bugfix

latex/chapters/chap1.tex

@@ -122,6 +122,8 @@ Other revised experiments (e.g., Aspect's experiment, calcium entangled photon e
 
 \subsection{The true model of light polarization}
 
+The contradiction above marks the point where classical probability stops being adequate. To continue, the sample-space picture must be replaced by states in a Hilbert space and by projections representing measurements. This operator model keeps the experimental probabilities but no longer forces incompatible measurements into a single classical joint distribution.
+    
 The full description of the light polarization is given below:
 
 State of polarization of a photon: $\psi=\alpha|0\rangle+\beta|1\rangle$, where $|0\rangle$ and $|1\rangle$ are the two orthogonal polarization states in $\mathbb{C}^2$.
@@ -160,8 +162,8 @@ The probability that a photon passes the first filter $P_{\alpha_i}$ is given by
 
 $$
 \operatorname{Prob}(P_i=1)
-=\operatorname{tr}(\rho P_{\alpha_i})
+=\operatorname{Tr}(\rho P_{\alpha_i})
-=\frac{1}{2} \operatorname{tr}(P_{\alpha_i})
+=\frac{1}{2} \operatorname{Tr}(P_{\alpha_i})
 =\frac{1}{2}
 $$
 
@@ -170,7 +172,7 @@ If the photon passes the first filter, the post-measurement state is given by th
 $$
 \rho \longmapsto
 \rho_i
-=\frac{P_{\alpha_i}\rho P_{\alpha_i}}{\operatorname{tr}(\rho P_{\alpha_i})}
+=\frac{P_{\alpha_i}\rho P_{\alpha_i}}{\operatorname{Tr}(\rho P_{\alpha_i})}
 = P_{\alpha_i}.
 $$
 
@@ -178,7 +180,7 @@ The probability that the photon then passes the second filter is
 
 $$
 \operatorname{Prob}(P_j=1 \mid P_i=1)
-=\operatorname{tr}(P_{\alpha_i} P_{\alpha_j})
+=\operatorname{Tr}(P_{\alpha_i} P_{\alpha_j})
 =\cos^2(\alpha_i-\alpha_j).
 $$
 
@@ -198,6 +200,8 @@ This agrees with the experimentally observed transmission probabilities, but it
 
 \section{Concentration of measure phenomenon}
 
+The operator model explains why entanglement is a meaningful observable, but it does not yet explain why large random systems are typically highly entangled. That is the role of concentration of measure. The next section moves from quantum motivation back to geometry and probability, where high-dimensional spheres already exhibit the same kind of rigidity that later reappears in the entropy of random bipartite states.
+
 \begin{defn}
 	$\eta$-Lipschitz function
 
@@ -251,9 +255,9 @@ To prove the lemma, we need to have a good understanding of the Riemannian geome
     $$
     \kappa_n(\epsilon)=\frac{\int_\epsilon^{\frac{\pi}{2}}\cos^{n-1}(t)dt}{\int_0^{\frac{\pi}{2}}\cos^{n-1}(t)dt}
     $$
-    $a_0$ is the \textbf{Levy mean} of function $f$, that is, the level set $f^{-1}:\mathbb{R}\to S^n$ divides the sphere into equal halves, characterized by the following equality:
+    $a_0$ is a \textbf{median} of $f$, characterized by the following inequalities:
     $$
-    \mu(f^{-1}(-\infty,a_0])\geq \frac{1}{2} \text{ and } \mu(f^{-1}[a_0,\infty))\geq \frac{1}{2}
+    \mu(f^{-1}((-\infty,a_0]))\geq \frac{1}{2} \text{ and } \mu(f^{-1}([a_0,\infty)))\geq \frac{1}{2}
     $$
 \end{theorem}
 
@@ -275,7 +279,7 @@ We will prove the theorem via the Maxwell-Boltzmann distribution law in this sec
 
 Basically, you can consider the Gaussian measure as the normalized Lebesgue measure on $\mathbb{R}^k$ with standard deviation $1$.
 
-It also has another name, the Projective limit theorem.~\cite{romanvershyni}
+It also has another name, the Poincar\'e limit theorem.~\cite{romanvershyni}
 
 If $X\sim \operatorname{Unif}(S^n(\sqrt{n}))$, then for any fixed unit vector $x$ we have $\langle X,x\rangle\to N(0,1)$ in distribution as $n\to \infty$.
 
@@ -348,7 +352,7 @@ Now we can prove Levy's concentration theorem, the proof is from~\cite{shioya201
     It is sufficient to show that,
 
     $$
-    U_{\epsilon_1}(\Omega_+)\cup U_{\epsilon_2}(\Omega_-)\subset \{x'-\epsilon_1\leq f_{n_i}\leq x'+\epsilon_2\}
+    U_{\epsilon_1}(\Omega_+)\cap U_{\epsilon_2}(\Omega_-)\subset \{x'-\epsilon_1\leq f_{n_i}\leq x'+\epsilon_2\}
     $$
 
     By 1-Lipschitz continuity of $f_{n_i}$, we have for all $\zeta\in U_{\epsilon_1}(\Omega_+)$, there is a point $\xi\in \Omega_+$ such that $d(\zeta,\xi)\leq \epsilon_1$. So $U_{\epsilon_1}(\Omega_+)\subset \{f_{n_i}\geq x'-\epsilon_1\}$. With the same argument, we have $U_{\epsilon_2}(\Omega_-)\subset \{f_{n_i}\leq x+\epsilon_2\}$.
@@ -359,7 +363,7 @@ Now we can prove Levy's concentration theorem, the proof is from~\cite{shioya201
     \begin{aligned}
     (f_{n_i})_*\sigma^{n_i}[x'-\epsilon_1,x'+\epsilon_2]&=\sigma^{n_i}(x'-\epsilon_1\leq f_{n_i}\leq x'+\epsilon_2)\\
     &\geq \sigma^{n_i}(U_{\epsilon_1}(\Omega_+)\cap U_{\epsilon_2}(\Omega_-))\\
-    &=\sigma^{n_i}(U_{\epsilon_1}(\Omega_+))+\sigma^{n_i}(U_{\epsilon_2}(\Omega_-))-1\\
+    &\geq \sigma^{n_i}(U_{\epsilon_1}(\Omega_+))+\sigma^{n_i}(U_{\epsilon_2}(\Omega_-))-1\\
     \end{aligned}
     $$
 
@@ -380,7 +384,7 @@ Now we can prove Levy's concentration theorem, the proof is from~\cite{shioya201
     $$
     \begin{aligned}
     \sigma_\infty[x'-\epsilon_1,x'+\epsilon_2]&\geq \liminf_{i\to \infty}(f_{n_i})_*\sigma^{n_i}[x'-\epsilon_1,x'+\epsilon_2]\\
-    &\geq \gamma^1[x'-\epsilon_1,\infty)\cap \gamma^1(-\infty,x+\epsilon_2]-1\\
+    &\geq \gamma^1[x'-\epsilon_1,\infty)+\gamma^1(-\infty,x+\epsilon_2]-1\\
     &=\gamma^1[x-\epsilon_1,x+\epsilon_2]
     \end{aligned}
     $$
@@ -391,6 +395,8 @@ The full proof of Levy's concentration theorem requires more digestion for cases
 
 \section{The application of the concentration of measure phenomenon in non-commutative probability theory}
 
+Having established concentration for Lipschitz observables on high-dimensional spheres, we can now return to quantum information. The remaining step is to identify a physically meaningful observable on pure states whose geometry is controlled well enough for Levy-type bounds to apply. In this thesis that observable is entanglement entropy, viewed after partial trace.
+
 In quantum communication, we can pass classical bits by sending quantum states. However, by the indistinguishability (Proposition~\ref{prop:indistinguishability}) of quantum states, we cannot send an infinite number of classical bits over a single qubit. There exists a bound for zero-error classical communication rate over a quantum channel.
 
 \begin{theorem}
@@ -424,6 +430,8 @@ To surpass the Holevo bound, we need to use the entanglement of quantum states.
 
 \subsection{Superdense coding and entanglement}
 
+Superdense coding is the operational reason entanglement matters in this chapter. It shows that entangled states are not merely algebraically interesting: they change communication capacity. Once that point is clear, the natural probabilistic question is whether highly entangled states are rare or typical, which leads directly to Hayden's concentration theorem.
+
 The description of the superdense coding can be found in~\cite{gupta2015functionalanalysisquantuminformation} and~\cite{Hayden}.
 
 Suppose $A$ and $B$ share a Bell state (or other maximally entangled state) $|\Phi^+\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$, where $A$ holds the first part and $B$ holds the second part.
@@ -453,6 +461,8 @@ Additionally, no information can be gained by measuring a pair of entangled qubi
 
 \subsection{Hayden's concentration of measure phenomenon}
 
+The geometric and communication-theoretic threads now meet. Random pure states live on a projective state space, partial trace sends them to mixed states, and entropy turns those mixed states into real numbers. Hayden's theorem is precisely the statement that this entropy observable is concentrated when the ambient dimension is large.
+
 The application of the concentration of measure phenomenon in the superdense coding can be realized in random sampling the entangled qubits~\cite{Hayden}:
 
 It is a theorem connecting the following mathematical structure:
@@ -468,8 +478,8 @@ It is a theorem connecting the following mathematical structure:
         % define nodes
         \node[main] (cp) {$\mathbb{C}P^{d_A d_B-1}$};
         \node[main] (pa) [left of=cp] {$\mathcal{P}(A\otimes B)$};
-        \node[main] (sa) [below of=pa] {$S_A$};
+        \node[main] (sa) [below of=pa] {$\mathcal{S}(A)$};
-        \node[main] (rng) [right of=sa] {$[0,\infty)\subset \mathbb{R}$};
+        \node[main] (rng) [right of=sa] {$[0,\log_2(d_A)]\subset \mathbb{R}$};
 
         % draw edges
         \draw[mutual] (cp) -- (pa);
@@ -483,7 +493,7 @@ It is a theorem connecting the following mathematical structure:
 
 \begin{itemize}
     \item The red arrow is the concentration of measure effect. $f=H(\operatorname{Tr}_B(\psi))$.
-    \item $S_A$ denotes the mixed states on $A$.
+    \item $\mathcal{S}(A)$ denotes the mixed states on $A$.
 \end{itemize}
 
 To prove the concentration of measure phenomenon, we need to analyze the following elements involved in figure~\ref{fig:Hayden_concentration_of_measure_phenomenon}:
@@ -491,7 +501,7 @@ To prove the concentration of measure phenomenon, we need to analyze the followi
     
 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.
 
-Due to time constrains of the projects, the following lemma is demonstrated but not investigated thoroughly through the research:
+Due to time constraints of the project, the following lemma is demonstrated but not investigated thoroughly through the research:
 
 
 \begin{lemma}
@@ -499,9 +509,9 @@ Due to time constrains of the projects, the following lemma is demonstrated but
 
     Page's lemma for expected entropy of mixed states
 
-    Choose a random pure state $\sigma=|\psi\rangle\langle\psi|$ from $A'\otimes A$.
+    Choose a random pure state $\sigma=|\psi\rangle\langle\psi|$ from $A\otimes B$.
 
-    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].
+    The expected value of the entropy of entanglement is known and is given by Page's formula~\cite{Pages_conjecture,Pages_conjecture_simple_proof,Bengtsson_Zyczkowski_2017}[15.72].
 
     $$
     \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}
@@ -546,12 +556,12 @@ Then we have bound for Lipschitz constant $\eta$ of the map $S(\varphi_A): \math
     Therefore 
     
     $$
-    \frac{\partial g}{\partial x_{jk}}=\frac{\partial g}{\partial p_j}\frac{\partial p_j}{x_{jk}}=-\frac{1+\ln p_j}{\ln 2}\cdot 2x_{jk}
+    \frac{\partial g}{\partial x_{jk}}=\frac{\partial g}{\partial p_j}\frac{\partial p_j}{\partial x_{jk}}=-\frac{1+\ln p_j}{\ln 2}\cdot 2x_{jk}
     \qquad 
     \frac{\partial g}{\partial y_{jk}}=-\frac{1+\ln p_j}{\ln 2}\cdot 2y_{jk}
     $$
     
-    Then the lipschitz constant of $g$ is
+    Then the Lipschitz constant of $g$ is
 
     $$
     \begin{aligned}
@@ -564,7 +574,7 @@ Then we have bound for Lipschitz constant $\eta$ of the map $S(\varphi_A): \math
 
     Note that $\sum_{k=1}^{d_B}|\varphi_{jk}|^2=p_j(\varphi)$, $\nabla g\cdot \nabla g=\frac{4}{(\ln 2)^2}\sum_{j=1}^{d_A}p_j(\varphi)(1+\ln p_j(\varphi))^2$.
 
-    Since $0\leq p_j\leq 1$, we have $\ln p_j(\varphi)\leq 0$, hence $\sum_{j=0}^{d_A}p_j(\varphi)\ln p_j(\varphi)\leq 0$.
+    Since $0\leq p_j\leq 1$, we have $\ln p_j(\varphi)\leq 0$, hence $\sum_{j=1}^{d_A}p_j(\varphi)\ln p_j(\varphi)\leq 0$.
     
     $$
     \begin{aligned}
@@ -583,7 +593,7 @@ Then we have bound for Lipschitz constant $\eta$ of the map $S(\varphi_A): \math
     \end{aligned}
     $$
 
-    Proving $\sum_j^{d_A} p_j(\varphi)\ln p_j(\varphi)\leq (\ln d_A)^2$ for $d_A\geq 3$ takes some efforts and we will continue that later.
+    Proving $\sum_{j=1}^{d_A} p_j(\varphi)(\ln p_j(\varphi))^2\leq (\ln d_A)^2$ for $d_A\geq 3$ takes some efforts and we will continue that later.
 
     Consider any two unit vectors $\ket{\varphi}$ and $\ket{\psi}$, assume $S(\varphi_A)\leq S(\psi_A)$. If we choose the measurement $M$ to be along the eigenbasis of $\varphi_A$, $H(M(\varphi_A))=S(\varphi_A)$ and we have
 
@@ -591,7 +601,7 @@ Then we have bound for Lipschitz constant $\eta$ of the map $S(\varphi_A): \math
     S(\psi_A)-S(\varphi_A)\leq H(M(\psi_A))-H(M(\varphi_A))\leq \eta\|\ket{\psi}-\ket{\varphi}\|
     $$
 
-    Thus the lipschitz constant of $S(\varphi_A)$ is upper bounded by $\sqrt{8}\log_2(d_A)$.
+    Thus the Lipschitz constant of $S(\varphi_A)$ is upper bounded by $\sqrt{8}\log_2(d_A)$.
 \end{proof}
 
 From Levy's lemma, we have

latex/chapters/chap2.tex

@@ -10,19 +10,21 @@
 \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 $D$.
+The concentration of measure phenomenon comes with an important notion called the observable diameter, which measures how tightly every $1$-Lipschitz function $f:X\to \R$ concentrates on a metric-measure space $X$ outside a set of measure at most $\kappa$.
 
-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.
+From Hayden's work, we know that for a random pure state in a bipartite system the entropy is nearly 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.
+Altogether, we study 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.
+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 they reveal for entropy functions.
 
 We will try to use the results from previous sections to estimate the observable diameter for complex projective spaces.
 
 \section{Observable diameters}
 
+This chapter starts from the qualitative concentration statements in Chapter 1 and asks for a geometric quantity that records the same phenomenon directly at the level of metric-measure spaces. Observable diameter is that quantity. It translates concentration of real observables into an invariant of the underlying space, which makes it possible to compare spheres and complex projective spaces on the same footing.
+
 Recall from previous sections, 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.
 
 \begin{defn}
@@ -31,9 +33,9 @@ Recall from previous sections, an arbitrary 1-Lipschitz function $f:S^n\to \math
   Let $X$ be a topological space with the following:
 
   \begin{enumerate}
-    \item $X$ is a complete (every Cauchy sequence converges)
+    \item $X$ is complete.
-    \item $X$ is a metric space with metric $d_X$
+    \item $X$ is a metric space with metric $d_X$.
-    \item $X$ has a Borel probability measure $\mu_X$
+    \item $X$ has a Borel probability measure $\mu_X$.
   \end{enumerate}
 
   Then $(X,d_X,\mu_X)$ is a \textbf{metric measure space}.
@@ -51,31 +53,30 @@ Recall from previous sections, an arbitrary 1-Lipschitz function $f:S^n\to \math
 \begin{defn}
   \label{defn:partial-diameter}
 
-  Let $(X,d_X,\mu_X)$ be a metric measure space, For any real number $\alpha\leq 1$, the \textbf{partial diameter} of $X$ is defined as
+  Let $(X,d_X,\mu_X)$ be a metric measure space. For any real number $\alpha\leq 1$, the \textbf{partial diameter} of $X$ is defined as
   $$
-  \diam(A;\alpha)=\inf_{A\subseteq X|\mu_X(A)\geq \alpha}\diam(A).
+  \diam(X;\alpha)=\inf_{A\subseteq X,\ \mu_X(A)\geq \alpha}\diam(A).
   $$
 \end{defn}
 
-This definition generalize the relation between the measure and metric in the metric-measure space. Intuitively, the space with smaller partial diameter can take more volume with the same diameter constrains.
+This definition generalizes the relation between the measure and metric in a metric-measure space. Intuitively, a space with smaller partial diameter can carry more mass inside the same diameter constraint.
-
-However, in higher dimensions, the volume may tend to concentrates more around a small neighborhood of the set, as we see in previous chapters with high dimensional sphere as example. We can safely cut $\kappa>0$ volume to significantly reduce the diameter, this yields better measure for concentration for shapes in spaces with high dimension.
 
+However, in higher dimensions, the volume may tend to concentrate more around a small neighborhood of a set, as we saw earlier for high-dimensional spheres. We can safely cut away $\kappa>0$ mass to reduce the diameter significantly, and this yields a better measure of concentration.
 
 \begin{defn}
   \label{defn:observable-diameter}
-  Let $X$ be a metric-measure space, $Y$ be a metric space, and $f:X\to Y$ be a 1-Lipschitz function. Then $f_*\mu_X=\mu_Y$ is a push forward measure on $Y$.
+  Let $X$ be a metric-measure space, $Y$ be a metric space, and $f:X\to Y$ be a 1-Lipschitz function. Then $f_*\mu_X=\mu_Y$ is a push-forward measure on $Y$.
   
   For any real number $\kappa>0$, the \textbf{$\kappa$-observable diameter with screen $Y$} is defined as
 
   $$
-  \obdiam_Y(X;\kappa)=\sup\{\diam(f_*\mu_X;1-\kappa)\}
+  \obdiam_Y(X;-\kappa)=\sup\{\diam(f_*\mu_X;1-\kappa)\mid f:X\to Y \text{ is 1-Lipschitz}\}.
   $$
 
-  And the \textbf{obbservable diameter with screen $Y$} is defined as
+  And the \textbf{observable diameter with screen $Y$} is defined as
 
   $$
-  \obdiam_Y(X)=\inf_{\kappa>0}\max\{\obdiam_Y(X;\kappa)\}
+  \obdiam_Y(X)=\inf_{\kappa>0}\max\{\obdiam_Y(X;-\kappa),\kappa\}.
   $$
 
   If $Y=\R$, we call it the \textbf{observable diameter}.
@@ -84,27 +85,27 @@ However, in higher dimensions, the volume may tend to concentrates more around a
 
 If we collapse it naively via
 $$
-  \inf_{\kappa>0}\obdiam_Y(X;\kappa),
+  \inf_{\kappa>0}\obdiam_Y(X;-\kappa),
 $$
-we typically get something degenerate: as $\kappa\to 1$, the condition ``mass $\ge 1-\kappa$'' becomes almost empty space, so $\diam(\nu;1-\kappa)$ can be forced to be $0$ (take a tiny set of positive mass), and hence the infimum tends to $0$ for essentially any non-atomic space.
+we typically get something degenerate: as $\kappa\to 1$, the condition ``mass $\ge 1-\kappa$'' becomes almost empty, so $\diam(\nu;1-\kappa)$ can be forced to be $0$ by taking a tiny set of positive mass. Hence the infimum tends to $0$ for essentially any non-atomic space.
 
 This is why one either:
 \begin{enumerate}
-  \item keeps $\obdiam_Y(X;\kappa)$ as a \emph{function of $\kappa$} (picking $\kappa$ to be small but not $0$), or
+  \item keeps $\obdiam_Y(X;-\kappa)$ as a \emph{function of $\kappa$} (picking $\kappa$ to be small but not $0$), or
-  \item if one insists on a single number, balances ``spread'' against ``exceptional mass'' by defining $\obdiam_Y(X)=\inf_{\kappa>0}\max\{\obdiam_Y(X;\kappa),\kappa\}$ as above.
+  \item if one insists on a single number, balances ``spread'' against ``exceptional mass'' by defining $\obdiam_Y(X)=\inf_{\kappa>0}\max\{\obdiam_Y(X;-\kappa),\kappa\}$ as above.
 \end{enumerate}
-The point of the $\max\{\cdot,\kappa\}$ is that it prevents cheating by taking $\kappa$ close to $1$: if $\kappa$ is large then the maximum is large regardless of how small $\obdiam_Y(X;\kappa)$ is, so the infimum is forced to occur where the exceptional mass and the observable spread are small.
+The point of the $\max\{\cdot,\kappa\}$ is that it prevents cheating by taking $\kappa$ close to $1$: if $\kappa$ is large then the maximum is large regardless of how small $\obdiam_Y(X;-\kappa)$ is, so the infimum is forced to occur where the exceptional mass and the observable spread are small.
 
-Few additional proposition in \cite{shioya2014metricmeasuregeometry} will help us to estimate the observable diameter for complex projective spaces.
+Few additional propositions in \cite{shioya2014metricmeasuregeometry} will help us estimate the observable diameter for complex projective spaces.
 
 \begin{prop}
   \label{prop:observable-diameter-domination}
-  Let $X$ and $Y$ be two metric-measure spaces and $\kappa>0$, and let $f:Y\to X$ be a 1-Lipschitz function ($Y$ dominates $X$, denoted as $X\prec Y$) then:
+  Let $X$ and $Y$ be two metric-measure spaces and $\kappa>0$, and let $f:Y\to X$ be a 1-Lipschitz function ($Y$ dominates $X$, denoted as $X\prec Y$). Then:
 
   \begin{enumerate}
     \item
     $
-    \diam(X,1-\kappa)\leq \diam(Y,1-\kappa)
+    \diam(X;1-\kappa)\leq \diam(Y;1-\kappa)
     $
     \item $\obdiam(X;-\kappa)\leq \diam(X;1-\kappa)$, and $\obdiam(X)$ is finite.
     \item
@@ -115,89 +116,92 @@ Few additional proposition in \cite{shioya2014metricmeasuregeometry} will help u
 \end{prop}
 
 \begin{proof}
-  Since $f$ is 1-Lipschitz, we have $f_*\mu Y=\mu_X$. Let $A$ be any Borel set of $Y$ with $\mu_Y(A)\geq 1-\kappa$ and $\overline{f(A)}$ be the closure of $f(A)$ in $X$. We have $\mu_X(\overline{f(A)})=\mu_Y(f^{-1}(\overline{f(A)}))\geq \mu_Y(A)\geq 1-\kappa$ and by the 1-lipschitz property, $\diam(\overline{f(A)})\leq \diam(A)$, so $\diam(X;1-\kappa)\leq \diam(A)\leq \diam(Y;1-\kappa)$.
+  Since $f$ is 1-Lipschitz, we have $f_*\mu_Y=\mu_X$. Let $A$ be any Borel set of $Y$ with $\mu_Y(A)\geq 1-\kappa$ and $\overline{f(A)}$ be the closure of $f(A)$ in $X$. We have $\mu_X(\overline{f(A)})=\mu_Y(f^{-1}(\overline{f(A)}))\geq \mu_Y(A)\geq 1-\kappa$ and by the 1-Lipschitz property, $\diam(\overline{f(A)})\leq \diam(A)$, so $\diam(X;1-\kappa)\leq \diam(A)\leq \diam(Y;1-\kappa)$.
 
-  Let $g:X\to \R$ be any 1-lipschitz function, since $(\R,|\cdot|,g_*\mu_X)$ is dominated by $X$, $\diam(\R;1-\kappa)\leq \diam(X;1-\kappa)$. Therefore, $\obdiam(X;-\kappa)\leq \diam(X;1-\kappa)$.
+  Let $g:X\to \R$ be any 1-Lipschitz function. Since $(\R,|\cdot|,g_*\mu_X)$ is dominated by $X$, $\diam(\R;1-\kappa)\leq \diam(X;1-\kappa)$. Therefore, $\obdiam(X;-\kappa)\leq \diam(X;1-\kappa)$.
  
-  and
+  And
   $$
-  \diam(g_*\mu_X;1-\kappa)\leq \diam((f\circ g)_*\mu_Y;1-\kappa)\leq \obdiam(Y;-\kappa)
+  \diam(g_*\mu_X;1-\kappa)\leq \diam((g\circ f)_*\mu_Y;1-\kappa)\leq \obdiam(Y;-\kappa).
   $$
 \end{proof}
 
 \begin{prop}
   \label{prop:observable-diameter-scale}
-  Let $X$ be an metric-measure space. Then for any real number $t>0$, we have
+  Let $X$ be a metric-measure space. Then for any real number $t>0$, we have
   
   $$
   \obdiam(tX;-\kappa)=t\obdiam(X;-\kappa)
   $$
 
-  Where $tX=(X,tdX,\mu X)$.
+  where $tX=(X,td_X,\mu_X)$.
 \end{prop}
 
 \begin{proof}
   $$
   \begin{aligned}
-    \obdiam(tX;-\kappa)&=\sup\{\diam(f_*\mu_X;1-\kappa)|f:tX\to \R \text{ is 1-Lipschitz}\}\\
+    \obdiam(tX;-\kappa)&=\sup\{\diam(f_*\mu_X;1-\kappa)\mid f:tX\to \R \text{ is 1-Lipschitz}\}\\
-    &=\sup\{\diam(f_*\mu_X;1-\kappa)|t^{-1}f:X\to \R \text{ is 1-Lipschitz}\}\\
+    &=\sup\{\diam(f_*\mu_X;1-\kappa)\mid t^{-1}f:X\to \R \text{ is 1-Lipschitz}\}\\
-    &=\sup\{\diam((tg)_*\mu_X;1-\kappa)|g:X\to \R \text{ is 1-Lipschitz}\}\\
+    &=\sup\{\diam((tg)_*\mu_X;1-\kappa)\mid g:X\to \R \text{ is 1-Lipschitz}\}\\
-    &=t\sup\{\diam(g_*\mu_X;1-\kappa)|g:X\to \R \text{ is 1-Lipschitz}\}\\
+    &=t\sup\{\diam(g_*\mu_X;1-\kappa)\mid g:X\to \R \text{ is 1-Lipschitz}\}\\
-    &=t\obdiam(X;-\kappa)
+    &=t\obdiam(X;-\kappa).
   \end{aligned}
   $$
 \end{proof}
 
 \subsection{Observable diameter for class of spheres}
 
-In this section, we will try to use the results from previous sections to estimate the observable diameter for class of spheres.
+With the basic formalism in place, the next step is to test it on the standard family where concentration is best understood: high-dimensional spheres. This is the model case in which Gaussian limits and scaling arguments can be made explicit, and it provides the comparison space that will later control complex projective space through the Hopf fibration.
+
+In this section, we will try to use the results from previous sections to estimate the observable diameter for the class of spheres.
 
 \begin{theorem}
   \label{thm:observable-diameter-sphere}
   For any real number $\kappa$ with $0<\kappa<1$, we have
   $$
-  \obdiam(S^n(1);-\kappa)=O(\sqrt{n})
+  \obdiam(S^n(1);-\kappa)=O\left(\frac{1}{\sqrt{n}}\right).
   $$
 \end{theorem}
 
 \begin{proof}
-  First, recall that by maxwell boltzmann distribution, we have that for any $n>0$, let $I(r)$ denote the measure of standard gaussian measure on the interval $[0,r]$. Then we have
+  First, recall that by the Maxwell-Boltzmann distribution law, for any $n>0$, if $I(r)$ denotes the measure of the standard Gaussian measure on the interval $[0,r]$, then we have
   
   $$
   \begin{aligned}
-    \lim_{n\to \infty} \obdiam(S^n(\sqrt{n});-\kappa)&=\lim_{n\to \infty} \sup\{\diam((\pi_{n,k})_*\sigma^n;1-\kappa)|\pi_{n,k} \text{ is 1-Lipschitz}\}\\
+    \lim_{n\to \infty} \obdiam(S^n(\sqrt{n});-\kappa)&=\lim_{n\to \infty} \sup\{\diam((\pi_{n,k})_*\sigma^n;1-\kappa)\mid \pi_{n,k} \text{ is 1-Lipschitz}\}\\
-    &=\lim_{n\to \infty} \sup\{\diam(\gamma^1;1-\kappa)|\gamma^1 \text{ is the standard gaussian measure}\}\\
+    &=\lim_{n\to \infty} \sup\{\diam(\gamma^1;1-\kappa)\mid \gamma^1 \text{ is the standard Gaussian measure}\}\\
     &=\diam(\gamma^1;1-\kappa)\\
-    &=2I^{-1}(\frac{1-\kappa}{2})\text { cutting the extremum for normal distribution}\\
+    &=2I^{-1}\left(\frac{1-\kappa}{2}\right).
   \end{aligned}
   $$
 
-  By proposition \ref{prop:observable-diameter-scale}, we have
+  By Proposition \ref{prop:observable-diameter-scale}, we have
 
   $$
-  \obdiam(S^n(\sqrt{n});-\kappa)=\sqrt{n}\obdiam(S^n(1);-\kappa)
+  \obdiam(S^n(\sqrt{n});-\kappa)=\sqrt{n}\obdiam(S^n(1);-\kappa).
   $$
 
-  So $\obdiam(S^n(1);-\kappa)=\sqrt{n}(2I^{-1}(\frac{1-\kappa}{2}))=O(\sqrt{n})$.
+  So
+  $$
+  \obdiam(S^n(1);-\kappa)=\frac{1}{\sqrt{n}}\,2I^{-1}\left(\frac{1-\kappa}{2}\right)=O\left(\frac{1}{\sqrt{n}}\right).
+  $$
 \end{proof}
 
-From the previous discussion, we see that the only remaining for finding observable diameter of $\C P^n$ is to find the lipchitz function that is isometric with consistent push-forward measure.
+From the previous discussion, we see that the only remaining step in finding the observable diameter of $\C P^n$ is to identify a $1$-Lipschitz map with the correct push-forward measure.
 
-To find such metric, we need some additional results.
+To find such a metric, we need some additional results.
 
 \begin{defn}
-  \label{defn:riemannian-metric}
+  \label{defn:riemannian-metric-chap2}
 
   Let $M$ be a smooth manifold. A \textit{\textbf{Riemannian metric}} on $M$ is a smooth covariant tensor field $g\in \mathcal{T}^2(M)$ such that for each $p\in M$, $g_p$ is an inner product on $T_pM$.
 
-  $g_p(v,v)\geq 0$ for each $p\in M$ and each $v\in T_pM$. equality holds if and only if $v=0$.
+  $g_p(v,v)\geq 0$ for each $p\in M$ and each $v\in T_pM$. Equality holds if and only if $v=0$.
 
 \end{defn}
 
-% TODO: There is a hidden chapter on group action on manifolds, can you find that?
-
 \begin{theorem}
-  \label{theorem:riemannian-submersion}
+  \label{theorem:riemannian-submersion-chap2}
 
   Let $(\tilde{M},\tilde{g})$ be a Riemannian manifold, let $\pi:\tilde{M}\to M$ be a surjective smooth submersion, and let $G$ be a group acting on $\tilde{M}$. If the \textbf{action} is
   \begin{enumerate}
@@ -209,29 +213,29 @@ To find such metric, we need some additional results.
 
 \end{theorem}
 
-A natural measure for $\C P^n$ is the normalized volume measure on $\C P^n$ induced from the Fubini-Study metric. \cite{lee_introduction_2018} Example 2.30
+A natural measure for $\C P^n$ is the normalized volume measure on $\C P^n$ induced from the Fubini-Study metric \cite[Example 2.30]{lee_introduction_2018}.
 
 \begin{defn}
   \label{defn:fubini-study-metric}
 
-  Let $n$ be a positive integer, and consider the complex projective space $\C P^n$ defined as the quotient space of $\C^{n+1}$ by the equivalence relation $z\sim z'$ if there exists $\lambda \in \C$ such that $z=\lambda z'$. The map $\pi:\C^{n+1}\setminus\{0\}\to \C P^n$ sending each point in $\C^{n+1}\setminus\{0\}$ to its span is surjective smooth submersion.
+  Let $n$ be a positive integer, and consider the complex projective space $\C P^n$ defined as the quotient space of $\C^{n+1}$ by the equivalence relation $z\sim z'$ if there exists $\lambda \in \C$ such that $z=\lambda z'$. The map $\pi:\C^{n+1}\setminus\{0\}\to \C P^n$ sending each point in $\C^{n+1}\setminus\{0\}$ to its span is a surjective smooth submersion.
 
-  Identifying $\C^{n+1}$ with $\R^{2n+2}$ with its Euclidean metric, we can view the unit sphere $S^{2n+1}$ with its roud metric $\mathring{g}$ as an embedded Riemannian submanifold of $\C^{n+1}\setminus\{0\}$. Let $p:S^{2n+1}\to \C P^n$ denote the restriction of the map $\pi$. Then $p$ is smooth, and its is surjective, because every 1-dimensional complex subspace contains elements of unit norm.
+  Identifying $\C^{n+1}$ with $\R^{2n+2}$ with its Euclidean metric, we can view the unit sphere $S^{2n+1}$ with its round metric $\mathring{g}$ as an embedded Riemannian submanifold of $\C^{n+1}\setminus\{0\}$. Let $p:S^{2n+1}\to \C P^n$ denote the restriction of the map $\pi$. Then $p$ is smooth, and it is surjective, because every 1-dimensional complex subspace contains elements of unit norm.
 
 \end{defn}
 
-There are many additional properties for such construction, we will check them just for curiosity.
+There are many additional properties for such a construction, and here we check only the point needed later.
 
 We need to show that it is a submersion.
 
 \begin{proof}
-   Let $z_0\in S^{2n+1}$ and set $\zeta_0=p(z_0)\in \C P^n$. Since $\pi$ is a smooth submersion, it has a smooth local section $\sigma: U\to \C^{n+1}$ defined on a neighborhood $U$ of $\zeta_0$ and satisfying $\sigma(\zeta_0)=z_0$ by the local section theorem (Theorem \ref{theorem:local_section_theorem}). Let $v:\C^{n+1}\setminus\{0\}\to S^{2n+1}$ be the radial projection on to the sphere:
+   Let $z_0\in S^{2n+1}$ and set $\zeta_0=p(z_0)\in \C P^n$. Since $\pi$ is a smooth submersion, it has a smooth local section $\sigma: U\to \C^{n+1}$ defined on a neighborhood $U$ of $\zeta_0$ and satisfying $\sigma(\zeta_0)=z_0$ by the local section theorem (Theorem \ref{theorem:local_section_theorem}). Let $v:\C^{n+1}\setminus\{0\}\to S^{2n+1}$ be the radial projection onto the sphere:
 
    $$
-   v(z)=\frac{z}{|z|}
+   v(z)=\frac{z}{|z|}.
    $$
 
-   Since dividing an element of $\C^{n+1}$ by a nonzero scalar does not change its span, it follows that $p\circ v=\pi$. Therefore, if we set $\tilde{\sigma}=v\circ \sigma$, then $\tilde{\sigma}$ is a smooth local section of $p$. Apply the local section theorem (Theorem \ref{theorem:local_section_theorem}) again, this shows that $p$ is a submersion.
+   Since dividing an element of $\C^{n+1}$ by a nonzero scalar does not change its span, it follows that $p\circ v=\pi$. Therefore, if we set $\tilde{\sigma}=v\circ \sigma$, then $\tilde{\sigma}$ is a smooth local section of $p$. Applying the local section theorem again shows that $p$ is a submersion.
 
    Define an action of $S^1$ on $S^{2n+1}$ by complex multiplication:
 
@@ -239,32 +243,39 @@ We need to show that it is a submersion.
    \lambda (z^1,z^2,\ldots,z^{n+1})=(\lambda z^1,\lambda z^2,\ldots,\lambda z^{n+1})
    $$
 
-   for $\lambda\in S^1$ (viewed as complex number of norm 1) and $z=(z^1,z^2,\ldots,z^{n+1})\in S^{2n+1}$. This is easily seen to be isometric, vertical, and transitive on fibers of $p$.
+   for $\lambda\in S^1$ (viewed as a complex number of norm $1$) and $z=(z^1,z^2,\ldots,z^{n+1})\in S^{2n+1}$. This action is isometric, vertical, and transitive on fibers of $p$.
 
-   By (Theorem \ref{theorem:riemannian-submersion}). Therefore, there is a unique metric on $\C P^n$ such that the map $p:S^{2n+1}\to \C P^n$ is a Riemannian submersion. This metric is called the Fubini-study metric.
+   By Theorem \ref{theorem:riemannian-submersion-chap2}, there is a unique metric on $\C P^n$ such that the map $p:S^{2n+1}\to \C P^n$ is a Riemannian submersion. This metric is called the Fubini--Study metric.
 \end{proof}
 
 \subsection{Observable diameter for complex projective spaces}
 
+The sphere estimate by itself is not yet a statement about quantum state space. The missing geometric bridge is the Hopf fibration, which realizes complex projective space as a quotient of the sphere by phase. Because this quotient map is a Riemannian submersion, the concentration scale on the sphere can be transferred to $\C P^n$.
+
 Using the projection map and Hopf's fibration, we can estimate the observable diameter for complex projective spaces from the observable diameter of spheres.
 
 \begin{theorem}
   \label{thm:observable-diameter-complex-projective-space}
   For any real number $\kappa$ with $0<\kappa<1$, we have
   $$
-  \obdiam(\mathbb{C}P^n(1);-\kappa)\leq O(\sqrt{n})
+  \obdiam(\mathbb{C}P^n(1);-\kappa)\leq O\left(\frac{1}{\sqrt{n}}\right).
   $$
 \end{theorem}
 
 \begin{proof}
-  Recall from Example 2.30 in \cite{lee_introduction_2018}, the Hopf fibration $f_n:S^{2n+1}(1)\to \C P^n$ is 1-Lipschitz continuous with respect to the Fubini-Study metric on $\C P^n$. and the push-forward $(f_n)_*\sigma^{2n+1}$ coincides with the normalized volume measure on $\C P^n$ induced from the Fubini-Study metric.
+  Recall from Example 2.30 in \cite{lee_introduction_2018} that the Hopf fibration $f_n:S^{2n+1}(1)\to \C P^n$ is $1$-Lipschitz with respect to the Fubini-Study metric on $\C P^n$. The push-forward $(f_n)_*\sigma^{2n+1}$ coincides with the normalized volume measure on $\C P^n$ induced from the Fubini-Study metric.
 
-  By proposition \ref{prop:observable-diameter-domination}, we have $\obdiam(\mathbb{C}P^n(1);-\kappa)\leq \obdiam(S^{2n+1}(1);-\kappa)\leq O(\sqrt{n})$.
+  By Proposition \ref{prop:observable-diameter-domination}, we have
+  $$
+  \obdiam(\mathbb{C}P^n(1);-\kappa)\leq \obdiam(S^{2n+1}(1);-\kappa)\leq O\left(\frac{1}{\sqrt{n}}\right).
+  $$
 
 \end{proof}
 
 \section{Use entropy function as estimator of observable diameter for complex projective spaces}
 
+The previous subsection gives an abstract upper bound. The next question is how to test that scale numerically on observables that arise naturally in quantum information. Entropy is the obvious candidate: it is physically meaningful, it is already controlled by Hayden's theorem, and after normalization by a Lipschitz constant it supplies a computable lower-bound proxy for observable diameter.
+
 In this section we describe a Monte Carlo pipeline for comparing concentration phenomena across three metric-measure spaces using real-valued entropy observables. The goal is not to compute the exact observable diameter
 $$
 \operatorname{ObsDiam}_{\mathbb{R}}(X;-\kappa)
@@ -332,6 +343,25 @@ f_{\mathrm{sphere}}(x^{(1)}),\dots,f_{\mathrm{sphere}}(x^{(N)}),
 $$
 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 L\'evy'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
@@ -387,54 +417,46 @@ $$
 $$
 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}
+\begin{figure}[ht]
+    \centering
+    \begin{minipage}{0.48\textwidth}
+        \centering
+        \includegraphics[width=\textwidth]{../results/exp-20260311-154003/hist_cp_16x16.png}
         
-The third family is the symmetric subspace
+        Entropy distribution for $(d_A,d_B)=(16,16)$
-$$
+    \end{minipage}
-\operatorname{Sym}^N(\mathbb{C}^2),
+    \hfill
-$$
+    \begin{minipage}{0.48\textwidth}
-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
+        \centering
-$$
+        \includegraphics[width=\textwidth]{../results/exp-20260311-154003/hist_cp_256x256.png}
-|\psi\rangle
-=
-\sum_{k=0}^{N} c_k |D^N_k\rangle,
-\qquad
-\sum_{k=0}^{N}|c_k|^2 = 1.
-$$
-The projective metric is again the Fubini--Study metric
-$$
-d_{FS}([\psi],[\phi])=\arccos |\langle \psi,\phi\rangle|.
-$$
         
-Sampling is performed by drawing a standard complex Gaussian vector
+        Entropy distribution for $(d_A,d_B)=(256,256)$
+    \end{minipage}
+\end{figure}
+
+\section{A conjecture on observable diameter for complex projective spaces}
+
+The numerical section does not compute $\obdiam(\mathbb{C}P^n(1);-\kappa)$ exactly, but it does produce a natural lower-bound proxy. If $f:\mathbb{C}P^n\to\mathbb{R}$ has Lipschitz constant $L_f>0$, then $L_f^{-1}f$ is $1$-Lipschitz, so
 $$
-(c_0,\dots,c_N)\in \mathbb{C}^{N+1}
+\frac{\diam(f_*\mu;1-\kappa)}{L_f}\leq \obdiam(\mathbb{C}P^n(1);-\kappa).
 $$
-and normalizing it. This gives the unitarily invariant measure on the projective symmetric state space.
+Taking $f$ to be the entropy observable means that the normalized entropy widths from the simulations cannot determine the full observable diameter, but they do give a computable lower bound on its scale. In that sense the entropy function is a probe of projective-space concentration, and the conjecture below asks that the upper bound coming from the Hopf-fibration argument has the same order as the concentration already suggested by entropy.
 
-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
+\begin{theorem}[Wu's conjecture]
+
+    For $0<\kappa<1$,
+    $$
+    \obdiam(\mathbb{C}P^n(1);-\kappa)= O\left(\frac{1}{\sqrt{n}}\right).
+    $$
+
+\end{theorem}
+
+\paragraph{Sketch of the proof.}
+The expected upper bound should come from the same geometric mechanism as on the sphere. The Hopf fibration $S^{2n+1}(1)\to \mathbb{C}P^n(1)$ is $1$-Lipschitz and sends the normalized spherical measure to the normalized Fubini--Study volume measure, so domination already gives
 $$
-f_{\mathrm{Maj}}([\psi])
+\obdiam(\mathbb{C}P^n(1);-\kappa)\leq \obdiam(S^{2n+1}(1);-\kappa).
-=
-S(\rho_1)
-=
--\operatorname{Tr}(\rho_1 \log_2 \rho_1).
 $$
-Since $\rho_1$ is a qubit state, this observable takes values in $[0,1]$.
+Thus the sphere estimate suggests the order $n^{-1/2}$ on projective space as well. To compare this upper bound with entropy data, one studies the entropy observable $f([\psi])=S(\operatorname{Tr}_B|\psi\rangle\langle\psi|)$ on the relevant projective state space, normalizes by a Lipschitz bound for $f$, and uses the resulting widths as lower-bound proxies for observable diameter. The remaining work is to relate these entropy-based lower bounds to intrinsic geometric concentration, possibly through curvature methods such as Fubini--Study geometry and L\'evy--Gromov type inequalities.
-
-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}
-
-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.
-
 
 \ifSubfilesClassLoaded{
   \printbibliography[title={References}]

latex/chapters/chap3.tex

@@ -1,4 +1,4 @@
-% chapters/chap2.tex
+% chapters/chap3.tex
 \documentclass[../main.tex]{subfiles}
 
 \ifSubfilesClassLoaded{
@@ -7,9 +7,9 @@
 
 \begin{document}
 
-\chapter{Seigel-Bargmann Space}
+\chapter{Segal-Bargmann Space}
 
-In this chapter, we will collect ideas and other perspective we have understanding the concentration of measure phenomenon. Especially with symmetric product of $\C P^1$ and see how it relates to Riemman surfaces and Seigel-Bargmann spaces.
+In this chapter, we collect ideas and another perspective on the concentration of measure phenomenon. In particular, we look at symmetric products of $\C P^1$ and how they relate to Riemann surfaces and Segal-Bargmann spaces.
 
 \begin{figure}[h]
     \centering
@@ -41,6 +41,8 @@ One might be interested in the random sampling over the $\operatorname{Sym}_n(\m
 
 \section{Majorana stellar representation of the quantum state}
 
+This branch continues the projective-space viewpoint from the previous chapters, but now through symmetric states and polynomial data. The advantage of this model is that it converts projective quantum states into configurations of points, making it plausible to compare concentration questions with geometry on symmetric products and holomorphic function spaces.
+
 \begin{defn}
     Let $n$ be a positive integer. The Majorana stellar representation of the quantum state is the set of all roots of a polynomial of degree $n$ in $\mathbb{C}$.
 
@@ -51,7 +53,7 @@ One might be interested in the random sampling over the $\operatorname{Sym}_n(\m
 
 
 \ifSubfilesClassLoaded{
-  \printbibliography[title={References for Chapter 2}]
+  \printbibliography[title={References for Chapter 3}]
 }
 
 \end{document}

latex/main.tex

@@ -110,7 +110,7 @@
 \mainmatter
 
 % Each chapter is in its own file and included as a subfile.
-% \subfile{preface}
+\subfile{preface}
 \subfile{chapters/chap0}
 \subfile{chapters/chap1}
 \subfile{chapters/chap2}

latex/preface.tex

@@ -1,4 +1,3 @@
-% preface.tex
 \documentclass[main.tex]{subfiles}
 
 \ifSubfilesClassLoaded{
@@ -10,72 +9,98 @@
 \chapter*{Preface}
 \addcontentsline{toc}{chapter}{Preface}
 
-Non-commutative probability theory is a branch of generalized probability theory that studies the probability of events in non-commutative algebras (e.g. the algebra of observables in quantum mechanics). In the 20th century, non-commutative probability theory has been applied to the study of quantum mechanics as the classical probability theory is not enough to describe quantum mechanics~\cite{kummer1998elements}.
+This thesis studies how concentration of measure enters quantum information through the geometry of pure states. The basic question is probabilistic but the answer is geometric: once the state space is viewed as a high-dimensional metric-measure space, many physically relevant observables become sharply concentrated. In particular, the entropy of a random bipartite pure state is typically close to its maximal value, which explains why high-dimensional random states are generically highly entangled.
 
-Recently, 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,
+The exposition is organized in layers. 
 
-$$
+Chapter 0 collects the algebraic, probabilistic, geometric, and quantum-mechanical background needed later. 
-\mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp\left(-\frac{(n-1)\epsilon^2}{2}\right)
-$$
 
-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.
+Chapter 1 develops the motivation from polarization experiments, recalls concentration on spheres, and then states the Hayden--Leung--Winter entanglement bound. 
 
-Mathematically, that is:
+Chapter 2 reformulates the same phenomenon in the language of observable diameter and uses entropy-based simulations as a concrete probe on spheres and complex projective spaces. 
 
-Let $\psi\in \mathcal{P}(A\otimes B)$ be a random pure state on $A\otimes B$.
+Chapter 3 records an exploratory direction through Majorana stellar representation and related holomorphic models. It branches out from our main narrative but is included here for completeness and future reference as our journey into the geometry of quantum states continues.
-
-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. We assume the reader is an undergraduate student in mathematics and is familiar with the basic concepts of probability theory, measure theory, linear algebra, and some basic skills of mathematical analysis. To make the report more self-contained, we will add detailed annotated proofs that I understand and references for the original sources.
 
 \section*{How to use the dependency graph}
 
-Since our topic integrates almost everything I've learned during undergraduate study, I will try to make some dependency graph for reader and for me to keep track of what are the necessary knowledge to understand part of the report.
+The graph below is meant to be read from top to bottom. The first row lists undergraduate courses that supply the basic language. The middle rows list the sections and subsection-level themes that are used in the body of the thesis. The bottom node records the concentration-of-measure theorem that motivates the main narrative. Not every reader needs every path: a reader interested mainly in the entanglement theorem can follow the linear algebra, probability, quantum, and concentration branches first, while a reader interested in the geometric reformulation can continue through manifolds, Riemannian geometry, Hopf fibration, and observable diameter.
-
-One can imagine the project as a big tree, where the root is in undergrad math and branches out to the topics of the report, including many advanced topics and motivation to study them.
 
 \bigskip
 
-% --- Dependency tree graph (TikZ) ---
 \begin{figure}[ht]
 \centering
 \begin{tikzpicture}[
-  node distance=10mm and 18mm,
+  x=1cm,
-  box/.style={draw, rectangle, fill=white, align=center, inner sep=4pt},
+  y=1cm,
-  arrow/.style={-Latex}
+  scale=0.74,
+  transform shape,
+  font=\small,
+  course/.style={draw, rounded corners, fill=gray!12, align=center, minimum width=2.6cm, minimum height=0.95cm},
+  topic/.style={draw, rounded corners, fill=blue!5, align=center, minimum width=2.9cm, minimum height=0.95cm},
+  advanced/.style={draw, rounded corners, fill=green!6, align=center, minimum width=3.1cm, minimum height=0.95cm},
+  final/.style={draw, rounded corners, fill=red!8, align=center, minimum width=4.0cm, minimum height=1.0cm},
+  edge/.style={-Latex, semithick}
 ]
 
-% \node[box] (lin) {Linear Algebra\\(bases, maps, eigenvalues)};
+\node[course] (lin) at (-8,0) {Linear algebra};
-% \node[box, right=of lin] (real) {Real Analysis\\(limits, continuity, measure-lite)};
+\node[course] (real) at (-4,0) {Real analysis};
-% \node[box, below=of lin] (prob) {Probability\\(expectation, variance, concentration)};
+\node[course] (measure) at (0,0) {Measure theory};
-% \node[box, below=of real] (top) {Topology/Geometry\\(metrics, compactness)};
+\node[course] (prob) at (4,0) {Probability theory};
+\node[course] (top) at (8,0) {Topology};
 
-% \node[box, below=12mm of prob] (func) {Functional Analysis\\($L^p$, Hilbert spaces, operators)};
+\node[topic] (cvec) at (-8,-2.3) {Complex vector\\ spaces};
-% \node[box, below=12mm of top] (quant) {Quantum Formalism\\(states, observables, partial trace)};
+\node[topic] (ncprob) at (-3.6,-2.3) {Non-commutative\\ probability theory};
+\node[topic] (man) at (0.8,-2.3) {Manifolds};
+\node[topic] (quant) at (5.6,-2.3) {Quantum physics\\ and terminologies};
+\node[topic] (mot) at (10.0,-2.3) {Motivation};
 
-% \node[box, below=14mm of func, xshift=25mm] (book) {This Book\\(Chapters 1--n)};
+\node[advanced] (smooth) at (-6.0,-4.8) {Smooth manifolds\\ and Lie groups};
-% % draw arrows behind nodes
+\node[advanced] (riem) at (-1.7,-4.8) {Riemannian\\ manifolds};
-% \begin{scope}[on background layer]
+\node[advanced] (hopf) at (2.6,-4.8) {Hopf fibration};
-%   \draw[arrow] (lin) -- (func);
+\node[advanced] (rand) at (6.9,-4.8) {Random quantum\\ states};
-%   \draw[arrow] (real) -- (func);
+\node[advanced] (conc) at (11.0,-4.8) {Concentration of\\ measure phenomenon};
-%   \draw[arrow] (prob) -- (func);
-%   \draw[arrow] (func) -- (quant);
-%   \draw[arrow] (lin) -- (quant);
-%   \draw[arrow] (top) -- (quant);
 
-%   \draw[arrow] (func) -- (book);
+\node[advanced] (app) at (-4.4,-7.3) {Application in\\ non-commutative probability};
-%   \draw[arrow] (quant) -- (book);
+\node[advanced] (obs) at (0.2,-7.3) {Observable\\ diameters};
-%   \draw[arrow] (prob) -- (book);
+\node[advanced] (entropy) at (4.9,-7.3) {Entropy estimator\\ on $\mathbb{C}P^n$};
-% \end{scope}
+\node[advanced] (majorana) at (9.6,-7.3) {Majorana stellar\\ representation};
 
+\node[final] (final) at (2.6,-9.9) {Concentration of measure};
+
+\draw[edge] (lin) -- (cvec);
+\draw[edge] (lin) -- (quant);
+\draw[edge] (real) -- (ncprob);
+\draw[edge] (measure) -- (ncprob);
+\draw[edge] (measure) -- (obs);
+\draw[edge] (prob) -- (mot);
+\draw[edge] (prob) -- (conc);
+\draw[edge] (top) -- (man);
+
+\draw[edge] (cvec) -- (ncprob);
+\draw[edge] (cvec) to[out=-25,in=165] (rand);
+\draw[edge] (ncprob) -- (app);
+\draw[edge] (man) -- (smooth);
+\draw[edge] (smooth) -- (riem);
+\draw[edge] (riem) -- (hopf);
+\draw[edge] (quant) -- (rand);
+\draw[edge] (quant) -- (mot);
+\draw[edge] (mot) -- (conc);
+
+\draw[edge] (rand) -- (app);
+\draw[edge] (conc) -- (app);
+\draw[edge] (conc) -- (obs);
+\draw[edge] (hopf) -- (obs);
+\draw[edge] (hopf) -- (entropy);
+\draw[edge] (obs) -- (entropy);
+\draw[edge] (rand) -- (entropy);
+\draw[edge] (quant) to[out=-70,in=110] (majorana);
+
+\draw[edge] (app) -- (final);
+\draw[edge] (obs) -- (final);
+\draw[edge] (entropy) -- (final);
+\draw[edge] (conc) to[out=-90,in=30] (final);
 \end{tikzpicture}
-\caption{Dependency tree: prerequisites and how they feed into the main text.}
+\caption{Dependency graph for the thesis. The central path runs from undergraduate background through geometry and quantum theory to concentration of measure, while the Majorana branch records a later exploratory direction.}
 \label{fig:dependency-tree}
 \end{figure}
 

presentation/ZheyuanWu_HonorThesis_Presentation.tex

@@ -551,7 +551,7 @@ Thus entanglement entropy is the von Neumann entropy of a subsystem, and it meas
     \begin{block}{Projective-space estimate from Gromov}
         For $0<\kappa<1$,
         $$
-        \obdiam(\mathbb{C}P^n(1);-\kappa)\leq O(\sqrt{n}).
+        \obdiam(\mathbb{C}P^n(1);-\kappa)\leq O\left(\frac{1}{\sqrt{n}}\right).
         $$
     \end{block}
 
@@ -562,7 +562,7 @@ Thus entanglement entropy is the von Neumann entropy of a subsystem, and it meas
     \end{itemize}
 
 
-    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 $O(\sqrt{n})$.
+    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 $O\left(\frac{1}{\sqrt{n}}\right)$.
 \end{frame}
 
 \begin{frame}{A conjecture}
@@ -570,7 +570,7 @@ Thus entanglement entropy is the von Neumann entropy of a subsystem, and it meas
 
         For $0<\kappa<1$,
         $$
-        \obdiam(\mathbb{C}P^n(1);-\kappa)= O(\sqrt{n}).
+        \obdiam(\mathbb{C}P^n(1);-\kappa)= O\left(\frac{1}{\sqrt{n}}\right).
         $$
     
     \end{block}