update>

latex/chapters/chap1.tex

@@ -28,6 +28,26 @@
 
 \chapter{Concentration of Measure And Quantum Entanglement}
 
+\begin{abstract}
+
+The concentration of measure phenomenon has been applied to the study of non-commutative probability theory. Basically, the non-trivial observation, citing from Gromov's work~\cite{MGomolovs}, states that an arbitrary 1-Lipschitz function $f:S^n\to \mathbb{R}$ concentrates near a single value $a_0\in \mathbb{R}$ as strongly as the distance function does. That is,
+$$
+\mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp\left(-\frac{(n-1)\epsilon^2}{2}\right)
+$$
+is applied to computing the probability that, given a bipartite system $A\otimes B$, assume $\dim(B)\geq \dim(A)\geq 3$, as the dimension of the smaller system $A$ increases, with very high probability, a random pure state $\sigma=|\psi\rangle\langle\psi|$ selected from $A\otimes B$ is almost as good as the maximally entangled state.
+
+Mathematically, that is:
+
+Let $\psi\in \mathcal{P}(A\otimes B)$ be a random pure state on $A\otimes B$.
+
+If we define $\beta=\frac{1}{\ln(2)}\frac{d_A}{d_B}$, then we have
+$$
+\operatorname{Pr}[H(\psi_A) < \log_2(d_A)-\alpha-\beta] \leq \exp\left(-\frac{1}{8\pi^2\ln(2)}\frac{(d_Ad_B-1)\alpha^2}{(\log_2(d_A))^2}\right)
+$$
+where $d_B\geq d_A\geq 3$~\cite{Hayden_2006}.
+
+In this report, we will show the process of my exploration of the concentration of measure phenomenon in the context of non-commutative probability theory.
+\end{abstract}
 
 First, we will build the mathematical model describing the behavior of quantum system and why they makes sense for physicists and meaningful for general publics.
 

latex/chapters/chap2.tex

@@ -9,6 +9,14 @@
 
 \chapter{Levy's family and observable diameters}
 
+\begin{abstract}
+The concentration of measure phenomenon come with important notation called the observable diameter, being the smallest number $D$ such that forall 1-lipschitz function $f$ from $X$ to $\R$, where $X$ is a metric measure space, except on a set with measure $\kappa$, the value of $f$ is concentrated in interval with length $D$.
+
+From Hayden's work, we know that a random pure state bipartite system, the entropy is nearly the maximal and the entropy function has Lipschitz constant upper bounded by $\sqrt{8}\log_2(d_A)$ for $d_A\geq 3$. We can use the entropy function as a proxy for the observable diameter.
+
+Altogether, we experimenting how entropy concentration reflects the geometry of high dimensional spheres, generic bipartite projective state spaces, and symmetric quantum state manifolds.
+\end{abstract}
+
 In this section, we will explore how the results from Hayden's concentration of measure theorem can be understood in terms of observable diameters from Gromov's perspective and what properties it reveals for entropy functions.
 
 We will try to use the results from previous sections to estimate the observable diameter for complex projective spaces.
@@ -113,7 +121,7 @@ Few additional proposition in \cite{shioya2014metricmeasuregeometry} will help u
  
   and
   $$
-  \diam(g_*\mu_X;1-\kappa)\leq \diam((f\circ g)_*\mu_Y;1-\kappa)\leq \obdiam(Y;1-\kappa)
+  \diam(g_*\mu_X;1-\kappa)\leq \diam((f\circ g)_*\mu_Y;1-\kappa)\leq \obdiam(Y;-\kappa)
   $$
 \end{proof}
 

latex/main.tex

@@ -88,6 +88,17 @@
 \newtheorem{prop}[theorem]{Proposition}
 \newtheorem{defn}[theorem]{Definition}
 
+%%%%%%%%%%%%%%%%%%%%%%
+% Add abstract for book
+\newenvironment{abstract}{
+  \begin{center}
+    \bfseries Abstract
+  \end{center}
+    \quotation
+}{
+  \endquotation
+}
+
 \title{Concentration of Measure And Quantum Entanglement}
 \author{Zheyuan Wu}
 \date{\today}