Trance-0/HonorThesis
1
0
Fork 0
Code Issues Pull Requests Actions Packages Projects Releases Wiki Activity

bugfix

Browse Source
This commit is contained in:
Zheyuan Wu
2026-03-29 18:08:48 -05:00
parent dd10a1969b
commit 2949c3e5b6
14 changed files with 720 additions and 663 deletions
Download Patch File Download Diff File Expand all files Collapse all files

172
latex/chapters/chap0.tex
View File

@@ -335,19 +335,21 @@ $$
Let $\rho$ be a state on $\mathscr{B}$ consisting of orthonormal basis $\{v_j\}$ and eigenvalue $\{\lambda_j\}$.
The partial trace of $T$ with respect to $\rho$ is the linear operator on $\mathscr{A}$ defined by
$$
\operatorname{Tr}_{\mathscr{A}}(T)=\sum_{j} \lambda_j L^*_{v_j}(T)L_{v_j}
$$
\end{defn}
The partial trace of $T$ with respect to $\rho$ is the linear operator on $\mathscr{A}$ defined by
$$
\operatorname{Tr}_{\mathscr{B},\rho}(T)=\sum_{j} \lambda_j L^*_{v_j}(T)L_{v_j}
$$
\end{defn}
This introduces a new model in mathematics explaining quantum mechanics: the non-commutative probability theory.
\section{Non-commutative probability theory}
The non-commutative probability theory is a branch of generalized probability theory that studies the probability of events in non-commutative algebras.
\section{Non-commutative probability theory}
The constructions above explain why tensor products and traces appear before probability is mentioned again: they are the algebraic devices that let composite quantum systems behave like probabilistic systems with marginals and expectations. The next section packages these operations into the operator-theoretic language of states, observables, and expectation values, which is the setting used later for random quantum states and entropy.
The non-commutative probability theory is a branch of generalized probability theory that studies the probability of events in non-commutative algebras.
There are several main components of the generalized probability theory; let's see how we can formulate them, comparing with the classical probability theory.
@@ -368,14 +370,14 @@ As a side note we will use later, we also defined the Borel measure on a space,
\label{defn:Borel_measure}
Borel measure:
Let $X$ be a topological space, then a Borel measure $\mu:\mathscr{B}(X)\to [0,\infty]$ on $X$ is a measure on the Borel $\sigma$-algebra of $X$ $\mathscr{B}(X)$ satisfying the following properties:
\begin{enumerate}
\item $X \in \mathscr{B}$.
\item Close under complement: If $A\subseteq X$, then $\mu(A^c)=\mu(X)-\mu(A)$
\item Close under countable unions; If $E_1,E_2,\cdots$ are disjoint sets, then $\mu(\bigcup_{i=1}^\infty E_i)=\sum_{i=1}^\infty \mu(E_i)$
\end{enumerate}
\end{defn}
Let $X$ be a topological space, then a Borel measure $\mu:\mathscr{B}(X)\to [0,\infty]$ on $X$ is a measure on the Borel $\sigma$-algebra of $X$ $\mathscr{B}(X)$ satisfying the following properties:
\begin{enumerate}
\item $X \in \mathscr{B}(X)$.
\item Close under complement: If $A\in \mathscr{B}(X)$, then $A^c\in \mathscr{B}(X)$.
\item Close under countable unions: If $E_1,E_2,\cdots$ are disjoint Borel sets, then $\mu(\bigcup_{i=1}^\infty E_i)=\sum_{i=1}^\infty \mu(E_i)$.
\end{enumerate}
\end{defn}
In later sections, we will use Lebesgue measure, and Haar measure for various circumstances, their detailed definition may be introduced in later sections.
@@ -832,23 +834,25 @@ Here is Table~\ref{tab:analog_of_classical_probability_theory_and_non_commutativ
\vspace{0.5cm}
\end{table}
\section{Manifolds}
In this section, we will introduce some basic definitions and theorems used in manifold theory that are relevant to our study. Assuming no prior knowledge of manifold theory but basic topology understanding. We will provide brief definitions and explanations for each term. From the most abstract Manifold definition to the Riemannian manifolds and related theorems.
\section{Manifolds}
Up to this point the emphasis has been algebraic and probabilistic. The concentration results used later, however, live naturally on curved spaces equipped with metrics and measures. For that reason the discussion now shifts from operator theory to manifold theory, starting with topological manifolds and then adding smooth and Riemannian structure until we can describe complex projective space as a genuine geometric state space.
In this section, we will introduce some basic definitions and theorems used in manifold theory that are relevant to our study. Assuming no prior knowledge of manifold theory but basic topology understanding. We will provide brief definitions and explanations for each term. From the most abstract Manifold definition to the Riemannian manifolds and related theorems.
\subsection{Manifolds}
\begin{defn}
\label{defn:m-manifold}
An $m$-manifold is a Topological space $X$ that is
\begin{enumerate}
\item Hausdroff: every distinct two points in $X$ can be separated by two disjoint open sets.
\item Second countable: $X$ has countable basis.
\item Every point $p$ has an open neighborhood $p\in U$ that is homeomorphic to an open subset of $\mathbb{R}^m$.
\end{enumerate}
\end{defn}
An $m$-manifold is a topological space $X$ that is
\begin{enumerate}
\item Hausdorff: every distinct two points in $X$ can be separated by two disjoint open sets.
\item Second countable: $X$ has a countable basis.
\item Every point $p$ has an open neighborhood $p\in U$ that is homeomorphic to an open subset of $\mathbb{R}^m$.
\end{enumerate}
\end{defn}
\begin{examples}
@@ -937,7 +941,7 @@ $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$.
Because $\sum_{i=1}^n \varphi_i(x_1)=1$, therefore there exists $\varphi_i(x_1)=\varphi_i(x_2)>0$.
Therefore $x1,x_2\in \operatorname{supp}(\phi_i)\subseteq U_i$.
@@ -961,9 +965,11 @@ Therefore $F$ is a homeomorphism.
\end{enumerate}
\end{proof}
\subsection{Smooth manifolds and Lie groups}
This section is adopted from \cite{lee_introduction_2012}
\subsection{Smooth manifolds and Lie groups}
This section is adopted from \cite{lee_introduction_2012}
The topological definition of a manifold tells us what the space looks like locally, but not how to differentiate on it. The next step is therefore to add charts with smooth transition maps. Once this smooth structure is available, notions such as differentials, submersions, and group actions can be stated precisely, and these are exactly the tools needed later for the Hopf fibration.
\begin{defn}
\label{defn:partial_derivative}
@@ -987,7 +993,7 @@ This section is adopted from \cite{lee_introduction_2012}
If for any $j\in \{1,\cdots,n\}$, the $j$-th partial derivative of $f$ is continuous at $a$, then $f$ is continuously differentiable at $a$.
If $\forall a\in U$, $\frac{\partial f}{\partial x_j}$ exists and is continuous at $a$, then $f$ is continuously differentiable on $U$. or $C^1$ map. (Note that $C^0$ map is just a continuous map.)
If $\forall a\in U$, $\frac{\partial f}{\partial x_j}$ exists and is continuous at $a$, then $f$ is continuously differentiable on $U$, or a $C^1$ map. (Note that $C^0$ map is just a continuous map.)
\end{defn}
@@ -1029,12 +1035,12 @@ This section is adopted from \cite{lee_introduction_2012}
$$
\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}
\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 $df_p:T_pM\to T_{f(p)}N$ is surjective.
Or equivalently $\operatorname{rank}(df_p)=\dim N$ for each $p\in M$.
\end{defn}
Here are some additional propositions that will be helpful for our study in later sections:
@@ -1046,10 +1052,12 @@ This one is from \cite{lee_introduction_2012} Theorem 4.26
Let $M$ and $N$ be smooth manifolds and $\pi:M\to N$ is a smooth map. Then $\pi$ is a smooth submersion if and only if every point of $M$ is in the image of a smooth local section of $\pi$ (a local section of $\pi$ is a map $\sigma:U\to M$ defined on some open subset $U\subseteq N$ with $\pi\circ \sigma=Id_U$).
\end{theorem}
\subsection{Riemannian manifolds}
\begin{defn}
\label{defn:riemannian-metric}
\subsection{Riemannian manifolds}
Smooth manifolds still do not measure lengths, angles, or volumes. To connect the manifold side of the thesis with concentration of measure, we need a metric structure that turns local smooth data into global geometric data. Riemannian metrics provide exactly that extra layer, and Riemannian submersions will later transfer this geometry from spheres to complex projective spaces.
\begin{defn}
\label{defn:riemannian-metric}
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 $T_pM$.
@@ -1057,9 +1065,9 @@ This one is from \cite{lee_introduction_2012} Theorem 4.26
\end{defn}
\begin{defn}
\label{defn:riemannian-submersion}
Suppose $(\tilde{M},\tilde{g})$ and $(M,g)$ are smooth Riemannian manifolds, and $\pi:\tilde{M}\to M$ is a smooth submersion. Then $\pi$ is said to be a \textit{\textbf{Riemannian submersion}} if for each $x\in \tilde{M}$, the differential $d\pi_x:\tilde{g}_x\to g_{\pi(x)}$ restricts to a linear isometry from $H_x$ onto $T_{\pi(x)}M$.
\begin{defn}
\label{defn:riemannian-submersion}
Suppose $(\tilde{M},\tilde{g})$ and $(M,g)$ are smooth Riemannian manifolds, and $\pi:\tilde{M}\to M$ is a smooth submersion. Then $\pi$ is said to be a \textit{\textbf{Riemannian submersion}} if for each $x\in \tilde{M}$, the differential $d\pi_x:T_x\tilde{M}\to T_{\pi(x)}M$ restricts to a linear isometry from $H_x$ onto $T_{\pi(x)}M$.
In other words, $\tilde{g}_x(v,w)=g_{\pi(x)}(d\pi_x(v),d\pi_x(w))$ whenever $v,w\in H_x$.
\end{defn}
@@ -1190,9 +1198,11 @@ This one is from \cite{lee_introduction_2012} Theorem 4.26
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:
\subsection{Hopf fibration}
The previous subsection gives the abstract mechanism for pushing a metric through a quotient map. The Hopf fibration is the concrete instance needed in this thesis: it explains why the geometry of the sphere descends to complex projective space, and therefore why concentration on spheres is relevant to the geometry of pure quantum states.
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
@@ -1235,16 +1245,18 @@ $$
$$
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.
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}
The geometric discussion above identifies the right state space, but the thesis ultimately studies physical observables on that space. We now return to quantum terminology and translate the geometric objects into the language of states, measurements, Haar sampling, and reduced density matrices. This is the point where the manifold picture and the operator picture meet.
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.
One might ask, what is the fundamental difference between a quantum system and a classical system, and why can we not directly apply those theorems in classical computers to a quantum computer? It turns out that quantum error-correcting codes are hard due to the following definitions and features for quantum computing.
@@ -1252,12 +1264,12 @@ One might ask, what is the fundamental difference between a quantum system and a
All quantum operations can be constructed by composing four kinds of transformations: (adapted from Chapter 10 of \cite{Bengtsson_Zyczkowski_2017})
\begin{enumerate}
\item Unitary operations. $U(\cdot)$ for any quantum state. It is possible to apply a non-unitary operation for an open quantum system, but that is usually not the focus for quantum computing and usually leads to non-recoverable loss of information that we wish to obtain.
\item Extend the system. Given a quantum state $\rho\in\mathcal{H}^N$, we can extend it to a larger quantum system by "entangle" (For this report, you don't need to worry for how quantum entanglement works) it with some new states $\sigma\in \mathcal{H}^K$ (The space where the new state dwells is usually called ancilla system) and get $\rho'=\rho\otimes\sigma\in \mathcal{H}^N\otimes \mathcal{K}$.
\item Partial trace. Given a quantum state $\rho\in\mathcal{H}^N$ and some reference state $\sigma\in\mathcal {H}^K$, we can trace out some subsystems and get a new state $\rho'\in\mathcal{H}^{N-K}$.
\item Selective measurement. Given a quantum state, we measure it and get a classical bit; unlike the classical case, the measurement is a probabilistic operation. (More specifically, this is some projection to a reference state corresponding to a classical bit output. For this report, you don't need to worry about how such a result is obtained and how the reference state is constructed.)
\end{enumerate}
\end{defn}
\item Unitary operations. $U(\cdot)$ for any quantum state. It is possible to apply a non-unitary operation for an open quantum system, but that is usually not the focus for quantum computing and usually leads to non-recoverable loss of information that we wish to obtain.
\item Extend the system. Given a quantum state $\rho\in\mathcal{H}^N$, we can extend it to a larger quantum system by "entangle" (For this report, you don't need to worry for how quantum entanglement works) it with some new states $\sigma\in \mathcal{H}^K$ (The space where the new state dwells is usually called ancilla system) and get $\rho'=\rho\otimes\sigma\in \mathcal{H}^N\otimes \mathcal{K}$.
\item Partial trace. Given a quantum state $\rho\in\mathcal{H}^N$ and some reference state $\sigma\in\mathcal {H}^K$, we can trace out some subsystems and get a reduced state on the remaining subsystem.
\item Selective measurement. Given a quantum state, we measure it and get a classical bit; unlike the classical case, the measurement is a probabilistic operation. (More specifically, this is some projection to a reference state corresponding to a classical bit output. For this report, you don't need to worry about how such a result is obtained and how the reference state is constructed.)
\end{enumerate}
\end{defn}
$U(n)$ is the group of all $n\times n$ \textbf{unitary matrices} over $\mathbb{C}$,
@@ -1271,14 +1283,14 @@ The uniqueness of such measurement came from the lemma below~\cite{Elizabeth_boo
\begin{lemma}
\label{lemma:haar_measure}
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}
Let $(U(n), \| \cdot \|, \mu)$ be a metric measure space where $\| \cdot \|$ is the Hilbert-Schmidt norm and $\mu$ is a Borel probability measure.
The Haar measure on $U(n)$ is the unique probability measure that is invariant under the action of $U(n)$ on itself.
That is, for every Borel set $E\subseteq U(n)$ and every $A\in U(n)$, $\mu(AE)=\mu(EA)=\mu(E)$.
The Haar measure is the unique probability measure that is invariant under the action of $U(n)$ on itself.
\end{lemma}
A finite-dimensional quantum system is modeled by a complex Hilbert space (a complete inner product space)
@@ -1348,9 +1360,11 @@ $$
Intuitively, if the two states are not orthogonal, then for any measurement (projection) there exists non-zero probability of getting the same outcome for both states.
\subsection{Random quantum states}
First, we need to define what is a random state in a bipartite system.
\subsection{Random quantum states}
The preceding material identifies the spaces and symmetries of quantum states. The next step is probabilistic: once Haar invariance is available, we can speak about random pure states and random mixed states in a way that matches the geometric viewpoint developed earlier. These definitions are the starting point for the concentration statements proved in the next chapter.
First, we need to define what is a random state in a bipartite system.
@@ -1363,14 +1377,14 @@ First, we need to define what is a random state in a bipartite system.
\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.
It is trivial that for the space of pure state, we can easily apply the Haar measure as the unitarily invariant probability measure by sampling unit vectors on $S^n$ for some $n$. However, for the case of mixed states, that is a bit complicated and we need to use partial tracing to define 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}
For a system $A$ and an integer $s\geq 1$, consider the distribution on the mixed states $\mathcal{S}(A)$ of $A$ induced by the partial trace over the second factor from 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 state, denoted by $\rho\in_R \mathcal{S}_s(A)$. And $\mathcal{P}(A)=\mathcal{S}_1(A)$.
\end{defn}
% When compiled standalone, print this chapter's references at the end.

BIN
latex/chapters/chap1.pdf
View File

Binary file not shown.

52
latex/chapters/chap1.tex
View File

@@ -121,6 +121,8 @@ $$
Other revised experiments (e.g., Aspect's experiment, calcium entangled photon experiment) are also conducted and the inequality is still violated.
\subsection{The true model of light polarization}
The contradiction above marks the point where classical probability stops being adequate. To continue, the sample-space picture must be replaced by states in a Hilbert space and by projections representing measurements. This operator model keeps the experimental probabilities but no longer forces incompatible measurements into a single classical joint distribution.
The full description of the light polarization is given below:
@@ -160,8 +162,8 @@ The probability that a photon passes the first filter $P_{\alpha_i}$ is given by
$$
\operatorname{Prob}(P_i=1)
=\operatorname{tr}(\rho P_{\alpha_i})
=\frac{1}{2} \operatorname{tr}(P_{\alpha_i})
=\operatorname{Tr}(\rho 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] (rng) [right of=sa] {$[0,\infty)\subset \mathbb{R}$};
\node[main] (sa) [below of=pa] {$\mathcal{S}(A)$};
\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

BIN
latex/chapters/chap2 copy.pdf Normal file
View File

Binary file not shown.

BIN
latex/chapters/chap2.pdf
View File

Binary file not shown.

924
latex/chapters/chap2.tex
View File

@@ -1,459 +1,465 @@
% chapters/chap2.tex
\documentclass[../main.tex]{subfiles}
\ifSubfilesClassLoaded{
\addbibresource{../main.bib}
}
\begin{document}
\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 $D$.
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.
Altogether, we experimenting 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.
We will try to use the results from previous sections to estimate the observable diameter for complex projective spaces.
\section{Observable diameters}
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}
\label{defn:mm-space}
Let $X$ be a topological space with the following:
\begin{enumerate}
\item $X$ is a complete (every Cauchy sequence converges)
\item $X$ is a metric space with metric $d_X$
\item $X$ has a Borel probability measure $\mu_X$
\end{enumerate}
Then $(X,d_X,\mu_X)$ is a \textbf{metric measure space}.
\end{defn}
\begin{defn}
\label{defn:diameter}
Let $(X,d_X)$ be a metric space. The \textbf{diameter} of a set $A\subset X$ is defined as
$$
\diam(A)=\sup_{x,y\in A}d_X(x,y).
$$
\end{defn}
\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
$$
\diam(A;\alpha)=\inf_{A\subseteq X|\mu_X(A)\geq \alpha}\diam(A).
$$
\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.
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.
\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 $Y$.
For any real number $\kappa>0$, the \textbf{$\kappa$-observable diameter with screen $Y$} is defined as
$$
\obdiam_Y(X;\kappa)=\sup\{\diam(f_*\mu_X;1-\kappa)\}
$$
And the \textbf{obbservable diameter with screen $Y$} is defined as
$$
\obdiam_Y(X)=\inf_{\kappa>0}\max\{\obdiam_Y(X;\kappa)\}
$$
If $Y=\R$, we call it the \textbf{observable diameter}.
\end{defn}
If we collapse it naively via
$$
\inf_{\kappa>0}\obdiam_Y(X;\kappa),
$$
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.
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 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.
Few additional proposition in \cite{shioya2014metricmeasuregeometry} will help us to 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:
\begin{enumerate}
\item
$
\diam(X,1-\kappa)\leq \diam(Y,1-\kappa)
$
\item $\obdiam(X;-\kappa)\leq \diam(X;1-\kappa)$, and $\obdiam(X)$ is finite.
\item
$
\obdiam(X;-\kappa)\leq \obdiam(Y;-\kappa)
$
\end{enumerate}
\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 $\diam(X;1-\kappa)\leq \diam(A)\leq \diam(Y;1-\kappa)$.
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
$$
\diam(g_*\mu_X;1-\kappa)\leq \diam((f\circ g)_*\mu_Y;1-\kappa)\leq \obdiam(Y;-\kappa)
$$
\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
$$
\obdiam(tX;-\kappa)=t\obdiam(X;-\kappa)
$$
Where $tX=(X,tdX,\mu X)$.
\end{prop}
\begin{proof}
$$
\begin{aligned}
\obdiam(tX;-\kappa)&=\sup\{\diam(f_*\mu_X;1-\kappa)|f:tX\to \R \text{ is 1-Lipschitz}\}\\
&=\sup\{\diam(f_*\mu_X;1-\kappa)|t^{-1}f:X\to \R \text{ is 1-Lipschitz}\}\\
&=\sup\{\diam((tg)_*\mu_X;1-\kappa)|g:X\to \R \text{ is 1-Lipschitz}\}\\
&=t\sup\{\diam(g_*\mu_X;1-\kappa)|g:X\to \R \text{ is 1-Lipschitz}\}\\
&=t\obdiam(X;-\kappa)
\end{aligned}
$$
\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.
\begin{theorem}
\label{thm:observable-diameter-sphere}
For any real number $\kappa$ with $0<\kappa<1$, we have
$$
\obdiam(S^n(1);-\kappa)=O(\sqrt{n})
$$
\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
$$
\begin{aligned}
\lim_{n\to \infty} \obdiam(S^n(\sqrt{n});-\kappa)&=\lim_{n\to \infty} \sup\{\diam((\pi_{n,k})_*\sigma^n;1-\kappa)|\pi_{n,k} \text{ is 1-Lipschitz}\}\\
&=\lim_{n\to \infty} \sup\{\diam(\gamma^1;1-\kappa)|\gamma^1 \text{ is the standard gaussian measure}\}\\
&=\diam(\gamma^1;1-\kappa)\\
&=2I^{-1}(\frac{1-\kappa}{2})\text { cutting the extremum for normal distribution}\\
\end{aligned}
$$
By proposition \ref{prop:observable-diameter-scale}, we have
$$
\obdiam(S^n(\sqrt{n});-\kappa)=\sqrt{n}\obdiam(S^n(1);-\kappa)
$$
So $\obdiam(S^n(1);-\kappa)=\sqrt{n}(2I^{-1}(\frac{1-\kappa}{2}))=O(\sqrt{n})$.
\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.
To find such metric, we need some additional results.
\begin{defn}
\label{defn:riemannian-metric}
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 $v=0$.
\end{defn}
% TODO: There is a hidden chapter on group action on manifolds, can you find that?
\begin{theorem}
\label{theorem:riemannian-submersion}
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}
\item isometric: the map $x\mapsto \varphi\cdot x$ is an isometry for each $\varphi\in G$.
\item vertical: every element $\varphi\in G$ takes each fiber to itself, that is $\pi(\varphi\cdot p)=\pi(p)$ for all $p\in \tilde{M}$.
\item transitive on fibers: for each $p,q\in \tilde{M}$ such that $\pi(p)=\pi(q)$, there exists $\varphi\in G$ such that $\varphi\cdot p = q$.
\end{enumerate}
Then there is a unique Riemannian metric on $M$ such that $\pi$ is a Riemannian submersion.
\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
\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.
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.
\end{defn}
There are many additional properties for such construction, we will check them just for curiosity.
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:
$$
v(z)=\frac{z}{|z|}
$$
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.
Define an action of $S^1$ on $S^{2n+1}$ by complex multiplication:
$$
\lambda (z^1,z^2,\ldots,z^{n+1})=(\lambda z^1,\lambda z^2,\ldots,\lambda z^{n+1})
$$
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 $p$.
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.
\end{proof}
\subsection{Observable diameter for complex projective spaces}
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
$$
\obdiam(\mathbb{C}P^n(1);-\kappa)\leq O(\sqrt{n})
$$
\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.
By proposition \ref{prop:observable-diameter-domination}, we have $\obdiam(\mathbb{C}P^n(1);-\kappa)\leq \obdiam(S^{2n+1}(1);-\kappa)\leq O(\sqrt{n})$.
\end{proof}
\section{Use entropy function as estimator of observable diameter for complex projective spaces}
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
$$
\operatorname{ObsDiam}_{\mathbb{R}}(X;-\kappa)
=
\sup_{f \in \operatorname{Lip}_1(X,\mathbb{R})}
\operatorname{diam}(f_*\mu_X;1-\kappa),
$$
but to estimate it by choosing a specific observable $f:X\to \mathbb{R}$ and then measuring the partial diameter of its push-forward distribution. Thus all numerical quantities below should be interpreted as \emph{entropy-based observable-diameter proxies}, not exact observable diameters in Gromov's sense \cite{MGomolovs,shioya2014metricmeasuregeometry}.
The screen is $\mathbb{R}$ equipped with the Euclidean metric, and for a fixed $\kappa \in (0,1)$ we set
$$
\alpha = 1-\kappa.
$$
Given sampled values $y_1,\dots,y_N \in \mathbb{R}$ of the observable, the code sorts them and computes the shortest interval $[a,b]$ containing at least $\lceil \alpha N \rceil$ samples. Its width
$$
b-a
$$
is the empirical partial diameter of the push-forward measure on $\mathbb{R}$.
To compare this width with the true observable diameter, the code also estimates an empirical Lipschitz constant of the chosen observable. If $x_i,x_j \in X$ are sampled states and $f(x_i),f(x_j)$ are the corresponding observable values, then the sampled slopes are
$$
\frac{|f(x_i)-f(x_j)|}{d_X(x_i,x_j)},
$$
where $d_X$ is the metric of the ambient space. The code records both the maximum sampled slope and the $0.99$-quantile of these slopes. Dividing the empirical partial diameter by these sampled Lipschitz constants gives two normalized proxies:
$$
\frac{\operatorname{diam}(f_*\mu_X;1-\kappa)}{L_{\max}}
\qquad \text{and} \qquad
\frac{\operatorname{diam}(f_*\mu_X;1-\kappa)}{L_{0.99}}.
$$
If the chosen observable were exactly $1$-Lipschitz, these normalized quantities would coincide with the raw width. In practice they should be viewed only as heuristic lower-scale corrections.
\subsection{Random sampling using standard uniform measure on the unit sphere}
The first family of spaces is the real unit sphere
$$
S^{m-1}
=
\left\{
x=(x_1,\dots,x_m)\in \mathbb{R}^m : \|x\|_2=1
\right\},
$$
equipped with the geodesic distance
$$
d_{S}(x,y)=\arccos \langle x,y\rangle
$$
and the normalized Riemannian volume measure. This is the standard metric-measure structure used in concentration of measure on spheres \cite{lee_introduction_2018,romanvershyni,shioya2014metricmeasuregeometry}.
Sampling is performed by drawing a standard Gaussian vector $g\in \mathbb{R}^m$ and normalizing:
$$
x=\frac{g}{\|g\|_2}.
$$
This produces the uniform distribution on $S^{m-1}$.
The observable is a Shannon entropy built from the squared coordinates:
$$
f_{\mathrm{sphere}}(x)
=
-\sum_{i=1}^m x_i^2 \log_2(x_i^2).
$$
Since $(x_1^2,\dots,x_m^2)$ is a probability vector, $f_{\mathrm{sphere}}$ takes values in $[0,\log_2 m]$, and the code records $\log_2 m$ as the natural upper bound of the observable.
For each chosen dimension $m$, the experiment generates $N$ independent samples $x^{(1)},\dots,x^{(N)}$, computes the values
$$
f_{\mathrm{sphere}}(x^{(1)}),\dots,f_{\mathrm{sphere}}(x^{(N)}),
$$
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
$$
\mathbb{C}P^{d_A d_B-1},
$$
viewed as the space of pure states in $\mathbb{C}^{d_A}\otimes \mathbb{C}^{d_B}$ modulo global phase. Geometrically, this space is equipped with the Fubini--Study metric and its associated normalized volume measure \cite{lee_introduction_2018,Bengtsson_Zyczkowski_2017}. Numerically, a projective point is represented by a unit vector
$$
\psi \in \mathbb{C}^{d_A d_B},
\qquad
\|\psi\|=1,
$$
and distances are computed by the Fubini--Study formula
$$
d_{FS}([\psi],[\phi])
=
\arccos |\langle \psi,\phi\rangle|.
$$
Sampling is implemented by drawing a complex Gaussian matrix
$$
G \in \mathbb{C}^{d_A \times d_B},
$$
with independent standard complex normal entries, and then normalizing it so that
$$
\psi = \frac{\operatorname{vec}(G)}{\|\operatorname{vec}(G)\|}.
$$
This is equivalent to Haar sampling on the unit sphere in $\mathbb{C}^{d_A d_B}$ and hence induces the standard unitarily invariant measure on $\mathbb{C}P^{d_A d_B-1}$ \cite{Bengtsson_Zyczkowski_2017,Nielsen_Chuang_2010}.
The real-valued observable is the bipartite entanglement entropy. Writing
$$
\rho_A = \operatorname{Tr}_B |\psi\rangle\langle \psi|,
$$
the code defines
$$
f_{\mathrm{CP}}([\psi])
=
S(\rho_A)
=
-\operatorname{Tr}(\rho_A \log_2 \rho_A).
$$
Equivalently, if $\lambda_1,\dots,\lambda_{d_A}$ are the eigenvalues of $\rho_A$, then
$$
f_{\mathrm{CP}}([\psi])
=
-\sum_{i=1}^{d_A}\lambda_i \log_2 \lambda_i.
$$
This observable takes values in $[0,\log_2 d_A]$.
For each dimension pair $(d_A,d_B)$, the experiment samples $N$ independent Haar-random pure states, computes the entropy values, and then forms the empirical push-forward distribution on $\mathbb{R}$. The shortest interval containing mass at least $1-\kappa$ is reported as the entropy-based observable-diameter proxy. In addition, the code plots histograms, upper-tail deficit plots for
$$
\log_2 d_A - S(\rho_A),
$$
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}.
\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 $0<\kappa<1$,
$$
\obdiam(\mathbb{C}P^n(1);-\kappa)= O(\sqrt{n}).
$$
\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}
\ifSubfilesClassLoaded{
\printbibliography[title={References}]
}
\end{document}
% chapters/chap2.tex
\documentclass[../main.tex]{subfiles}
\ifSubfilesClassLoaded{
\addbibresource{../main.bib}
}
\begin{document}
\chapter{Levy's family and observable diameters}
\begin{abstract}
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 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 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 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}
\label{defn:mm-space}
Let $X$ be a topological space with the following:
\begin{enumerate}
\item $X$ is complete.
\item $X$ is a metric space with metric $d_X$.
\item $X$ has a Borel probability measure $\mu_X$.
\end{enumerate}
Then $(X,d_X,\mu_X)$ is a \textbf{metric measure space}.
\end{defn}
\begin{defn}
\label{defn:diameter}
Let $(X,d_X)$ be a metric space. The \textbf{diameter} of a set $A\subset X$ is defined as
$$
\diam(A)=\sup_{x,y\in A}d_X(x,y).
$$
\end{defn}
\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
$$
\diam(X;\alpha)=\inf_{A\subseteq X,\ \mu_X(A)\geq \alpha}\diam(A).
$$
\end{defn}
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 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 $Y$.
For any real number $\kappa>0$, the \textbf{$\kappa$-observable diameter with screen $Y$} is defined as
$$
\obdiam_Y(X;-\kappa)=\sup\{\diam(f_*\mu_X;1-\kappa)\mid f:X\to Y \text{ is 1-Lipschitz}\}.
$$
And the \textbf{observable diameter with screen $Y$} is defined as
$$
\obdiam_Y(X)=\inf_{\kappa>0}\max\{\obdiam_Y(X;-\kappa),\kappa\}.
$$
If $Y=\R$, we call it the \textbf{observable diameter}.
\end{defn}
If we collapse it naively via
$$
\inf_{\kappa>0}\obdiam_Y(X;-\kappa),
$$
we typically get something degenerate: as $\kappa\to 1$, the condition ``mass $\ge 1-\kappa$'' becomes almost empty, 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 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.
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:
\begin{enumerate}
\item
$
\diam(X;1-\kappa)\leq \diam(Y;1-\kappa)
$
\item $\obdiam(X;-\kappa)\leq \diam(X;1-\kappa)$, and $\obdiam(X)$ is finite.
\item
$
\obdiam(X;-\kappa)\leq \obdiam(Y;-\kappa)
$
\end{enumerate}
\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 $\diam(X;1-\kappa)\leq \diam(A)\leq \diam(Y;1-\kappa)$.
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
$$
\diam(g_*\mu_X;1-\kappa)\leq \diam((g\circ f)_*\mu_Y;1-\kappa)\leq \obdiam(Y;-\kappa).
$$
\end{proof}
\begin{prop}
\label{prop:observable-diameter-scale}
Let $X$ be a metric-measure space. Then for any real number $t>0$, we have
$$
\obdiam(tX;-\kappa)=t\obdiam(X;-\kappa)
$$
where $tX=(X,td_X,\mu_X)$.
\end{prop}
\begin{proof}
$$
\begin{aligned}
\obdiam(tX;-\kappa)&=\sup\{\diam(f_*\mu_X;1-\kappa)\mid f:tX\to \R \text{ is 1-Lipschitz}\}\\
&=\sup\{\diam(f_*\mu_X;1-\kappa)\mid t^{-1}f:X\to \R \text{ is 1-Lipschitz}\}\\
&=\sup\{\diam((tg)_*\mu_X;1-\kappa)\mid g:X\to \R \text{ is 1-Lipschitz}\}\\
&=t\sup\{\diam(g_*\mu_X;1-\kappa)\mid g:X\to \R \text{ is 1-Lipschitz}\}\\
&=t\obdiam(X;-\kappa).
\end{aligned}
$$
\end{proof}
\subsection{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
$$
\obdiam(S^n(1);-\kappa)=O\left(\frac{1}{\sqrt{n}}\right).
$$
\end{theorem}
\begin{proof}
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
$$
\begin{aligned}
\lim_{n\to \infty} \obdiam(S^n(\sqrt{n});-\kappa)&=\lim_{n\to \infty} \sup\{\diam((\pi_{n,k})_*\sigma^n;1-\kappa)\mid \pi_{n,k} \text{ is 1-Lipschitz}\}\\
&=\lim_{n\to \infty} \sup\{\diam(\gamma^1;1-\kappa)\mid \gamma^1 \text{ is the standard Gaussian measure}\}\\
&=\diam(\gamma^1;1-\kappa)\\
&=2I^{-1}\left(\frac{1-\kappa}{2}\right).
\end{aligned}
$$
By Proposition \ref{prop:observable-diameter-scale}, we have
$$
\obdiam(S^n(\sqrt{n});-\kappa)=\sqrt{n}\obdiam(S^n(1);-\kappa).
$$
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 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 a metric, we need some additional results.
\begin{defn}
\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 $v=0$.
\end{defn}
\begin{theorem}
\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}
\item isometric: the map $x\mapsto \varphi\cdot x$ is an isometry for each $\varphi\in G$.
\item vertical: every element $\varphi\in G$ takes each fiber to itself, that is $\pi(\varphi\cdot p)=\pi(p)$ for all $p\in \tilde{M}$.
\item transitive on fibers: for each $p,q\in \tilde{M}$ such that $\pi(p)=\pi(q)$, there exists $\varphi\in G$ such that $\varphi\cdot p = q$.
\end{enumerate}
Then there is a unique Riemannian metric on $M$ such that $\pi$ is a Riemannian submersion.
\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[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 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 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 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 onto the sphere:
$$
v(z)=\frac{z}{|z|}.
$$
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 $p$. 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:
$$
\lambda (z^1,z^2,\ldots,z^{n+1})=(\lambda z^1,\lambda z^2,\ldots,\lambda z^{n+1})
$$
for $\lambda\in S^1$ (viewed as 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-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
$$
\obdiam(\mathbb{C}P^n(1);-\kappa)\leq O\left(\frac{1}{\sqrt{n}}\right).
$$
\end{theorem}
\begin{proof}
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
$$
\obdiam(\mathbb{C}P^n(1);-\kappa)\leq \obdiam(S^{2n+1}(1);-\kappa)\leq O\left(\frac{1}{\sqrt{n}}\right).
$$
\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
$$
\operatorname{ObsDiam}_{\mathbb{R}}(X;-\kappa)
=
\sup_{f \in \operatorname{Lip}_1(X,\mathbb{R})}
\operatorname{diam}(f_*\mu_X;1-\kappa),
$$
but to estimate it by choosing a specific observable $f:X\to \mathbb{R}$ and then measuring the partial diameter of its push-forward distribution. Thus all numerical quantities below should be interpreted as \emph{entropy-based observable-diameter proxies}, not exact observable diameters in Gromov's sense \cite{MGomolovs,shioya2014metricmeasuregeometry}.
The screen is $\mathbb{R}$ equipped with the Euclidean metric, and for a fixed $\kappa \in (0,1)$ we set
$$
\alpha = 1-\kappa.
$$
Given sampled values $y_1,\dots,y_N \in \mathbb{R}$ of the observable, the code sorts them and computes the shortest interval $[a,b]$ containing at least $\lceil \alpha N \rceil$ samples. Its width
$$
b-a
$$
is the empirical partial diameter of the push-forward measure on $\mathbb{R}$.
To compare this width with the true observable diameter, the code also estimates an empirical Lipschitz constant of the chosen observable. If $x_i,x_j \in X$ are sampled states and $f(x_i),f(x_j)$ are the corresponding observable values, then the sampled slopes are
$$
\frac{|f(x_i)-f(x_j)|}{d_X(x_i,x_j)},
$$
where $d_X$ is the metric of the ambient space. The code records both the maximum sampled slope and the $0.99$-quantile of these slopes. Dividing the empirical partial diameter by these sampled Lipschitz constants gives two normalized proxies:
$$
\frac{\operatorname{diam}(f_*\mu_X;1-\kappa)}{L_{\max}}
\qquad \text{and} \qquad
\frac{\operatorname{diam}(f_*\mu_X;1-\kappa)}{L_{0.99}}.
$$
If the chosen observable were exactly $1$-Lipschitz, these normalized quantities would coincide with the raw width. In practice they should be viewed only as heuristic lower-scale corrections.
\subsection{Random sampling using standard uniform measure on the unit sphere}
The first family of spaces is the real unit sphere
$$
S^{m-1}
=
\left\{
x=(x_1,\dots,x_m)\in \mathbb{R}^m : \|x\|_2=1
\right\},
$$
equipped with the geodesic distance
$$
d_{S}(x,y)=\arccos \langle x,y\rangle
$$
and the normalized Riemannian volume measure. This is the standard metric-measure structure used in concentration of measure on spheres \cite{lee_introduction_2018,romanvershyni,shioya2014metricmeasuregeometry}.
Sampling is performed by drawing a standard Gaussian vector $g\in \mathbb{R}^m$ and normalizing:
$$
x=\frac{g}{\|g\|_2}.
$$
This produces the uniform distribution on $S^{m-1}$.
The observable is a Shannon entropy built from the squared coordinates:
$$
f_{\mathrm{sphere}}(x)
=
-\sum_{i=1}^m x_i^2 \log_2(x_i^2).
$$
Since $(x_1^2,\dots,x_m^2)$ is a probability vector, $f_{\mathrm{sphere}}$ takes values in $[0,\log_2 m]$, and the code records $\log_2 m$ as the natural upper bound of the observable.
For each chosen dimension $m$, the experiment generates $N$ independent samples $x^{(1)},\dots,x^{(N)}$, computes the values
$$
f_{\mathrm{sphere}}(x^{(1)}),\dots,f_{\mathrm{sphere}}(x^{(N)}),
$$
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
$$
\mathbb{C}P^{d_A d_B-1},
$$
viewed as the space of pure states in $\mathbb{C}^{d_A}\otimes \mathbb{C}^{d_B}$ modulo global phase. Geometrically, this space is equipped with the Fubini--Study metric and its associated normalized volume measure \cite{lee_introduction_2018,Bengtsson_Zyczkowski_2017}. Numerically, a projective point is represented by a unit vector
$$
\psi \in \mathbb{C}^{d_A d_B},
\qquad
\|\psi\|=1,
$$
and distances are computed by the Fubini--Study formula
$$
d_{FS}([\psi],[\phi])
=
\arccos |\langle \psi,\phi\rangle|.
$$
Sampling is implemented by drawing a complex Gaussian matrix
$$
G \in \mathbb{C}^{d_A \times d_B},
$$
with independent standard complex normal entries, and then normalizing it so that
$$
\psi = \frac{\operatorname{vec}(G)}{\|\operatorname{vec}(G)\|}.
$$
This is equivalent to Haar sampling on the unit sphere in $\mathbb{C}^{d_A d_B}$ and hence induces the standard unitarily invariant measure on $\mathbb{C}P^{d_A d_B-1}$ \cite{Bengtsson_Zyczkowski_2017,Nielsen_Chuang_2010}.
The real-valued observable is the bipartite entanglement entropy. Writing
$$
\rho_A = \operatorname{Tr}_B |\psi\rangle\langle \psi|,
$$
the code defines
$$
f_{\mathrm{CP}}([\psi])
=
S(\rho_A)
=
-\operatorname{Tr}(\rho_A \log_2 \rho_A).
$$
Equivalently, if $\lambda_1,\dots,\lambda_{d_A}$ are the eigenvalues of $\rho_A$, then
$$
f_{\mathrm{CP}}([\psi])
=
-\sum_{i=1}^{d_A}\lambda_i \log_2 \lambda_i.
$$
This observable takes values in $[0,\log_2 d_A]$.
For each dimension pair $(d_A,d_B)$, the experiment samples $N$ independent Haar-random pure states, computes the entropy values, and then forms the empirical push-forward distribution on $\mathbb{R}$. The shortest interval containing mass at least $1-\kappa$ is reported as the entropy-based observable-diameter proxy. In addition, the code plots histograms, upper-tail deficit plots for
$$
\log_2 d_A - S(\rho_A),
$$
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}.
\begin{figure}[ht]
\centering
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{../results/exp-20260311-154003/hist_cp_16x16.png}
Entropy distribution for $(d_A,d_B)=(16,16)$
\end{minipage}
\hfill
\begin{minipage}{0.48\textwidth}
\centering
\includegraphics[width=\textwidth]{../results/exp-20260311-154003/hist_cp_256x256.png}
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
$$
\frac{\diam(f_*\mu;1-\kappa)}{L_f}\leq \obdiam(\mathbb{C}P^n(1);-\kappa).
$$
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.
\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
$$
\obdiam(\mathbb{C}P^n(1);-\kappa)\leq \obdiam(S^{2n+1}(1);-\kappa).
$$
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.
\ifSubfilesClassLoaded{
\printbibliography[title={References}]
}
\end{document}

BIN
latex/chapters/chap3.pdf
View File

Binary file not shown.

30
latex/chapters/chap3.tex
View File

@@ -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}
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.
\chapter{Segal-Bargmann Space}
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
@@ -37,21 +37,23 @@ Basically, there is a bijection between the complex projective space $\mathbb{C}
We can use a symmetric group of permutations of $n$ complex numbers (or $S^2$) to represent the $\mathbb{C}P^n$, that is, $\mathbb{C}P^n=S^2\times S^2\times \cdots \times S^2/S_n$.
One might be interested in the random sampling over the $\operatorname{Sym}_n(\mathbb{C}P^1)$ and the concentration of measure phenomenon on that.
\section{Majorana stellar representation of the quantum state}
\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 $\mathbb{C}$.
One might be interested in the random sampling over the $\operatorname{Sym}_n(\mathbb{C}P^1)$ and the concentration of measure phenomenon on that.
\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 $\mathbb{C}$.
\end{defn}
\section{Space of complex valued functions and pure states}
\ifSubfilesClassLoaded{
\printbibliography[title={References for Chapter 2}]
}
\ifSubfilesClassLoaded{
\printbibliography[title={References for Chapter 3}]
}
\end{document}

BIN
latex/main.pdf
View File

Binary file not shown.

2
latex/main.tex
View File

@@ -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}

BIN
latex/preface.pdf
View File

Binary file not shown.

197
latex/preface.tex
View File

@@ -1,86 +1,111 @@
% preface.tex
\documentclass[main.tex]{subfiles}
\ifSubfilesClassLoaded{
\addbibresource{main.bib}
}
\begin{document}
\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}.
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,
$$
\mu\{x\in S^n: |f(x)-a_0|\geq\epsilon\} < \kappa_n(\epsilon)\leq 2\exp\left(-\frac{(n-1)\epsilon^2}{2}\right)
$$
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. 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.
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,
box/.style={draw, rectangle, fill=white, align=center, inner sep=4pt},
arrow/.style={-Latex}
]
% \node[box] (lin) {Linear Algebra\\(bases, maps, eigenvalues)};
% \node[box, right=of lin] (real) {Real Analysis\\(limits, continuity, measure-lite)};
% \node[box, below=of lin] (prob) {Probability\\(expectation, variance, concentration)};
% \node[box, below=of real] (top) {Topology/Geometry\\(metrics, compactness)};
% \node[box, below=12mm of prob] (func) {Functional Analysis\\($L^p$, Hilbert spaces, operators)};
% \node[box, below=12mm of top] (quant) {Quantum Formalism\\(states, observables, partial trace)};
% \node[box, below=14mm of func, xshift=25mm] (book) {This Book\\(Chapters 1--n)};
% % draw arrows behind nodes
% \begin{scope}[on background layer]
% \draw[arrow] (lin) -- (func);
% \draw[arrow] (real) -- (func);
% \draw[arrow] (prob) -- (func);
% \draw[arrow] (func) -- (quant);
% \draw[arrow] (lin) -- (quant);
% \draw[arrow] (top) -- (quant);
% \draw[arrow] (func) -- (book);
% \draw[arrow] (quant) -- (book);
% \draw[arrow] (prob) -- (book);
% \end{scope}
\end{tikzpicture}
\caption{Dependency tree: prerequisites and how they feed into the main text.}
\label{fig:dependency-tree}
\end{figure}
\ifSubfilesClassLoaded{
\printbibliography[title={References}]
}
\end{document}
\documentclass[main.tex]{subfiles}
\ifSubfilesClassLoaded{
\addbibresource{main.bib}
}
\begin{document}
\chapter*{Preface}
\addcontentsline{toc}{chapter}{Preface}
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.
The exposition is organized in layers.
Chapter 0 collects the algebraic, probabilistic, geometric, and quantum-mechanical background needed later.
Chapter 1 develops the motivation from polarization experiments, recalls concentration on spheres, and then states the Hayden--Leung--Winter entanglement bound.
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.
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.
\section*{How to use the dependency graph}
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.
\bigskip
\begin{figure}[ht]
\centering
\begin{tikzpicture}[
x=1cm,
y=1cm,
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[course] (lin) at (-8,0) {Linear algebra};
\node[course] (real) at (-4,0) {Real analysis};
\node[course] (measure) at (0,0) {Measure theory};
\node[course] (prob) at (4,0) {Probability theory};
\node[course] (top) at (8,0) {Topology};
\node[topic] (cvec) at (-8,-2.3) {Complex vector\\ spaces};
\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[advanced] (smooth) at (-6.0,-4.8) {Smooth manifolds\\ and Lie groups};
\node[advanced] (riem) at (-1.7,-4.8) {Riemannian\\ manifolds};
\node[advanced] (hopf) at (2.6,-4.8) {Hopf fibration};
\node[advanced] (rand) at (6.9,-4.8) {Random quantum\\ states};
\node[advanced] (conc) at (11.0,-4.8) {Concentration of\\ measure phenomenon};
\node[advanced] (app) at (-4.4,-7.3) {Application in\\ non-commutative probability};
\node[advanced] (obs) at (0.2,-7.3) {Observable\\ diameters};
\node[advanced] (entropy) at (4.9,-7.3) {Entropy estimator\\ on $\mathbb{C}P^n$};
\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 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}
\ifSubfilesClassLoaded{
\printbibliography[title={References}]
}
\end{document}