complie

latex/chapters/chap0.tex

@@ -869,10 +869,97 @@ An $m$-manifold is a Topological space $X$ that is
 
 \begin{theorem}
     \label{Theorem of imbedded space}
+
+    Whithney's Embedding Theorem:
+
     If $X$ is a compact $m$-manifold, then $X$ can be imbedded in $\mathbb{R}^n$ for some $n$.
 \end{theorem}
 
-This theorem might save you from imagining abstract structures back to real dimension. Good news, at least you stay in some real numbers.
+This proof is from topology course, and use additional one lemma:
+
+\begin{lemma}
+    \label{lemma:finite_partition_of_unity}
+
+    Let $\{U_i\}_{i=1}^n$ be a finite open cover of a normal space $X$ (Every pair of closed sets in $X$ can be separated by two open sets in $X$).
+
+Then there exists a partition of unity dominated by $\{U_i\}_{i=1}^n$.
+\end{lemma}
+
+\begin{proof}
+    
+Since $X$ is a $m$ compact manifold, $\forall x\in X$, there is an open neighborhood $U_x$ of $x$ such that $U_x$ is homeomorphic to $\mathbb{R}^m$. That means there exists $\varphi_i:U_x\to \varphi(U_x)\subseteq \mathbb{R}^m$.
+
+Where $\{U_x\}_{x\in X}$ is an open cover of $X$. Since $X$ is compact, there is a finite subcover $\bigcup_{i=1}^k U_{x_i}=X$.
+
+Apply the existence of a finite partition of unity, we can find a partition of unity dominated by $\{U_{x_i}\}_{i=1}^k$. With family of functions $\phi_i:\mathbb{R}^d\to[0,1]$.
+
+Define $h_i:X\to \mathbb{R}^m$ by
+
+$$
+h_i(x)=\begin{cases}
+\phi_i(x)\varphi_i(x) & \text{if }x=x_i\\
+0 & \text{otherwise}
+\end{cases}
+$$
+
+We claim that $h_i$ is continuous using pasting lemma.
+
+On $U_i$, $h_i=\phi_i\varphi_i$ is product of two continuous functions therefore continuous.
+
+On $X-\operatorname{supp}(\phi_i)$, $h_i=0$ is continuous.
+
+By pasting lemma, $h_i$ is continuous.
+
+Define
+
+$$
+F: X\to (\mathbb{R}^m\times \mathbb{R})^n
+$$
+
+where $x\mapsto (h_1(x),\varphi_1(x),h_2(x),\varphi_2(x),\dots,h_n(x),\varphi_n(x))$
+
+We want to show that $F$ is imbedding map.
+
+\begin{enumerate}
+
+    \item $F$ is continuous
+
+
+since it is a product of continuous functions.
+
+\item $F$ is injective
+
+that is, if $F(x_1)=F(x_2)$, then $x_1=x_2$.
+
+By partition of unity, we have, 
+
+$h_1(x_1)=h_1(x_2), h_2(x_1)=h_2(x_2), \dots, h_n(x_1)=h_n(x_2)$.
+
+And $\varphi_1(x_1)=\varphi_1(x_2), \varphi_2(x_1)=\varphi_2(x_2), \dots, \varphi_n(x_1)=\varphi_n(x_2)$.
+
+Because $\sum_{i=1}^n \varphi_i(x_1)=1$, therefore the exists $\varphi_i(x_1)=\varphi_i(x_2)>0$.
+
+Therefore $x1,x_2\in \operatorname{supp}(\phi_i)\subseteq U_i$.
+
+By definition of $h$, $h_i(x_1)=h_i(x_2)$, $\varphi_i(x_1)\phi_i(x_1)=\varphi_i(x_2)\phi_i(x_2)$.
+
+Using cancellation, $\phi_i(x_1)=\phi_i(x_2)$.
+
+Therefore $x_1=x_2$ since $\phi_i(x_1)=\phi_i(x_2)$ is a homeomorphism.
+
+\textit{In this proof, $\varphi$ ensures the imbedding is properly defined on the open sets}
+
+\item  $F$ is a homeomorphism.
+
+Note that if $f:X\to Y$ is continuous and $X$ is compact, $Y$ is Hausdorff, then $f$ is a closed map.
+
+$F:X\to F(X)$ is a bijective map from a compact space to a Hausdorff space, therefore $F$ is a closed map.
+
+Since $F$ is continuous, then $F^{-1}(C)$ where $C$ is a closed set in $F(X)$, $F^{-1}(C)$ is closed in $X$.
+
+Therefore $F$ is a homeomorphism.
+\end{enumerate} 
+\end{proof}
 
 \subsection{Smooth manifolds and Lie groups}
 
@@ -906,26 +993,26 @@ $$
 
 \begin{defn}
     \label{defn:smooth_map}
-    A function $f:U\to \mathbb{R}^n$ is smooth if it is of class $C^k$ for every $k\geq 0$ on $U$. Such function is called a diffeomorphism if it is also a \textbf{\texttt{bijection}} and its \textbf{\texttt{inverse is also smooth}}.
+    A function $f:U\to \mathbb{R}^n$ is smooth if it is of class $C^k$ for every $k\geq 0$ on $U$. Such function is called a diffeomorphism if it is also a \textbf{bijection} and its \textbf{inverse is also smooth}.
 \end{defn}
 
 
 \begin{defn}
     \label{defn:chart}
 
-    Let $M$ be a smooth manifold. A \textbf{\texttt{chart}} is a pair $(U,\varphi)$ where $U\subseteq M$ is an open subset and $\varphi:U\to \hat{U}\subseteq \mathbb{R}^n$ is a homeomorphism (a continuous bijection map and its inverse is also continuous).
+    Let $M$ be a smooth manifold. A \textbf{chart} is a pair $(U,\varphi)$ where $U\subseteq M$ is an open subset and $\varphi:U\to \hat{U}\subseteq \mathbb{R}^n$ is a homeomorphism (a continuous bijection map and its inverse is also continuous).
 
     If $p\in U$ and $\varphi(p)=0$, then we say that $p$ is the origin of the chart $(U,\varphi)$.
 
-    For $p\in U$, we note that the continuous function $\varphi(p)=(x_1(p),\cdots,x_n(p))$ gives a vector in $\mathbb{R}^n$. The $(x_1(p),\cdots,x_n(p))$ is called the \textbf{\texttt{local coordinates}} of $p$ in the chart $(U,\varphi)$.
+    For $p\in U$, we note that the continuous function $\varphi(p)=(x_1(p),\cdots,x_n(p))$ gives a vector in $\mathbb{R}^n$. The $(x_1(p),\cdots,x_n(p))$ is called the \textbf{local coordinates} of $p$ in the chart $(U,\varphi)$.
 
 \end{defn}
 
 \begin{defn}
     \label{defn:atlas}
-    Let $M$ be a smooth manifold. An \textbf{\texttt{atlas}} is a collection of charts $\mathcal{A}=\{(U_\alpha,\phi_\alpha)\}_{\alpha\in I}$ such that $M=\bigcup_{\alpha\in I} U_\alpha$.
+    Let $M$ be a smooth manifold. An \textbf{atlas} is a collection of charts $\mathcal{A}=\{(U_\alpha,\phi_\alpha)\}_{\alpha\in I}$ such that $M=\bigcup_{\alpha\in I} U_\alpha$.
 
-    An atlas is said to be \textbf{\texttt{smooth}} if the transition maps $\phi_\alpha\circ \phi_\beta^{-1}:\phi_\beta(U_\alpha\cap U_\beta)\to \phi_\alpha(U_\alpha\cap U_\beta)$ are smooth for all $\alpha, \beta\in I$.
+    An atlas is said to be \textbf{smooth} if the transition maps $\phi_\alpha\circ \phi_\beta^{-1}:\phi_\beta(U_\alpha\cap U_\beta)\to \phi_\alpha(U_\alpha\cap U_\beta)$ are smooth for all $\alpha, \beta\in I$.
 \end{defn}
 
 
@@ -936,12 +1023,17 @@ $$
 
 \begin{defn}
     \label{defn:differential}
-
+    Let $M$ and $N$ be smooth manifolds, and $f:M\to N$ be a smooth map. For each $p\in M$, the \textbf{differential} of $f$ at $p$ is the linear map
+    $$    
+    df_p:T_pM\to T_{f(p)}N
+    $$
 \end{defn}
 
 \begin{defn}
     \label{defn:smooth-submersion}
+     A smooth map $f:M\to N$ is a \textbf{smooth submersion} if for each $p\in M$, the differential $F:M\to N$ is surjective.
 
+     Or equivalently $\operatorname{rank}(F)=\dim N$ for each $p\in M$.
 \end{defn}
 
 Here are some additional propositions that will be helpful for our study in later sections:
@@ -1098,6 +1190,58 @@ Hence $g'=g$.
     Therefore there exists a unique Riemannian metric on $M$ such that $\pi$ is a Riemannian submersion.
 \end{proof}
 
+\subsection{Hopf fibration}
+
+There are some remaining steps for showing how the metric on Sphere induces the metric on complex projective space, now we will just drop the conclusion here so that we can continue our discussion:
+
+\begin{itemize}
+    \item Every nonzero vector can be normalized, so each pure state has a representative on the unit sphere
+          $$
+              S^{2n+1} \subset \mathbb C^{n+1}.
+          $$
+    \item Two unit vectors represent the same pure state exactly when they differ by a phase:
+          $$
+              z \sim e^{i\theta} z.
+          $$
+    \item Therefore
+          $$
+              \mathbb C P^n = S^{2n+1}/S^1.
+          $$
+\end{itemize}
+
+\vspace{0.4em}
+The quotient map
+$$
+    p:S^{2n+1}\to \mathbb C P^n, \qquad p(z)=[z]=\{\lambda z : \lambda \in \mathbb C^\times\},
+$$
+is the \textbf{Hopf fibration}.
+
+
+The geometric picture is
+$$
+    S^{2n+1}
+    \xrightarrow{\text{Hopf fibration}}
+    \mathbb C P^n,
+    \qquad
+    \text{round metric}
+    \rightsquigarrow
+    \text{Fubini--Study metric}.
+$$
+
+
+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 $\mathbb R^{2n+2}$.
+
+In homogeneous coordinates $[z]\in\mathbb C P^n$, the \textbf{Fubini--Study metric} is
+$$
+    g_{FS}
+    =
+    \frac{\langle dz,dz\rangle \langle z,z\rangle-|\langle z,dz\rangle|^2}{\langle z,z\rangle^2},
+$$
+
 \section{Quantum physics and terminologies}
 
 In this section, we will introduce some terminologies and theorems used in quantum physics that are relevant to our study. Assuming no prior knowledge of quantum physics, we will provide brief definitions and explanations for each term.
@@ -1137,21 +1281,50 @@ The uniqueness of such measurement came from the lemma below~\cite{Elizabeth_boo
 \end{lemma}
 
 
-\begin{defn}
-    \label{defn:pure_state}
-    Pure state:
+A finite-dimensional quantum system is modeled by a complex Hilbert space (a complete inner product space)
+$$
+    \mathcal H \cong \mathbb C^{n+1}.
+$$
 
-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 $\varphi\in_R\mathcal{P}(A)$.
-\end{defn}
+A \textbf{pure state} is represented by a unit vector
+$$
+    \psi \in \mathcal H, \qquad \|\psi\|=1.
+$$
 
-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.
+A \textbf{mixed state} is represented by a density matrix
+$$
+    \rho=\sum_{j=1}^n p_j|\psi_j\rangle\langle\psi_j|, \qquad \sum_{j=1}^n p_j=1, \qquad p_j\geq 0.
+$$
 
-\begin{defn}
-    \label{defn:rank_s_random_state}
-    Rank-$s$ random state.
+Some key comparisons between pure states and mixed states:
+
+Pure states describe maximal information; mixed states describe probabilistic mixtures or partial information.
+
+Pure states form a curved geometric space; mixed states form a convex set inside the space of matrices.
+
+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:
+          $$
+              \psi \sim \lambda \psi, \qquad \lambda \in \mathbb C^\times.
+          $$
+    \item In particular, multiplying by a phase $e^{i\theta}$ does not change any physical predictions.
+    \item Therefore the physical pure state is not a single vector, but the \emph{complex line} spanned by that vector.
+\end{itemize}
+
+Hence the space of pure states (denoted by $\mathcal{P}(\mathcal H)$) is
+$$
+    \mathcal{P}(\mathcal H)
+    =
+    (\mathcal H \setminus \{0\})/\mathbb C^\times.
+$$
+
+After choosing a basis $\mathcal H \cong \mathbb C^{n+1}$, this becomes
+$$
+    \mathcal{P}(\mathcal H) \cong \mathbb C P^n.
+$$
 
-    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}
 
 
 \begin{prop}
@@ -1180,6 +1353,26 @@ Intuitively, if the two states are not orthogonal, then for any measurement (pro
 First, we need to define what is a random state in a bipartite system.
 
 
+
+
+\begin{defn}
+    \label{defn:random_pure_state}
+    Pure state:
+
+    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 $\varphi\in_R\mathcal{P}(A)$.
+\end{defn}
+
+
+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}
+    \label{defn:rank_s_random_state}
+    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}
+
+
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latex/chapters/chap2.tex

@@ -332,6 +332,26 @@ f_{\mathrm{sphere}}(x^{(1)}),\dots,f_{\mathrm{sphere}}(x^{(N)}),
 $$
 and then evaluates the shortest interval containing mass at least $1-\kappa$. This gives an empirical observable-diameter proxy for the sphere family. The code also computes the empirical mean, median, standard deviation, and the normalized proxies obtained from sampled Lipschitz ratios.
 
+The experiment produces histograms of the observable values, upper-tail deficit plots for $\log_2 m - f_{\mathrm{sphere}}(x)$, and family-wise comparisons of partial diameter, standard deviation, and mean deficit. When available, these plots are overlaid with theoretical concentration scales derived from Lévy's lemma and related results \cite{lee_introduction_2018}.
+
+\begin{figure}[ht]
+    \centering
+    \begin{minipage}{0.48\textwidth}
+        \centering
+        \includegraphics[width=\textwidth]{../results/exp-20260311-154003/hist_sphere_16.png}
+        
+        Entropy distribution for $S^{15}$
+    \end{minipage}
+    \hfill
+    \begin{minipage}{0.48\textwidth}
+        \centering
+        \includegraphics[width=\textwidth]{../results/exp-20260311-154003/hist_sphere_256.png}
+        
+        Entropy distribution for $S^{255}$
+    \end{minipage}
+\end{figure}
+
+
 \subsection{Visualized the concentration of measure phenomenon on complex projective space}
 
 The second family is complex projective space
@@ -385,56 +405,52 @@ For each dimension pair $(d_A,d_B)$, the experiment samples $N$ independent Haar
 $$
 \log_2 d_A - S(\rho_A),
 $$
+
 and family-wise comparisons of partial diameter, standard deviation, and mean deficit. When available, these plots are overlaid with the Page average entropy and with Hayden-style concentration scales, which serve as theoretical guides rather than direct outputs of the simulation \cite{Hayden,Hayden_2006,Pages_conjecture_simple_proof}.
 
-\subsection{Random sampling using Majorana Stellar representation}
 
-The third family is the symmetric subspace
+\begin{figure}[ht]
+    \centering
+    \begin{minipage}{0.48\textwidth}
+        \centering
+        \includegraphics[width=\textwidth]{../results/exp-20260311-154003/hist_cp_16x16.png}
+        
+        Entropy distribution for $\mathbb{C}P^{15}\otimes\mathbb{C}P^{15}$
+    \end{minipage}
+    \hfill
+    \begin{minipage}{0.48\textwidth}
+        \centering
+        \includegraphics[width=\textwidth]{../results/exp-20260311-154003/hist_cp_256x256.png}
+        
+        Entropy distribution for $\mathbb{C}P^{255}\otimes\mathbb{C}P^{255}$
+    \end{minipage}
+\end{figure}
+
+
+\section{A conjecture on observable diameter for complex projective spaces}
+
+Given all the simulations so far, what does the concentration 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 $n\to \infty$?
+
+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 $O(\sqrt{n})$.
+
+\begin{theorem}{Wu's conjecture}
+
+    For $0<\kappa<1$,
     $$
-\operatorname{Sym}^N(\mathbb{C}^2),
-$$
-which is naturally identified with $\mathbb{C}P^N$ after projectivization. In this model, a pure symmetric $N$-qubit state is written in the Dicke basis as
-$$
-|\psi\rangle
-=
-\sum_{k=0}^{N} c_k |D^N_k\rangle,
-\qquad
-\sum_{k=0}^{N}|c_k|^2 = 1.
-$$
-The projective metric is again the Fubini--Study metric
-$$
-d_{FS}([\psi],[\phi])=\arccos |\langle \psi,\phi\rangle|.
+    \obdiam(\mathbb{C}P^n(1);-\kappa)= O(\sqrt{n}).
     $$
 
-Sampling is performed by drawing a standard complex Gaussian vector
-$$
-(c_0,\dots,c_N)\in \mathbb{C}^{N+1}
-$$
-and normalizing it. This gives the unitarily invariant measure on the projective symmetric state space.
+\end{theorem}
 
-The observable used by the code is the one-particle entropy of the symmetric state. From the coefficient vector $(c_0,\dots,c_N)$ one constructs the one-qubit reduced density matrix $\rho_1$, and then defines
-$$
-f_{\mathrm{Maj}}([\psi])
-=
-S(\rho_1)
-=
--\operatorname{Tr}(\rho_1 \log_2 \rho_1).
-$$
-Since $\rho_1$ is a qubit state, this observable takes values in $[0,1]$.
+The sketch for the proof is as follows:
 
-To visualize the same states in Majorana form, the code also associates to a sampled symmetric state its Majorana polynomial and computes its roots. After stereographic projection, these roots define $N$ points on $S^2$, called the Majorana stars \cite{Bengtsson_Zyczkowski_2017}. The resulting star plots are included only as geometric visualizations; they are not used to define the metric or the observable. The metric-measure structure used in the actual simulation remains the Fubini--Study metric and the unitarily invariant measure on the projective symmetric state space.
+\begin{itemize}
+    \item Entropy function is not globally lipschitz, so we need to bound the deficit of the entropy function.
 
-Thus, for each $N$, the simulation produces:
-\begin{enumerate}
-  \item a sample of symmetric states,
-  \item the corresponding one-body entropy values,
-  \item the shortest interval containing mass at least $1-\kappa$ in the push-forward distribution on $\mathbb{R}$,
-  \item empirical Lipschitz-normalized versions of this width,
-  \item and a separate Majorana-star visualization of representative samples.
-\end{enumerate}
-
-Taken together, these three families allow us to compare how entropy-based concentration behaves on a real sphere, on a general complex projective space carrying bipartite entanglement entropy, and on the symmetric subspace described by Majorana stellar data.
+    \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}
 
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