update

chapters/chap1.tex

@@ -468,41 +468,9 @@ It is a theorem connecting the following mathematical structure:
 
 To prove the concentration of measure phenomenon, we need to analyze the following elements involved in figure~\ref{fig:Hayden_concentration_of_measure_phenomenon}:
 
-First, we need to define what is a random state in a bipartite system. In fact, for pure states, there is a unique uniform distribution under Haar measure that is unitarily invariant.
-
-$U(n)$ is the group of all $n\times n$ \textbf{unitary matrices} over $\mathbb{C}$, 
-
-$$
-U(n)=\{A\in \mathbb{C}^{n\times n}: A^*A=AA^*=I_n\}
-$$
-
-The uniqueness of such measurement came from the lemma below~\cite{Elizabeth_book}
-
-\begin{lemma}
-
-    Let $(U(n), \| \cdot \|, \mu)$ be a metric measure space where $\| \cdot \|$ is the Hilbert-Schmidt norm and $\mu$ is the measure function.
-
-    The Haar measure on $U(n)$ is the unique probability measure that is invariant under the action of $U(n)$ on itself.
-    
-    That is, fixing $B\in U(n)$, $\forall A\in U(n)$, $\mu(A\cdot B)=\mu(B\cdot A)=\mu(B)$.
-    
-    The Haar measure is the unique probability measure that is invariant under the action of $U(n)$ on itself.
-
-\end{lemma}
-
     
 The existence and uniqueness of the Haar measure is a theorem in compact lie group theory. For this research topic, we will not prove it.
 
-A random pure state $\varphi$ is any random variable distributed according to the unitarily invariant probability measure on the pure states $\mathcal{P}(A)$ of the system $A$, denoted by $\varphi\in_R\mathcal{P}(A)$.
-
-It is trivial that for the space of pure state, we can easily apply the Haar measure as the unitarily invariant probability measure since the space of pure state is $S^n$ for some $n$. However, for the case of mixed states, that is a bit complicated and we need to use partial tracing to defined the rank-$s$ random states.
-
-\begin{defn}
-    Rank-$s$ random state.
-
-    For a system $A$ and an integer $s\geq 1$, consider the distribution onn the mixed states $\mathcal{S}(A)$ of A induced by the partial trace over the second factor form the uniform distribution on pure states of $A\otimes\mathbb{C}^s$. Any random variable $\rho$ distributed as such will be called a rank-$s$ random states; denoted as $\rho\in_R \mathcal{S}_s(A)$. And $\mathcal{P}(A)=\mathcal{S}_1(A)$.
-\end{defn}
-
 Due to time constrains of the projects, the following lemma is demonstrated but not investigated thoroughly through the research:
 
 
@@ -513,11 +481,11 @@ Due to time constrains of the projects, the following lemma is demonstrated but
 
     Choose a random pure state $\sigma=|\psi\rangle\langle\psi|$ from $A'\otimes A$.
 
-    The expected value of the entropy of entanglement is known and satisfies a concentration inequality known as Page's formula~\cite{Pages_conjecture,Pages_conjecture_simple_proof,Bengtsson_Życzkowski_2017}[15.72]. The detailed proof is not fully explored in this project and is intended to be done in the next semester.
+    The expected value of the entropy of entanglement is known and satisfies a concentration inequality known as Page's formula~\cite{Pages_conjecture,Pages_conjecture_simple_proof,Bengtsson_Zyczkowski_2017}[15.72].
 
-    \[
-    \mathbb{E}[H(\psi_A)] \geq \log_2(d_A)-\frac{1}{2\ln(2)}\frac{d_A}{d_B}
-    \]
+    $$
+    \mathbb{E}[H(\psi_A)]=\frac{1}{\ln(2)}\left(\sum_{j=d_B+1}^{d_Ad_B}\frac{1}{j}-\frac{d_A-1}{2d_B}\right) \geq \log_2(d_A)-\frac{1}{2\ln(2)}\frac{d_A}{d_B}
+    $$
 
 \end{lemma}
 
@@ -530,18 +498,27 @@ It basically provides a lower bound for the expected entropy of entanglement. Ex
 	\label{fig:entropy_vs_dim}
 \end{figure}
 
-Then we have bound for Lipschitz constant $\eta$ of the map $H(\varphi_A)$
+Then we have bound for Lipschitz constant $\eta$ of the map $S(\varphi_A): \mathcal{P}(A\otimes B)\to \R$
 
 \begin{lemma}
     The Lipschitz constant $\eta$ of $S(\varphi_A)$ is upper bounded by $\sqrt{8}\log_2(d_A)$ for $d_A\geq 3$.
 \end{lemma}
 
+\begin{proof}
+    The proof use lagrange multiplier method to find the maximum of the gradient of $S(\varphi_A)$.
+    %
+    TODO: use lagrange multiplier method to find the maximum of the gradient of $S(\varphi_A)$.
+    %
+\end{proof}
+
 From Levy's lemma, we have
 
 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}.
 
 Experimentally, we can have the following result: