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zI5V0q&w=I|$bG_Dm6K*EUT6Pxu?@pZ=+fh7CAwG1iI%p}eN!bu^Hs3@?m84AE;{i`PLoB={wRw9_fFaxcb`?5Lg)s{ z>hXFIao|6L)tgFmx9O*MS`{~_Ee#=L%F@l=ll4R>)HPp*4X)Q^yBEPLv6qQZc(r=7 z2>VkCHQRj2)oRNjcpvh3$@2Ojl#;BDJpOpagx%IdEA5H&R)^5($JNVAmdCr4HvpmT zQMG^N6N3;sx!xh1{U=^Bv;9yY7(mkP^uUl8$Vbg^+Z4fGmup&-Z`G z5)jS#y(|&YiXE~DqAizYNr*n!^=uT;-A)!m^x0-vGNLCY%Tf@%*tJ?5(d$*RZiwY7 zhx}B;zV)@NJ7QPUWj&Bw(NC6!4E8-m{F2?4;#blQ zSI=mF$~s+am`y&j$!9kC%qE}N6aNI4IK0$, $$ @@ -569,13 +462,10 @@ Thus entanglement entropy is the von Neumann entropy of a subsystem, and it meas $$ \end{block} - \begin{block}{Lipschitz estimate} - $$ - \|H(\psi_A)\|_{\mathrm{Lip}} - \leq - \sqrt{8}\,\log_2(d_A), - \qquad d_A\geq 3. - $$ + \begin{block}{Lipschitz estimate for $H(\psi_A)$} + The Lipschitz constant for the function $ + H(\psi_A)$ should be upper bounded by $ + \sqrt{8}\,\log_2(d_A)$, for $d_A\geq 3$. \end{block} Levy concentration plus these two estimates produces the exponential entropy tail bound. @@ -605,7 +495,7 @@ Thus entanglement entropy is the von Neumann entropy of a subsystem, and it meas \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} +\begin{frame}{A natural 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} @@ -614,13 +504,10 @@ Thus entanglement entropy is the von Neumann entropy of a subsystem, and it meas \begin{frame}{Observable diameter: the inner definition} \begin{block}{Partial diameter on $\mathbb{R}$} - Let $\nu$ (nu) be a Borel probability measure on $\mathbb{R}$ and let $\alpha \in (0,1]$. - The \textbf{partial diameter} of $\nu$ at mass level $\alpha$ is + Let $\mu$ be a Borel probability measure on $\mathbb{R}$ and let $\alpha \in (0,1]$. + The \textbf{partial diameter} of $\mu$ at mass level $\alpha$ is $$ - \diameter(\nu;\alpha):= - \{\diameter(A):A \subseteq \mathcal{B}(\mathbb{R}), - \nu(A)\ge \alpha - \}, + \diameter(\mu;\alpha):=\inf_{A\subset \mathbb{R}}\left\{ \diameter(A): \mu(A)\geq \alpha \right\}. $$ where $$ @@ -630,8 +517,8 @@ Thus entanglement entropy is the von Neumann entropy of a subsystem, and it meas \vspace{0.4em} \begin{itemize} - \item This asks for the shortest interval-like region containing at least $\alpha$ of the total mass. - \item So $\diameter(\nu;1-\kappa)$ measures how tightly we can capture \emph{most} of the distribution, allowing us to discard a set of mass at most $\kappa$. + \item The partial diameter asks for: what is the shortest interval I need to capture at least $\alpha$ of the mass (measure)? + \item If $1$ is the total measure of the space, $\diameter(\mu;1-\kappa)$, measures how tightly we can capture \emph{most} of the distribution, allowing us to discard a set of mass at most $\kappa$. \end{itemize} \end{frame} diff --git a/presentation/backgrounds.tex b/presentation/backgrounds.tex new file mode 100644 index 0000000..f351fc6 --- /dev/null +++ b/presentation/backgrounds.tex @@ -0,0 +1,107 @@ + +\begin{frame}{Backgrounds: Motivation of Tensor product} + +Recall from the traditional notation of product space of two vector spaces $V$ and $W$, that is, $V\times W$, is the set of all ordered pairs $(\ket{v},\ket{w})$ where $\ket{v}\in V$ and $\ket{w}\in W$. + +The space has dimension $\dim V+\dim W$. + +We want to define a vector space with the notation of multiplication of two vectors from different vector spaces. + +That is + +$$ + (\ket{v_1}+\ket{v_2})\otimes \ket{w}=(\ket{v_1}\otimes \ket{w})+(\ket{v_2}\otimes \ket{w}) +$$ +$$ + \ket{v}\otimes (\ket{w_1}+\ket{w_2})=(\ket{v}\otimes \ket{w_1})+(\ket{v}\otimes \ket{w_2}) +$$ + +and enables scalar multiplication by + +$$ + \lambda (\ket{v}\otimes \ket{w})=(\lambda \ket{v})\otimes \ket{w}=\ket{v}\otimes (\lambda \ket{w}) +$$ + +And we wish to build a way to associate the basis of $V$ and $W$ with the basis of $V\otimes W$. That makes the tensor product a vector space with dimension $\dim V\times \dim W$. + +\end{frame} + +\begin{frame}{Backgrounds: Tensor product of vectors} + + \begin{block}{Definition of Bilinear functional} + A bilinear functional is a bilinear function $\beta:V\times W\to \mathbb{F}$ satisfying the condition that $\ket{v}\to \beta(\ket{v},\ket{w})$ is a linear functional for all $\ket{w}\in W$ and $\ket{w}\to \beta(\ket{v},\ket{w})$ is a linear functional for all $\ket{v}\in V$. + + \end{block} + + The vector space of all bilinear functionals is denoted by $\mathcal{B}(V, W)$. + \begin{block}{Definition of Tensor product of vectors} + Let $V, W$ be two vector spaces. + + Let $V'$ and $W'$ be the dual spaces of $V$ and $W$, respectively, that is $V'=\{\psi:V\to \mathbb{F}\}$ and $W'=\{\phi:W\to \mathbb{F}\}$, $\psi, \phi$ are linear functionals. + + The \textbf{tensor product of vectors} $v\in V$ and $w\in W$ is the bilinear functional defined by $\forall (\psi,\phi)\in V'\times W'$ given by the notation + + $$ + (v\otimes w)(\psi,\phi)=\psi(v)\phi(w) + $$ + \end{block} +\end{frame} + +\begin{frame}{Backgrounds: Tensor product of vector spaces} + \begin{block}{Definition of Tensor product of vector spaces} + The tensor product of two vector spaces $V$ and $W$ is the vector space $\mathcal{B}(V',W')$ + + Notice that the basis of such vector space is the linear combination of the basis of $V'$ and $W'$, that is, if $\{e_i\}$ is the basis of $V'$ and $\{f_j\}$ is the basis of $W'$, then $\{e_i\otimes f_j\}$ is the basis of $\mathcal{B}(V', W')$. + + Since $\{e_i\}$ and $\{f_j\}$ are bases of $V'$ and $W'$, respectively, then we can always find a set of linear functionals $\{\phi_i\}$ and $\{\psi_j\}$ such that $\phi_i(e_j)=\delta_{ij}$ and $\psi_j(f_i)=\delta_{ij}$. (Here $\delta_{ij}=1$ if $i=j$ and $0$ otherwise.) + + $$ + V\otimes W=\left\{\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w): \phi_i\in V', \psi_j\in W'\right\} + $$ + \end{block} + +Note that $\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w)$ is a bilinear functional that maps $V'\times W'$ to $\mathbb{F}$. +\end{frame} + +\begin{frame}{Backgrounds: Trace} + +\label{defn:trace} + +\begin{block}{Trace} +Let $T$ be a linear operator on $\mathscr{H}$, $(e_1,e_2,\cdots,e_n)$ be a basis of $\mathscr{H}$ and $(\epsilon_1,\epsilon_2,\cdots,\epsilon_n)$ be a basis of dual space $\mathscr{H}^*$. Then the trace of $T$ is defined by + +$$ +\operatorname{Tr}(T)=\sum_{i=1}^n \epsilon_i(T(e_i))=\sum_{i=1}^n \langle e_i,T(e_i)\rangle +$$ + +\end{block} + +This is equivalent to the sum of the diagonal elements of $T$. + +\vspace{1em} +Q: How we generalize the trace to a subsystem of a larger, entangled quantum system $A\otimes B$? +\end{frame} + +\begin{frame}{Backgrounds: Partial trace} + +\begin{block}{Definition of Partial trace} + +Let $T$ be a linear operator on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$, where $\mathscr{A}$ and $\mathscr{B}$ are finite-dimensional Hilbert spaces. + +An operator $T$ on $\mathscr{H}=\mathscr{A}\otimes \mathscr{B}$ can be written as + +$$ +T=\sum_{i=1}^n a_i A_i\otimes B_i +$$ + +where $A_i$ is a linear operator on $\mathscr{A}$ and $B_i$ is a linear operator on $\mathscr{B}$. + +The $\mathscr{B}$-partial trace of $T$ ($\operatorname{Tr}_{\mathscr{B}}(T):\mathcal{L}(\mathscr{A}\otimes \mathscr{B})\to \mathcal{L}(\mathscr{A})$) is the linear operator on $\mathscr{A}$ defined by + +$$ +\operatorname{Tr}_{\mathscr{B}}(T)=\sum_{i=1}^n a_i \operatorname{Tr}(B_i) A_i +$$ + +\end{block} + +\end{frame}