update

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
-$$
-\operatorname{Sym}^N(\mathbb{C}^2),
-$$
-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
-$$
-d_{FS}([\psi],[\phi])=\arccos |\langle \psi,\phi\rangle|.
-$$
+\begin{figure}[ht]
+    \centering
+    \begin{minipage}{0.48\textwidth}
+        \centering
+        \includegraphics[width=\textwidth]{../results/exp-20260311-154003/hist_cp_16x16.png}
         
-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.
+        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}
         
-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 $[0,1]$.
+        Entropy distribution for $\mathbb{C}P^{255}\otimes\mathbb{C}P^{255}$
+    \end{minipage}
+\end{figure}
 
-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.
 
-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}
+\section{A conjecture on observable diameter for complex projective spaces}
 
-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.
+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 $0<\kappa<1$,
+    $$
+    \obdiam(\mathbb{C}P^n(1);-\kappa)= O(\sqrt{n}).
+    $$
+
+\end{theorem}
+
+The sketch for the proof is as follows:
+
+\begin{itemize}
+    \item Entropy function is not globally lipschitz, so we need to bound the deficit of the entropy function.
+
+    \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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