A=\sum_{\lambda\in \operatorname{sp}(A)}\lambda P(\lambda)$ \\
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\hline
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Probability measure $\mu$ on $\Omega$ & Density operator $\rho$ on $\mathscr{H}$ \\
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\hline
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\end{tabular}
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}
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\end{table}
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\end{frame}
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\begin{frame}{Quantum states}
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Given a non-commutative probability space $(\mathscr{B}(\mathscr{H}),\mathscr{P})$,
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\begin{block}{Definition of (Quantum) State}
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A state is a unit vector $\ket{\psi}$ in the Hilbert space $\mathscr{H}$, such that $\bra{\psi}\ket{\psi}=1$.
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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:
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\begin{itemize}
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\item Light passing through a polarizer becomes polarized in the direction of that filter.
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\item If two filters are placed with relative angle $\alpha$, the transmitted intensity decreases as $\alpha$ increases.
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\item In particular, the transmitted intensity vanishes when $\alpha=\pi/2$.
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\item $\rho(O)=0$, where $O$ is the zero projection, and $\rho(I)=1$, where $I$ is the identity projection.
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\item If $P_1,P_2,\ldots,P_n$ are pairwise disjoint orthogonal projections, then $\rho(P_1 + P_2 + \cdots + P_n) = \sum_{i=1}^n \rho(P_i)$.
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\end{itemize}
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\end{block}
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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 $\ket{\psi})$.
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\end{frame}
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\begin{frame}{Polarization experiment}
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\begin{frame}{Quantum measurements}
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\vspace{0.5em}
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Now consider three filters $F_1,F_2,F_3$ with directions
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$$
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\alpha_1,\alpha_2,\alpha_3.
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$$
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Testing them pairwise suggests introducing three $0$--$1$ random variables
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$$
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P_1,P_2,P_3,
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$$
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where $P_i=1$ means that the photon passes filter $F_i$.
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\begin{block}{Definition of Quantum Measurement}
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\vspace{0.5em}
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If these were classical random variables on one probability space, they would satisfy a Bell-type inequality.
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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 $\mathscr{H}$.
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Given the initial state (pure state, unit vector) $u$, the probability of measurement outcome $x$ is given by:
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$$
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p(x)=\|M_xu\|^2
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$$
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Note that to make sense of this definition, the collection of measurement operators $\{M_x\}$ must satisfy the completeness requirement:
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$$
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1=\sum_{x\in X} p(x)=\sum_{x\in X}\|M_xu\|^2=\sum_{x\in X}\langle M_xu,M_xu\rangle=\langle u,(\sum_{x\in X}M_x^*M_x)u\rangle
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$$
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So $\sum_{x\in X}M_x^*M_x=I$ (Law of total probability).
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\end{block}
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\end{frame}
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\begin{frame}{A classical Bell-type inequality}
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\begin{block}{Bell-type inequality}
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For any classical random variables $P_1,P_2,P_3\in\{0,1\}$,
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\begin{frame}{Backgrounds: Motivation of Tensor product}
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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 $\ket{w}\in W$.
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The space has dimension $\dim V+\dim W$.
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We want to define a vector space with the notation of multiplication of two vectors from different vector spaces.
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That is
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$$
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(\ket{v_1}+\ket{v_2})\otimes \ket{w}=(\ket{v_1}\otimes \ket{w})+(\ket{v_2}\otimes \ket{w})
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$$
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$$
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\ket{v}\otimes (\ket{w_1}+\ket{w_2})=(\ket{v}\otimes \ket{w_1})+(\ket{v}\otimes \ket{w_2})
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$$
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and enables scalar multiplication by
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$$
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\lambda (\ket{v}\otimes \ket{w})=(\lambda \ket{v})\otimes \ket{w}=\ket{v}\otimes (\lambda \ket{w})
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$$
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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 $\dim V\times \dim W$.
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\end{frame}
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\begin{frame}{Backgrounds: Tensor product of vectors}
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\begin{block}{Definition of Bilinear functional}
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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 $\ket{v}\in V$.
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\end{block}
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The vector space of all bilinear functionals is denoted by $\mathcal{B}(V, W)$.
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\begin{block}{Definition of Tensor product of vectors}
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Let $V, W$ be two vector spaces.
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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.
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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
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$$
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(v\otimes w)(\psi,\phi)=\psi(v)\phi(w)
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$$
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\end{block}
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\end{frame}
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\begin{frame}{Backgrounds: Tensor product of vector spaces}
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\begin{block}{Definition of Tensor product of vector spaces}
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The tensor product of two vector spaces $V$ and $W$ is the vector space $\mathcal{B}(V',W')$
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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 $\mathcal{B}(V', W')$.
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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.)
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$$
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\operatorname{Prob}(P_1=1,P_3=0)
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\leq
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\operatorname{Prob}(P_1=1,P_2=0)
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+
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\operatorname{Prob}(P_2=1,P_3=0).
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V\otimes W=\left\{\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w): \phi_i\in V', \psi_j\in W'\right\}
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$$
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\end{block}
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\vspace{0.5em}
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\begin{proof}
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The event $\{P_1=1,P_3=0\}$ splits into two disjoint cases according to whether $P_2=0$ or $P_2=1$:
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$$
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\{P_1=1,P_3=0\}
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=
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\{P_1=1,P_2=0,P_3=0\}
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\sqcup
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\{P_1=1,P_2=1,P_3=0\}.
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$$
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Therefore,
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$$
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\begin{aligned}
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\operatorname{Prob}(P_1=1,P_3=0)
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&=
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\operatorname{Prob}(P_1=1,P_2=0,P_3=0) \\
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&\quad+
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\operatorname{Prob}(P_1=1,P_2=1,P_3=0) \\
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&\leq
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\operatorname{Prob}(P_1=1,P_2=0)
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+
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\operatorname{Prob}(P_2=1,P_3=0).
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\end{aligned}
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$$
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\end{proof}
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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 $\mathbb{F}$.
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\end{frame}
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\begin{frame}{Experimental law}
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For unpolarized incoming light, the \textbf{observed transition law} for a pair of filters is
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$$
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\operatorname{Prob}(P_i=1,P_j=0)
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=
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\operatorname{Prob}(P_i=1)-\operatorname{Prob}(P_i=1,P_j=1).
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$$
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Using the polarization law,
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$$
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\operatorname{Prob}(P_i=1)=\frac12,
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\qquad
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\operatorname{Prob}(P_i=1,P_j=1)=\frac12\cos^2(\alpha_i-\alpha_j),
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$$
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hence
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$$
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\operatorname{Prob}(P_i=1,P_j=0)
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=
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\frac12-\frac12\cos^2(\alpha_i-\alpha_j)
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=
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\frac12\sin^2(\alpha_i-\alpha_j).
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$$
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\vspace{0.5em}
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So the experimentally observed probabilities depend only on the angle difference $\alpha_i-\alpha_j$.
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\end{frame}
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\begin{frame}{Violation of the classical inequality}
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Substituting the experimental law into the classical inequality gives
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$$
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\frac12\sin^2(\alpha_1-\alpha_3)
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\leq
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\frac12\sin^2(\alpha_1-\alpha_2)
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+
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\frac12\sin^2(\alpha_2-\alpha_3).
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$$
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Choose
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$$
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\alpha_1=0,\qquad
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\alpha_2=\frac{\pi}{6},\qquad
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\alpha_3=\frac{\pi}{3}.
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$$
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Then
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$$
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\begin{aligned}
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\frac12\sin^2\!\left(-\frac{\pi}{3}\right)
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&\leq
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\frac12\sin^2\!\left(-\frac{\pi}{6}\right)
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+
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\frac12\sin^2\!\left(-\frac{\pi}{6}\right) \\
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\frac38 &\leq \frac18+\frac18 \\
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\frac38 &\leq \frac14,
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\end{aligned}
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$$
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which is false.
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\vspace{0.5em}
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Therefore the pairwise polarization data cannot come from one classical probability model with random variables $P_1,P_2,P_3$.
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\end{frame}
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\begin{frame}{The quantum model of polarization}
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The correct model uses a Hilbert space rather than classical events.
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\begin{itemize}
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\item A pure polarization state is a vector
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$$
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\psi=\alpha|0\rangle+\beta|1\rangle \in \mathbb{C}^2.
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$$
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\item A filter at angle $\alpha$ is represented by the orthogonal projection
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$$
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P_\alpha=
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\begin{pmatrix}
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\cos^2\alpha & \cos\alpha\sin\alpha \\
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\cos\alpha\sin\alpha & \sin^2\alpha
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\end{pmatrix}.
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$$
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\item For a pure state $\psi$, the probability of passing the filter is
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$$
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\langle P_\alpha\psi,\psi\rangle.
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$$
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\end{itemize}
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\vspace{0.4em}
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The key point is that sequential measurements are described by \emph{ordered products} of projections, and these need not commute.
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\end{frame}
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\begin{frame}{Recovering the observed law from the operator model}
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Assume the incoming light is unpolarized, so its state is the density matrix
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$$
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\rho=\frac12 I.
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$$
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The probability of passing the first filter $P_{\alpha_i}$ is
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$$
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\operatorname{Prob}(P_i=1)
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=
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\operatorname{tr}(\rho P_{\alpha_i})
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=
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\frac12\operatorname{tr}(P_{\alpha_i})
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=
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\frac12.
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$$
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If the photon passes the first filter, the post-measurement state is
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$$
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\rho_i
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=
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\frac{P_{\alpha_i}\rho P_{\alpha_i}}{\operatorname{tr}(\rho P_{\alpha_i})}
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=
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P_{\alpha_i}.
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$$
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||||
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||||
|
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$$
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P_\alpha=
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\begin{pmatrix}
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\cos^2\alpha & \cos\alpha\sin\alpha \\
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\cos\alpha\sin\alpha & \sin^2\alpha
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\end{pmatrix}.
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$$
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Therefore
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$$
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\operatorname{Prob}(P_j=1\mid P_i=1)
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=
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\operatorname{tr}(\rho_i P_{\alpha_j})
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=
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\operatorname{tr}(P_{\alpha_i}P_{\alpha_j})
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=
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\cos^2(\alpha_i-\alpha_j).
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||||
$$
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||||
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||||
\end{frame}
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\begin{frame}{Recovering the observed law from the operator model (cont.)}
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\begin{frame}{Backgrounds: Trace}
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||||
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\label{defn:trace}
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||||
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$$
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||||
\begin{aligned}
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\operatorname{Prob}(P_i=1,P_j=0)
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&=
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\operatorname{Prob}(P_i=1)
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\bigl(1-\operatorname{Prob}(P_j=1\mid P_i=1)\bigr) \\
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&=
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||||
\frac12\bigl(1-\cos^2(\alpha_i-\alpha_j)\bigr) \\
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&=
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\frac12\sin^2(\alpha_i-\alpha_j).
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||||
\end{aligned}
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$$
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\begin{block}{Trace}
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||||
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
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||||
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||||
This matches the experiment exactly.
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||||
$$
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||||
\operatorname{Tr}(T)=\sum_{i=1}^n \epsilon_i(T(e_i))=\sum_{i=1}^n \langle e_i,T(e_i)\rangle
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||||
$$
|
||||
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||||
\end{block}
|
||||
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||||
This is equivalent to the sum of the diagonal elements of $T$.
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||||
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||||
\vspace{1em}
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||||
Q: How we generalize the trace to a subsystem of a larger, entangled quantum system $A\otimes B$?
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||||
\end{frame}
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||||
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||||
\begin{frame}{Conclusion}
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||||
\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}
|
||||
|
||||
Reference in New Issue
Block a user