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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.