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README.md
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where $d_B\geq d_A\geq 3$.
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.
## About bibliography for the report
Since we are most referencing books, to the future self who want to separate the content, don't do so unless your bib exceeds 100 entries.

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% chapters/chap0.tex
\documentclass[../main.tex]{subfiles}
% If this chapter is compiled *by itself*, we must load only its own .bib
% and print its bibliography at the end of the chapter.
\ifSubfilesClassLoaded{
\addbibresource{\subfix{../main.bib}}
}
\begin{document}
\chapter*{Chapter 0: Brief definitions and basic concepts}
\addcontentsline{toc}{chapter}{Chapter 0: Brief definitions and basic concepts}
\markboth{Chapter 0: Brief definitions and basic concepts}{}
This section serve as reference for definitions and theorems that we will use later. This section can be safely ignored if you are already familiar with the definitions and theorems.
But for the future self who might have no idea what I'm talking about, we will provided detailed definitions to you to understand the concepts.
\section{Complex vector spaces}
The main vector space we are interested in is $\mathbb{C}^n$; therefore, all the linear operators we defined are from $\mathbb{C}^n$ to $\mathbb{C}^n$.
We denote a vector in vector space as $\ket{\psi}=(z_1,\ldots,z_n)$ (might also be infinite dimensional, and $z_i\in\mathbb{C}$).
A natural inner product space defined on $\mathbb{C}^n$ is given by the Hermitian inner product:
$$
\langle\psi|\varphi\rangle=\sum_{i=1}^n z_iz_i^*
$$
This satisfies the following properties:
\begin{enumerate}
\item $\bra{\psi}\sum_i \lambda_i\ket{\varphi}=\sum_i \lambda_i \langle\psi|\varphi\rangle$ (linear on the second argument. Note that in physics \cite{Nielsen_Chuang_2010} we use linear on the second argument and conjugate linear on the first argument. But in math, we use linear on the first argument and conjugate linear on the second argument \cite{Axler_2024}. As promised in the beginning, we will use the physics convention in this report.)
\item $\langle\varphi|\psi\rangle=(\langle\psi|\varphi\rangle)^*$
\item $\langle\psi|\psi\rangle\geq 0$ with equality if and only if $\ket{\psi}=0$
\end{enumerate}
Here $\psi$ is just a label for the vector, and you don't need to worry about it too much. This is also called the ket, where the counterpart:
\begin{itemize}
\item $\langle\psi\rangle$ is called the bra, used to denote the vector dual to $\psi$; such an element is a linear functional if you really want to know what that is.
\item $\langle\psi|\varphi\rangle$ is the inner product between two vectors, and $\bra{\psi} A\ket{\varphi}$ is the inner product between $A\ket{\varphi}$ and $\bra{\psi}$, or equivalently $A^\dagger \bra{\psi}$ and $\ket{\varphi}$.
\item Given a complex matrix $A=\mathbb{C}^{n\times n}$,
\begin{enumerate}
\item $A^*$ is the complex conjugate of $A$.
i.e.,
$$
A=\begin{bmatrix}
1+i & 2+i & 3+i\\
4+i & 5+i & 6+i\\
7+i & 8+i & 9+i\end{bmatrix},
A^*=\begin{bmatrix}
1-i & 2-i & 3-i\\
4-i & 5-i & 6-i\\
7-i & 8-i & 9-i
\end{bmatrix}
$$
\item $A^\top$ denotes the transpose of $A$.
i.e.,
$$
A=\begin{bmatrix}
1+i & 2+i & 3+i\\
4+i & 5+i & 6+i\\
7+i & 8+i & 9+i
\end{bmatrix},
A^\top=\begin{bmatrix}
1+i & 4+i & 7+i\\
2+i & 5+i & 8+i\\
3+i & 6+i & 9+i
\end{bmatrix}
$$
\item $A^\dagger=(A^*)^\top$ denotes the complex conjugate transpose, referred to as the adjoint, or Hermitian conjugate of $A$.
i.e.,
$$
A=\begin{bmatrix}
1+i & 2+i & 3+i\\
4+i & 5+i & 6+i\\
7+i & 8+i & 9+i
\end{bmatrix},
A^\dagger=\begin{bmatrix}
1-i & 4-i & 7-i\\
2-i & 5-i & 8-i\\
3-i & 6-i & 9-i
\end{bmatrix}
$$
\item $A$ is unitary if $A^\dagger A=AA^\dagger=I$.
\item $A$ is hermitian (self-adjoint in mathematics literature) if $A^\dagger=A$.
\end{enumerate}
\end{itemize}
\subsubsection{Motivation of Tensor product}
Recall from the traditional notation of product space of two vector spaces $V$ and $W$, that is, $V\times W$, is the set of all ordered pairs $(\ket{v},\ket{w})$ where $\ket{v}\in V$ and $\ket{w}\in W$.
The space has dimension $\dim V+\dim W$.
We want to define a vector space with the notation of multiplication of two vectors from different vector spaces.
That is
$$
(\ket{v_1}+\ket{v_2})\otimes \ket{w}=(\ket{v_1}\otimes \ket{w})+(\ket{v_2}\otimes \ket{w})
$$
$$
\ket{v}\otimes (\ket{w_1}+\ket{w_2})=(\ket{v}\otimes \ket{w_1})+(\ket{v}\otimes \ket{w_2})
$$
and enables scalar multiplication by
$$
\lambda (\ket{v}\otimes \ket{w})=(\lambda \ket{v})\otimes \ket{w}=\ket{v}\otimes (\lambda \ket{w})
$$
And we wish to build a way to associate the basis of $V$ and $W$ with the basis of $V\otimes W$. That makes the tensor product a vector space with dimension $\dim V\times \dim W$.
\begin{defn}
Definition of linear functional
A linear functional is a linear map from $V$ to $\mathbb{F}$.
\end{defn}
Note the difference between a linear functional and a linear map.
A generalized linear map is a function $f: V\to W$ satisfying the condition.
\begin{itemize}
\item $f(\ket{u}+\ket{v})=f(\ket{u})+f(\ket{v})$
\item $f(\lambda \ket{v})=\lambda f(\ket{v})$
\end{itemize}
\begin{defn}
A bilinear functional is a bilinear function $\beta:V\times W\to \mathbb{F}$ satisfying the condition that $\ket{v}\to \beta(\ket{v},\ket{w})$ is a linear functional for all $\ket{w}\in W$ and $\ket{w}\to \beta(\ket{v},\ket{w})$ is a linear functional for all $\ket{v}\in V$.
\end{defn}
The vector space of all bilinear functionals is denoted by $\mathcal{B}(V, W)$.
\begin{defn}
Let $V, W$ be two vector spaces.
Let $V'$ and $W'$ be the dual spaces of $V$ and $W$, respectively, that is $V'=\{\psi:V\to \mathbb{F}\}$ and $W'=\{\phi:W\to \mathbb{F}\}$, $\psi, \phi$ are linear functionals.
The tensor product of vectors $v\in V$ and $w\in W$ is the bilinear functional defined by $\forall (\psi,\phi)\in V'\times W'$ given by the notation
$$
(v\otimes w)(\psi,\phi)\coloneqq\psi(v)\phi(w)
$$
The tensor product of two vector spaces $V$ and $W$ is the vector space $\mathcal{B}(V',W')$
Notice that the basis of such vector space is the linear combination of the basis of $V'$ and $W'$, that is, if $\{e_i\}$ is the basis of $V'$ and $\{f_j\}$ is the basis of $W'$, then $\{e_i\otimes f_j\}$ is the basis of $\mathcal{B}(V', W')$.
That is, every element of $\mathcal{B}(V', W')$ can be written as a linear combination of the basis.
Since $\{e_i\}$ and $\{f_j\}$ are bases of $V'$ and $W'$, respectively, then we can always find a set of linear functionals $\{\phi_i\}$ and $\{\psi_j\}$ such that $\phi_i(e_j)=\delta_{ij}$ and $\psi_j(f_i)=\delta_{ij}$.
Here $\delta_{ij}=\begin{cases}
1 & \text{if } i=j \\
0 & \text{otherwise}
\end{cases}$ is the Kronecker delta.
\end{defn}
$$
V\otimes W=\left\{\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w): \phi_i\in V', \psi_j\in W'\right\}
$$
Note that $\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w)$ is a bilinear functional that maps $V'\times W'$ to $\mathbb{F}$.
This enables basis-free construction of vector spaces with proper multiplication and scalar multiplication.
This vector space is equipped with the unique inner product $\langle v\otimes w, u\otimes x\rangle_{V\otimes W}$ defined by
$$
\langle v\otimes w, u\otimes x\rangle=\langle v,u\rangle_V\langle w,x\rangle_W
$$
In practice, we ignore the subscript of the vector space and just write $\langle v\otimes w, u\otimes x\rangle=\langle v,u\rangle\langle w,x\rangle$.
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.
There are several main components of the generalized probability theory; let's see how we can formulate them, comparing with the classical probability theory.
First, we define the Hilbert space in case one did not make the step from the linear algebra courses like me.
\begin{defn}
\label{defn:Hilbert_space}
Hilbert space:
A Hilbert space is a complete inner product space.
\end{defn}
That is, a vector space equipped with an inner product that is complete (every Cauchy sequence converges to a limit).
To introduce an example of Hilbert space we use when studying quantum mechanics, we need to introduce a common inner product used in $\mathbb{C}^n$.
\begin{defn}
\label{defn:Hermitian_inner_product}
Hermitian inner product:
On $\mathbb{C}^n$, the Hermitian inner product is defined by
$$
\langle u,v\rangle=\sum_{i=1}^n \overline{u_i}v_i
$$
\end{defn}
\begin{prop}
\label{prop:Hermitian_inner_product_with_complex_vectorspace}
The Hermitian inner product on the complex vector space $\C^n$ makes it a Hilbert space.
\end{prop}
\begin{proof}
We first verify that the Hermitian inner product
$$
\langle u,v\rangle = \sum_{i=1}^n \overline{u_i} v_i
$$
on $\C^n$ satisfies the axioms of an inner product:
\begin{enumerate}
\item \textbf{Conjugate symmetry:} For all $u,v\in\C^n$,
$$
\langle u,v\rangle =\sum_{i=1}^n \overline{u_i} v_i=\overline{\sum_{i=1}^n \overline{v_i} u_i}=\overline{\langle v,u\rangle}.
$$
\item \textbf{Linearity:} For any $u,v,w\in\C^n$ and scalars $a,b\in\C$, we have
$$
\langle u, av + bw\rangle = \sum_{i=1}^n \overline{u_i} (av_i + bw_i)=a\langle u,v\rangle + b\langle u,w\rangle.
$$
\item \textbf{Positive definiteness:} For every $u=(u_1,u_2,\cdots,u_n)\in\C^n$, let $u_j=a_j+b_ji$, where $a_j,b_j\in\mathbb{R}$.
$$
\langle u,u\rangle = \sum_{j=1}^n \overline{u_j} u_j=\sum_{i=1}^n (a_i^2+b_i^2)\geq 0,
$$
with equality if and only if $u=0$.
Therefore, the Hermitian inner product is an inner product.
\end{enumerate}
Next, we show that $\C^n$ is complete with respect to the norm induced by this inner product:
$$
\|u\| = \sqrt{\langle u,u\rangle}.
$$
Since $\C^n$ is finite-dimensional, every Cauchy sequence (with respect to any norm) converges in $\C^n$. This is a standard result in finite-dimensional normed spaces, which implies that $\C^n$ is indeed complete.
Therefore, since the Hermitian inner product fulfills the inner product axioms and $\C^n$ is complete, the complex vector space $\C^n$ with the Hermitian inner product is a Hilbert space.
\end{proof}
Another classical example of Hilbert space is $L^2(\Omega, \mathscr{F}, P)$, where $(\Omega, \mathscr{F}, P)$ is a measure space ($\Omega$ is a set, $\mathscr{F}$ is a $\sigma$-algebra on $\Omega$, and $P$ is a measure on $\mathscr{F}$). The $L^2$ space is the space of all function on $\Omega$ that is
\begin{enumerate}
\item \textbf{square integrable}: square integrable functions are the functions $f:\Omega\to \mathbb{C}$ such that
$$
\int_\Omega |f(\omega)|^2 dP(\omega)<\infty
$$
with inner product defined by
$$
\langle f,g\rangle=\int_\Omega \overline{f(\omega)}g(\omega)dP(\omega)
$$
\item \textbf{complex-valued}: functions are complex-valued measurable. $f=u+v i$ is complex-valued if $u$ and $v$ are real-valued measurable.
\end{enumerate}
\begin{prop}
\label{prop:L2_space_is_a_Hilbert_space}
$L^2(\Omega, \mathscr{F}, P)$ is a Hilbert space.
\end{prop}
\begin{proof}
We check the two conditions of the Hilbert space:
\begin{itemize}
\item Completeness:
Let $(f_n)$ be a Cauchy sequence in $L^2(\Omega, \mathscr{F}, P)$. Then for any $\epsilon>0$, there exists an $N$ such that for all $m,n\geq N$, we have
$$
\int_\Omega |f_m(\omega)-f_n(\omega)|^2 dP(\omega)<\epsilon^2
$$
This means that $(f_n)$ is a Cauchy sequence in the norm of $L^2(\Omega, \mathscr{F}, P)$.
\item Inner product:
The inner product is defined by
$$
\langle f,g\rangle=\int_\Omega \overline{f(\omega)}g(\omega)dP(\omega)
$$
This is a well-defined inner product on $L^2(\Omega, \mathscr{F}, P)$. We can check the properties of the inner product:
\begin{itemize}
\item Linearity:
$$
\langle af+bg,h\rangle=a\langle f,h\rangle+b\langle g,h\rangle
$$
\item Conjugate symmetry:
$$
\langle f,g\rangle=\overline{\langle g,f\rangle}
$$
\item Positive definiteness:
$$
\langle f,f\rangle\geq 0
$$
\end{itemize}
\end{itemize}
\end{proof}
Let $\mathscr{H}$ be a Hilbert space. $\mathscr{H}$ consists of complex-valued functions on a finite set $\Omega=\{1,2,\cdots,n\}$, and the functions $(e_1,e_2,\cdots,e_n)$ form an orthonormal basis of $\mathscr{H}$. (We use Dirac notation $|k\rangle$ to denote the basis vector $e_k$~\cite{parthasarathy1992quantum}.)
As an analog to the classical probability space $(\Omega,\mathscr{F},\mu)$, which consists of a sample space $\Omega$ and a probability measure $\mu$ on the state space $\mathscr{F}$, the non-commutative probability space $(\mathscr{H},\mathscr{P},\rho)$ consists of a Hilbert space $\mathscr{H}$ and a state $\rho$ on the space of all orthogonal projections $\mathscr{P}$.
The detailed definition of the non-commutative probability space is given below:
\begin{defn}
\label{defn:non-commutative_probability_space}
Non-commutative probability space:
A non-commutative probability space is a pair $(\mathscr{B}(\mathscr{H}),\mathscr{P})$, where $\mathscr{B}(\mathscr{H})$ is the set of all \textbf{bounded} linear operators on $\mathscr{H}$.
A linear operator on $\mathscr{H}$ is \textbf{bounded} if for all $u$ such that $\|u\|\leq 1$, we have $\|Au\|\leq M$ for some $M>0$.
$\mathscr{P}$ is the set of all orthogonal projections on $\mathscr{B}(\mathscr{H})$.
The set $\mathscr{P}=\{P\in\mathscr{B}(\mathscr{H}):P^*=P=P^2\}$ is the set of all orthogonal projections on $\mathscr{B}(\mathscr{H})$.
\end{defn}
Recall from classical probability theory, we call the initial probability distribution for possible outcomes in the classical probability theory as our \textit{state}, simillarly, we need to define the \textit{state} in the non-commutative probability theory.
\begin{defn}
\label{defn:state}
Non-commutative probability state:
A state on $(\mathscr{B}(\mathscr{H}),\mathscr{P})$ is a map $\rho:\mathscr{P}\to[0,1]$, (commonly named as density operator) such that:
\begin{itemize}
\item $\rho(O)=0$, where $O$ is the zero projection, and $\rho(I)=1$, where $I$ is the identity projection.
\item If $P_1,P_2,\ldots,P_n$ are pairwise disjoint orthogonal projections, then $\rho(P_1 + P_2 + \cdots + P_n) = \sum_{i=1}^n \rho(P_i)$.
\end{itemize}
\end{defn}
An example of a density operator can be given as follows:
If $(|\psi_1\rangle,|\psi_2\rangle,\cdots,|\psi_n\rangle)$ is an orthonormal basis of $\mathscr{H}$ consisting of eigenvectors of $\rho$, for the eigenvalues $p_1,p_2,\cdots,p_n$, then $p_j\geq 0$ and $\sum_{j=1}^n p_j=1$.
We can write $\rho$ as
\[
\rho=\sum_{j=1}^n p_j|\psi_j\rangle\langle\psi_j|
\]
(Under basis $|\psi_j\rangle$, it is a diagonal matrix with $p_j$ on the diagonal.)
% Then we need to introduce a theorem that ensures that every state on the space of all orthogonal projections on $\mathscr{H}$ can be represented by a density operator.
% \begin{theorem}
% \label{theorem:Gleason's_theorem}
% Gleason's theorem (Theorem 1.1.15 in~\cite{parthasarathy2005mathematical})
% Let $\mathscr{H}$ be a Hilbert space over $\mathbb{C}$ or $\mathbb{R}$ of dimension $n\geq 3$. Let $\mu$ be a state on the space $\mathscr{P}$ of projections on $\mathscr{H}$. Then there exists a unique density operator $\rho$ such that
% \[
% \mu(P)=\operatorname{Tr}(\rho P)
% \]
% for all $P\in\mathscr{P}$. $\mathscr{P}$ is the space of all orthogonal projections on $\mathscr{H}$.
% \end{theorem}
% This proof came from~\cite{parthasarathy2005mathematical}.
% \begin{proof}
% % TODO: FILL IN THE PROOF
% \end{proof}
% This theorem is a very important theorem in non-commutative probability theory; it states that any state on the space of all orthogonal projections on $\mathscr{H}$ can be represented by a density operator.
The counterpart of the random variable in the non-commutative probability theory is called an observable, which is a Hermitian operator on $\mathscr{H}$ (for all $\psi,\phi$ in the domain of $A$, we have $\langle A\psi,\phi\rangle=\langle\psi,A\phi\rangle$. This kind of operator ensures that our outcome interpreted as probability is a real number).
\begin{defn}
\label{defn:observable}
Observable:
Let $\mathscr{B}(\mathbb{R})$ be the set of all Borel sets on $\mathbb{R}$.
A random variable on the Hilbert space $\mathscr{H}$ is a projection-valued map (measure) $P:\mathscr{B}(\mathbb{R})\to\mathscr{P}$.
With the following properties:
\begin{itemize}
\item $P(\emptyset)=O$ (the zero projection)
\item $P(\mathbb{R})=I$ (the identity projection)
\item For any sequence $A_1,A_2,\cdots,A_n\in \mathscr{B}(\mathbb{R})$, the following holds:
\begin{itemize}
\item $P(\bigcup_{i=1}^n A_i)=\bigvee_{i=1}^n P(A_i)$
\item $P(\bigcap_{i=1}^n A_i)=\bigwedge_{i=1}^n P(A_i)$
\item $P(A^c)=I-P(A)$
\item If $A_j$ are mutually disjoint (that is $P(A_i)P(A_j)=P(A_j)P(A_i)=O$ for $i\neq j$), then $P(\bigcup_{j=1}^n A_j)=\sum_{j=1}^n P(A_j)$
\end{itemize}
\end{itemize}
\end{defn}
\begin{defn}
\label{defn:probability_of_random_variable}
Probability of a random variable:
For a system prepared in state $\rho$, the probability that the random variable given by the projection-valued measure $P$ is in the Borel set $A$ is $\operatorname{Tr}(\rho P(A))$.
\end{defn}
When operators commute, we recover classical probability measures.
\begin{defn}
\label{defn:measurement}
Definition of measurement:
A measurement (observation) of a system prepared in a given state produces an outcome $x$, $x$ is a physical event that is a subset of the set of all possible outcomes. For each $x$, we associate a measurement operator $M_x$ on $\mathscr{H}$.
Given the initial state (pure state, unit vector) $u$, the probability of measurement outcome $x$ is given by:
\[
p(x)=\|M_xu\|^2
\]
Note that to make sense of this definition, the collection of measurement operators $\{M_x\}$ must satisfy the completeness requirement:
\[
1=\sum_{x\in X} p(x)=\sum_{x\in X}\|M_xu\|^2=\sum_{x\in X}\langle M_xu,M_xu\rangle=\langle u,(\sum_{x\in X}M_x^*M_x)u\rangle
\]
So $\sum_{x\in X}M_x^*M_x=I$.
\end{defn}
\begin{prop}
\label{prop:indistinguishability}
Proposition of indistinguishability:
Suppose that we have two systems $u_1,u_2\in \mathscr{H}_1$, the two states are distinguishable if and only if they are orthogonal.
\end{prop}
\begin{proof}
Ways to distinguish the two states:
\begin{enumerate}
\item Set $X=\{0,1,2\}$ and $M_i=|u_i\rangle\langle u_i|$, $M_0=I-M_1-M_2$
\item Then $\{M_0,M_1,M_2\}$ is a complete collection of measurement operators on $\mathscr{H}$.
\item Suppose the prepared state is $u_1$, then $p(1)=\|M_1u_1\|^2=\|u_1\|^2=1$, $p(2)=\|M_2u_1\|^2=0$, $p(0)=\|M_0u_1\|^2=0$.
\end{enumerate}
If they are not orthogonal, then there is no choice of measurement operators to perfectly distinguish the two states.
\end{proof}
\textit{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.}
Here is Table~\ref{tab:analog_of_classical_probability_theory_and_non_commutative_probability_theory} summarizing the analog of classical probability theory and non-commutative (\textit{quantum}) probability theory~\cite{Feres}:
\begin{table}[H]
\centering
\renewcommand{\arraystretch}{1.5}
\caption{Analog of classical probability theory and non-commutative (\textit{quantum}) probability theory}
\label{tab:analog_of_classical_probability_theory_and_non_commutative_probability_theory}
{\tiny
\begin{tabular}{|p{0.5\linewidth}|p{0.5\linewidth}|}
\hline
\textbf{Classical probability} & \textbf{Non-commutative probability} \\
\hline
Sample space $\Omega$, cardinality $\vert\Omega\vert=n$, example: $\Omega=\{0,1\}$ & Complex Hilbert space $\mathscr{H}$, dimension $\dim\mathscr{H}=n$, example: $\mathscr{H}=\mathbb{C}^2$ \\
\hline
Common algebra of $\mathbb{C}$ valued functions & Algebra of bounded operators $\mathscr{B}(\mathscr{H})$ \\
\hline
$f\mapsto \bar{f}$ complex conjugation & $P\mapsto P^*$ adjoint \\
\hline
Events: indicator functions of sets & Projections: space of orthogonal projections $\mathscr{P}\subseteq\mathscr{B}(\mathscr{H})$ \\
\hline
functions $f$ such that $f^2=f=\overline{f}$ & orthogonal projections $P$ such that $P^*=P=P^2$ \\
\hline
$\mathbb{R}$-valued functions $f=\overline{f}$ & self-adjoint operators $A=A^*$ \\
\hline
$\mathbb{I}_{f^{-1}(\{\lambda\})}$ is the indicator function of the set $f^{-1}(\{\lambda\})$ & $P(\lambda)$ is the orthogonal projection to eigenspace \\
\hline
$f=\sum_{\lambda\in \operatorname{Range}(f)}\lambda \mathbb{I}_{f^{-1}(\{\lambda\})}$ & $A=\sum_{\lambda\in \operatorname{sp}(A)}\lambda P(\lambda)$ \\
\hline
Probability measure $\mu$ on $\Omega$ & Density operator $\rho$ on $\mathscr{H}$ \\
\hline
Delta measure $\delta_\omega$ & Pure state $\rho=\vert\psi\rangle\langle\psi\vert$ \\
\hline
$\mu$ is non-negative measure and $\sum_{i=1}^n\mu(\{i\})=1$ & $\rho$ is positive semi-definite and $\operatorname{Tr}(\rho)=1$ \\
\hline
Expected value of random variable $f$ is $\mathbb{E}_{\mu}(f)=\sum_{i=1}^n f(i)\mu(\{i\})$ & Expected value of operator $A$ is $\mathbb{E}_\rho(A)=\operatorname{Tr}(\rho A)$ \\
\hline
Variance of random variable $f$ is $\operatorname{Var}_\mu(f)=\sum_{i=1}^n (f(i)-\mathbb{E}_\mu(f))^2\mu(\{i\})$ & Variance of operator $A$ is $\operatorname{Var}_\rho(A)=\operatorname{Tr}(\rho A^2)-\operatorname{Tr}(\rho A)^2$ \\
\hline
Covariance of random variables $f$ and $g$ is $\operatorname{Cov}_\mu(f,g)=\sum_{i=1}^n (f(i)-\mathbb{E}_\mu(f))(g(i)-\mathbb{E}_\mu(g))\mu(\{i\})$ & Covariance of operators $A$ and $B$ is $\operatorname{Cov}_\rho(A,B)=\operatorname{Tr}(\rho A\circ B)-\operatorname{Tr}(\rho A)\operatorname{Tr}(\rho B)$ \\
\hline
Composite system is given by Cartesian product of the sample spaces $\Omega_1\times\Omega_2$ & Composite system is given by tensor product of the Hilbert spaces $\mathscr{H}_1\otimes\mathscr{H}_2$ \\
\hline
Product measure $\mu_1\times\mu_2$ on $\Omega_1\times\Omega_2$ & Tensor product of space $\rho_1\otimes\rho_2$ on $\mathscr{H}_1\otimes\mathscr{H}_2$ \\
\hline
Marginal distribution $\pi_*v$ & Partial trace $\operatorname{Tr}_2(\rho)$ \\
\hline
\end{tabular}
}
\vspace{0.5cm}
\end{table}
% When compiled standalone, print this chapter's references at the end.
\ifSubfilesClassLoaded{
\printbibliography[title={References}]
}
\end{document}

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% If this chapter is compiled *by itself*, we must load only its own .bib
% and print its bibliography at the end of the chapter.
\ifSubfilesClassLoaded{
\addbibresource{chap1.bib}
\addbibresource{../main.bib}
}
\usepackage{amsmath, amsfonts, amsthm}
@@ -28,25 +28,9 @@
\chapter{Concentration of Measure And Quantum Entanglement}
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}.
As the future version of me might forgot everything we have over the summer, as I did for now, I will make a review again from the simple definition to recall the necessary information to tell you why we are here and how we are going to proceed.
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.
First, we will build the mathematical model describing the behavior of quantum system and why they makes sense for physicists and meaningful for general publics.
\section{Motivation}
@@ -58,59 +42,62 @@ The light which comes through a polarizer is polarized in a certain direction. I
The light intensity decreases with $\alpha$ (the angle between the two filters). The light should vanish when $\alpha=\pi/2$.
\begin{figure}
However, for a system of 3 polarizing filters $F_1,F_2,F_3$, having directions $\alpha_1,\alpha_2,\alpha_3$, if we put them on the optical bench in pairs, then we will have three random variables $P_1,P_2,P_3$.
\begin{figure}[H]
\centering
\includegraphics[width=0.8\textwidth]{Filter_figure.png}
\includegraphics[width=0.7\textwidth]{Filter_figure.png}
\caption{The light polarization experiment, image from \cite{kummer1998elements}}
\label{fig:Filter_figure}
\end{figure}
However, for a system of 3 polarizing filters $F_1,F_2,F_3$, having directions $\alpha_1,\alpha_2,\alpha_3$, if we put them on the optical bench in pairs, then we will have three random variables $P_1,P_2,P_3$.
\begin{theorem}
\label{theorem:Bell's_3_variable_inequality}
Bell's 3 variable inequality:
For any three random variables $P_1,P_2,P_3$ in a classical probability space, we have
\[
$$
\operatorname{Prob}(P_1=1,P_3=0)\leq \operatorname{Prob}(P_1=1,P_2=0)+\operatorname{Prob}(P_2=1,P_3=0)
\]
$$
\end{theorem}
\begin{proof}
By the law of total probability (the event that the photon passes through the first filter but not the third filter is the union of the event that the photon did not pass through the second filter and the event that the photon passed the second filter and did not pass through the third filter), we have
\[
By the law of total probability there are only two possibility if we don't observe any light passing the filter pair $F_i,F_j$, it means the photon is either blocked by $F_i$ or $F_j$, it means
$$
\begin{aligned}
\operatorname{Prob}(P_1=1,P_3=0)&=\operatorname{Prob}(P_1=1,P_2=0,P_3=0)\\
&+\operatorname{Prob}(P_1=1,P_2=1,P_3=0)\\
&\leq\operatorname{Prob}(P_1=1,P_2=0)+\operatorname{Prob}(P_2=1,P_3=0)
\end{aligned}
\]
$$
\end{proof}
However, according to our experimental measurement, for any pair of polarizers $F_i,F_j$, by the complement rule, we have
\[
$$
\begin{aligned}
\operatorname{Prob}(P_i=1,P_j=0)&=\operatorname{Prob}(P_i=1)-\operatorname{Prob}(P_i=1,P_j=1)\\
&=\frac{1}{2}-\frac{1}{2}\cos^2(\alpha_i-\alpha_j)\\
&=\frac{1}{2}\sin^2(\alpha_i-\alpha_j)
\end{aligned}
\]
$$
This leads to a contradiction if we apply the inequality to the experimental data.
\[
$$
\frac{1}{2}\sin^2(\alpha_1-\alpha_3)\leq\frac{1}{2}\sin^2(\alpha_1-\alpha_2)+\frac{1}{2}\sin^2(\alpha_2-\alpha_3)
\]
$$
If $\alpha_1=0,\alpha_2=\frac{\pi}{6},\alpha_3=\frac{\pi}{3}$, then
\[
$$
\begin{aligned}
\frac{1}{2}\sin^2(-\frac{\pi}{3})&\leq\frac{1}{2}\sin^2(-\frac{\pi}{6})+\frac{1}{2}\sin^2(\frac{\pi}{6}-\frac{\pi}{3})\\
\frac{3}{8}&\leq\frac{1}{8}+\frac{1}{8}\\
\frac{3}{8}&\leq\frac{1}{4}
\end{aligned}
\]
$$
Other revised experiments (e.g., Aspect's experiment, calcium entangled photon experiment) are also conducted and the inequality is still violated.
@@ -123,17 +110,184 @@ State of polarization of a photon: $\psi=\alpha|0\rangle+\beta|1\rangle$, where
Polarization filter (generalized 0,1 valued random variable): orthogonal projection $P_\alpha$ on $\mathbb{C}^2$ corresponding to the direction $\alpha$ (operator satisfies $P_\alpha^*=P_\alpha=P_\alpha^2$).
The matrix representation of $P_\alpha$ is given by
\[
$$
P_\alpha=\begin{pmatrix}
\cos^2(\alpha) & \cos(\alpha)\sin(\alpha)\\
\cos(\alpha)\sin(\alpha) & \sin^2(\alpha)
\end{pmatrix}
\]
$$
Probability of a photon passing through the filter $P_\alpha$ is given by $\langle P_\alpha\psi,\psi\rangle$; this is $\cos^2(\alpha)$ if we set $\psi=|0\rangle$.
Since the probability of a photon passing through the three filters is not commutative, it is impossible to discuss $\operatorname{Prob}(P_1=1,P_3=0)$ in the classical setting.
The main vector space we are interested in is $\mathbb{C}^n$; therefore, all the linear operators we defined are from $\mathbb{C}^n$ to $\mathbb{C}^n$.
We denote a vector in vector space as $\ket{\psi}=(z_1,\ldots,z_n)$ (might also be infinite dimensional, and $z_i\in\mathbb{C}$).
A natural inner product space defined on $\mathbb{C}^n$ is given by the Hermitian inner product:
$$
\langle\psi|\varphi\rangle=\sum_{i=1}^n z_iz_i^*
$$
This satisfies the following properties:
\begin{enumerate}
\item $\bra{\psi}\sum_i \lambda_i\ket{\varphi}=\sum_i \lambda_i \langle\psi|\varphi\rangle$ (linear on the second argument. Note that in physics \cite{Nielsen_Chuang_2010} we use linear on the second argument and conjugate linear on the first argument. But in math, we use linear on the first argument and conjugate linear on the second argument \cite{Axler_2024}. As promised in the beginning, we will use the physics convention in this report.)
\item $\langle\varphi|\psi\rangle=(\langle\psi|\varphi\rangle)^*$
\item $\langle\psi|\psi\rangle\geq 0$ with equality if and only if $\ket{\psi}=0$
\end{enumerate}
Here $\psi$ is just a label for the vector, and you don't need to worry about it too much. This is also called the ket, where the counterpart:
\begin{itemize}
\item $\langle\psi\rangle$ is called the bra, used to denote the vector dual to $\psi$; such an element is a linear functional if you really want to know what that is.
\item $\langle\psi|\varphi\rangle$ is the inner product between two vectors, and $\bra{\psi} A\ket{\varphi}$ is the inner product between $A\ket{\varphi}$ and $\bra{\psi}$, or equivalently $A^\dagger \bra{\psi}$ and $\ket{\varphi}$.
\item Given a complex matrix $A=\mathbb{C}^{n\times n}$,
\begin{enumerate}
\item $A^*$ is the complex conjugate of $A$.
i.e.,
$$
A=\begin{bmatrix}
1+i & 2+i & 3+i\\
4+i & 5+i & 6+i\\
7+i & 8+i & 9+i\end{bmatrix},
A^*=\begin{bmatrix}
1-i & 2-i & 3-i\\
4-i & 5-i & 6-i\\
7-i & 8-i & 9-i
\end{bmatrix}
$$
\item $A^\top$ denotes the transpose of $A$.
i.e.,
$$
A=\begin{bmatrix}
1+i & 2+i & 3+i\\
4+i & 5+i & 6+i\\
7+i & 8+i & 9+i
\end{bmatrix},
A^\top=\begin{bmatrix}
1+i & 4+i & 7+i\\
2+i & 5+i & 8+i\\
3+i & 6+i & 9+i
\end{bmatrix}
$$
\item $A^\dagger=(A^*)^\top$ denotes the complex conjugate transpose, referred to as the adjoint, or Hermitian conjugate of $A$.
i.e.,
$$
A=\begin{bmatrix}
1+i & 2+i & 3+i\\
4+i & 5+i & 6+i\\
7+i & 8+i & 9+i
\end{bmatrix},
A^\dagger=\begin{bmatrix}
1-i & 4-i & 7-i\\
2-i & 5-i & 8-i\\
3-i & 6-i & 9-i
\end{bmatrix}
$$
\item $A$ is unitary if $A^\dagger A=AA^\dagger=I$.
\item $A$ is hermitian (self-adjoint in mathematics literature) if $A^\dagger=A$.
\end{enumerate}
\end{itemize}
\subsubsection{Motivation of Tensor product}
Recall from the traditional notation of product space of two vector spaces $V$ and $W$, that is, $V\times W$, is the set of all ordered pairs $(\ket{v},\ket{w})$ where $\ket{v}\in V$ and $\ket{w}\in W$.
The space has dimension $\dim V+\dim W$.
We want to define a vector space with the notation of multiplication of two vectors from different vector spaces.
That is
$$
(\ket{v_1}+\ket{v_2})\otimes \ket{w}=(\ket{v_1}\otimes \ket{w})+(\ket{v_2}\otimes \ket{w})
$$
$$
\ket{v}\otimes (\ket{w_1}+\ket{w_2})=(\ket{v}\otimes \ket{w_1})+(\ket{v}\otimes \ket{w_2})
$$
and enables scalar multiplication by
$$
\lambda (\ket{v}\otimes \ket{w})=(\lambda \ket{v})\otimes \ket{w}=\ket{v}\otimes (\lambda \ket{w})
$$
And we wish to build a way to associate the basis of $V$ and $W$ with the basis of $V\otimes W$. That makes the tensor product a vector space with dimension $\dim V\times \dim W$.
\begin{defn}
Definition of linear functional
A linear functional is a linear map from $V$ to $\mathbb{F}$.
\end{defn}
Note the difference between a linear functional and a linear map.
A generalized linear map is a function $f: V\to W$ satisfying the condition.
\begin{itemize}
\item $f(\ket{u}+\ket{v})=f(\ket{u})+f(\ket{v})$
\item $f(\lambda \ket{v})=\lambda f(\ket{v})$
\end{itemize}
\begin{defn}
A bilinear functional is a bilinear function $\beta:V\times W\to \mathbb{F}$ satisfying the condition that $\ket{v}\to \beta(\ket{v},\ket{w})$ is a linear functional for all $\ket{w}\in W$ and $\ket{w}\to \beta(\ket{v},\ket{w})$ is a linear functional for all $\ket{v}\in V$.
\end{defn}
The vector space of all bilinear functionals is denoted by $\mathcal{B}(V, W)$.
\begin{defn}
Let $V, W$ be two vector spaces.
Let $V'$ and $W'$ be the dual spaces of $V$ and $W$, respectively, that is $V'=\{\psi:V\to \mathbb{F}\}$ and $W'=\{\phi:W\to \mathbb{F}\}$, $\psi, \phi$ are linear functionals.
The tensor product of vectors $v\in V$ and $w\in W$ is the bilinear functional defined by $\forall (\psi,\phi)\in V'\times W'$ given by the notation
$$
(v\otimes w)(\psi,\phi)\coloneqq\psi(v)\phi(w)
$$
The tensor product of two vector spaces $V$ and $W$ is the vector space $\mathcal{B}(V',W')$
Notice that the basis of such vector space is the linear combination of the basis of $V'$ and $W'$, that is, if $\{e_i\}$ is the basis of $V'$ and $\{f_j\}$ is the basis of $W'$, then $\{e_i\otimes f_j\}$ is the basis of $\mathcal{B}(V', W')$.
That is, every element of $\mathcal{B}(V', W')$ can be written as a linear combination of the basis.
Since $\{e_i\}$ and $\{f_j\}$ are bases of $V'$ and $W'$, respectively, then we can always find a set of linear functionals $\{\phi_i\}$ and $\{\psi_j\}$ such that $\phi_i(e_j)=\delta_{ij}$ and $\psi_j(f_i)=\delta_{ij}$.
Here $\delta_{ij}=\begin{cases}
1 & \text{if } i=j \\
0 & \text{otherwise}
\end{cases}$ is the Kronecker delta.
\end{defn}
$$
V\otimes W=\left\{\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w): \phi_i\in V', \psi_j\in W'\right\}
$$
Note that $\sum_{i=1}^n \sum_{j=1}^m a_{ij} \phi_i(v)\psi_j(w)$ is a bilinear functional that maps $V'\times W'$ to $\mathbb{F}$.
This enables basis-free construction of vector spaces with proper multiplication and scalar multiplication.
This vector space is equipped with the unique inner product $\langle v\otimes w, u\otimes x\rangle_{V\otimes W}$ defined by
$$
\langle v\otimes w, u\otimes x\rangle=\langle v,u\rangle_V\langle w,x\rangle_W
$$
In practice, we ignore the subscript of the vector space and just write $\langle v\otimes w, u\otimes x\rangle=\langle v,u\rangle\langle w,x\rangle$.
This introduces a new model in mathematics explaining quantum mechanics: the non-commutative probability theory.
\section{Non-commutative probability theory}
@@ -160,9 +314,9 @@ To introduce an example of Hilbert space we use when studying quantum mechanics,
Hermitian inner product:
On $\mathbb{C}^n$, the Hermitian inner product is defined by
\[
$$
\langle u,v\rangle=\sum_{i=1}^n \overline{u_i}v_i
\]
$$
\end{defn}
\begin{prop}
@@ -172,47 +326,51 @@ To introduce an example of Hilbert space we use when studying quantum mechanics,
\begin{proof}
We first verify that the Hermitian inner product
\[
$$
\langle u,v\rangle = \sum_{i=1}^n \overline{u_i} v_i
\]
$$
on $\C^n$ satisfies the axioms of an inner product:
\begin{enumerate}
\item \textbf{Conjugate symmetry:} For all $u,v\in\C^n$,
\[
$$
\langle u,v\rangle =\sum_{i=1}^n \overline{u_i} v_i=\overline{\sum_{i=1}^n \overline{v_i} u_i}=\overline{\langle v,u\rangle}.
\]
$$
\item \textbf{Linearity:} For any $u,v,w\in\C^n$ and scalars $a,b\in\C$, we have
\[
$$
\langle u, av + bw\rangle = \sum_{i=1}^n \overline{u_i} (av_i + bw_i)=a\langle u,v\rangle + b\langle u,w\rangle.
\]
$$
\item \textbf{Positive definiteness:} For every $u=(u_1,u_2,\cdots,u_n)\in\C^n$, let $u_j=a_j+b_ji$, where $a_j,b_j\in\mathbb{R}$.
\[
$$
\langle u,u\rangle = \sum_{j=1}^n \overline{u_j} u_j=\sum_{i=1}^n (a_i^2+b_i^2)\geq 0,
\]
$$
with equality if and only if $u=0$.
Therefore, the Hermitian inner product is an inner product.
\end{enumerate}
Next, we show that $\C^n$ is complete with respect to the norm induced by this inner product:
\[
$$
\|u\| = \sqrt{\langle u,u\rangle}.
\]
$$
Since $\C^n$ is finite-dimensional, every Cauchy sequence (with respect to any norm) converges in $\C^n$. This is a standard result in finite-dimensional normed spaces, which implies that $\C^n$ is indeed complete.
Therefore, since the Hermitian inner product fulfills the inner product axioms and $\C^n$ is complete, the complex vector space $\C^n$ with the Hermitian inner product is a Hilbert space.
\end{proof}
Another classical example of Hilbert space is $L^2(\Omega, \mathscr{F}, P)$, where $(\Omega, \mathscr{F}, P)$ is a measure space ($\Omega$ is a set, $\mathscr{F}$ is a $\sigma$-algebra on $\Omega$, and $P$ is a measure on $\mathscr{F}$). The $L^2$ space is the space of all square integrable, complex-valued measurable functions on $\Omega$.
Another classical example of Hilbert space is $L^2(\Omega, \mathscr{F}, P)$, where $(\Omega, \mathscr{F}, P)$ is a measure space ($\Omega$ is a set, $\mathscr{F}$ is a $\sigma$-algebra on $\Omega$, and $P$ is a measure on $\mathscr{F}$). The $L^2$ space is the space of all function on $\Omega$ that is
The square integrable functions are the functions $f:\Omega\to \mathbb{C}$ such that
\[
\int_\Omega |f(\omega)|^2 dP(\omega)<\infty
\]
with inner product defined by
\[
\langle f,g\rangle=\int_\Omega \overline{f(\omega)}g(\omega)dP(\omega)
\]
\begin{enumerate}
\item \textbf{square integrable}: square integrable functions are the functions $f:\Omega\to \mathbb{C}$ such that
$$
\int_\Omega |f(\omega)|^2 dP(\omega)<\infty
$$
with inner product defined by
$$
\langle f,g\rangle=\int_\Omega \overline{f(\omega)}g(\omega)dP(\omega)
$$
\item \textbf{complex-valued}: functions are complex-valued measurable. $f=u+v i$ is complex-valued if $u$ and $v$ are real-valued measurable.
\end{enumerate}
\begin{prop}
\label{prop:L2_space_is_a_Hilbert_space}
@@ -224,29 +382,29 @@ with inner product defined by
\begin{itemize}
\item Completeness:
Let $(f_n)$ be a Cauchy sequence in $L^2(\Omega, \mathscr{F}, P)$. Then for any $\epsilon>0$, there exists an $N$ such that for all $m,n\geq N$, we have
\[
$$
\int_\Omega |f_m(\omega)-f_n(\omega)|^2 dP(\omega)<\epsilon^2
\]
$$
This means that $(f_n)$ is a Cauchy sequence in the norm of $L^2(\Omega, \mathscr{F}, P)$.
\item Inner product:
The inner product is defined by
\[
$$
\langle f,g\rangle=\int_\Omega \overline{f(\omega)}g(\omega)dP(\omega)
\]
$$
This is a well-defined inner product on $L^2(\Omega, \mathscr{F}, P)$. We can check the properties of the inner product:
\begin{itemize}
\item Linearity:
\[
$$
\langle af+bg,h\rangle=a\langle f,h\rangle+b\langle g,h\rangle
\]
$$
\item Conjugate symmetry:
\[
$$
\langle f,g\rangle=\overline{\langle g,f\rangle}
\]
$$
\item Positive definiteness:
\[
$$
\langle f,f\rangle\geq 0
\]
$$
\end{itemize}
\end{itemize}
\end{proof}
@@ -254,30 +412,30 @@ with inner product defined by
Let $\mathscr{H}$ be a Hilbert space. $\mathscr{H}$ consists of complex-valued functions on a finite set $\Omega=\{1,2,\cdots,n\}$, and the functions $(e_1,e_2,\cdots,e_n)$ form an orthonormal basis of $\mathscr{H}$. (We use Dirac notation $|k\rangle$ to denote the basis vector $e_k$~\cite{parthasarathy1992quantum}.)
The detailed definition of the non-commutative probability space is given below:
As an analog to the classical probability space $(\Omega,\mathscr{F},\mu)$, which consists of a sample space $\Omega$ and a probability measure $\mu$ on the state space $\mathscr{F}$, the non-commutative probability space $(\mathscr{H},\mathscr{P},\rho)$ consists of a Hilbert space $\mathscr{H}$ and a state $\rho$ on the space of all orthogonal projections $\mathscr{P}$.
The detailed definition of the non-commutative probability space is given below:
\begin{defn}
\label{defn:non-commutative_probability_space}
Non-commutative probability space:
A non-commutative probability space is a pair $(\mathscr{B}(\mathscr{H}),\mathscr{P})$, where $\mathscr{B}(\mathscr{H})$ is the set of all bounded linear operators on $\mathscr{H}$.
A non-commutative probability space is a pair $(\mathscr{B}(\mathscr{H}),\mathscr{P})$, where $\mathscr{B}(\mathscr{H})$ is the set of all \textbf{bounded} linear operators on $\mathscr{H}$.
A linear operator on $\mathscr{H}$ is bounded if for all $u$ such that $\|u\|\leq 1$, we have $\|Au\|\leq M$ for some $M>0$.
A linear operator on $\mathscr{H}$ is \textbf{bounded} if for all $u$ such that $\|u\|\leq 1$, we have $\|Au\|\leq M$ for some $M>0$.
$\mathscr{P}$ is the set of all orthogonal projections on $\mathscr{B}(\mathscr{H})$.
The set $\mathscr{P}=\{P\in\mathscr{B}(\mathscr{H}):P^*=P=P^2\}$ is the set of all orthogonal projections on $\mathscr{B}(\mathscr{H})$.
\end{defn}
As a counterpart for the initial probability distribution in the classical probability theory, we need to define the state in the non-commutative probability theory.
Recall from classical probability theory, we call the initial probability distribution for possible outcomes in the classical probability theory as our \textit{state}, simillarly, we need to define the \textit{state} in the non-commutative probability theory.
\begin{defn}
\label{defn:state}
Non-commutative probability state:
A state on $(\mathscr{B}(\mathscr{H}),\mathscr{P})$ is a map $\rho:\mathscr{P}\to[0,1]$ such that:
A state on $(\mathscr{B}(\mathscr{H}),\mathscr{P})$ is a map $\rho:\mathscr{P}\to[0,1]$, (commonly named as density operator) such that:
\begin{itemize}
\item $\rho(O)=0$, where $O$ is the zero projection, and $\rho(I)=1$, where $I$ is the identity projection.
\item If $P_1,P_2,\ldots,P_n$ are pairwise disjoint orthogonal projections, then $\rho(P_1 + P_2 + \cdots + P_n) = \sum_{i=1}^n \rho(P_i)$.
@@ -294,26 +452,26 @@ We can write $\rho$ as
\]
(Under basis $|\psi_j\rangle$, it is a diagonal matrix with $p_j$ on the diagonal.)
Then we need to introduce a theorem that ensures that every state on the space of all orthogonal projections on $\mathscr{H}$ can be represented by a density operator.
% Then we need to introduce a theorem that ensures that every state on the space of all orthogonal projections on $\mathscr{H}$ can be represented by a density operator.
\begin{theorem}
\label{theorem:Gleason's_theorem}
Gleason's theorem (Theorem 1.1.15 in~\cite{parthasarathy2005mathematical})
% \begin{theorem}
% \label{theorem:Gleason's_theorem}
% Gleason's theorem (Theorem 1.1.15 in~\cite{parthasarathy2005mathematical})
Let $\mathscr{H}$ be a Hilbert space over $\mathbb{C}$ or $\mathbb{R}$ of dimension $n\geq 3$. Let $\mu$ be a state on the space $\mathscr{P}$ of projections on $\mathscr{H}$. Then there exists a unique density operator $\rho$ such that
\[
\mu(P)=\operatorname{Tr}(\rho P)
\]
for all $P\in\mathscr{P}$. $\mathscr{P}$ is the space of all orthogonal projections on $\mathscr{H}$.
\end{theorem}
% Let $\mathscr{H}$ be a Hilbert space over $\mathbb{C}$ or $\mathbb{R}$ of dimension $n\geq 3$. Let $\mu$ be a state on the space $\mathscr{P}$ of projections on $\mathscr{H}$. Then there exists a unique density operator $\rho$ such that
% \[
% \mu(P)=\operatorname{Tr}(\rho P)
% \]
% for all $P\in\mathscr{P}$. $\mathscr{P}$ is the space of all orthogonal projections on $\mathscr{H}$.
% \end{theorem}
This proof came from~\cite{parthasarathy2005mathematical}.
% This proof came from~\cite{parthasarathy2005mathematical}.
% \begin{proof}
% % TODO: FILL IN THE PROOF
% \end{proof}
This theorem is a very important theorem in non-commutative probability theory; it states that any state on the space of all orthogonal projections on $\mathscr{H}$ can be represented by a density operator.
% This theorem is a very important theorem in non-commutative probability theory; it states that any state on the space of all orthogonal projections on $\mathscr{H}$ can be represented by a density operator.
The counterpart of the random variable in the non-commutative probability theory is called an observable, which is a Hermitian operator on $\mathscr{H}$ (for all $\psi,\phi$ in the domain of $A$, we have $\langle A\psi,\phi\rangle=\langle\psi,A\phi\rangle$. This kind of operator ensures that our outcome interpreted as probability is a real number).
@@ -819,8 +977,6 @@ As the dimension of the Hilbert space increases, the chance of getting an almost
In Hayden's work, the result is also extended to the multiparty case~\cite{Hayden}, and the result is still under research and I will show the result in the final report if I have enough time.
\subsection{Majorana stellar representation of the quantum state}
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@@ -11,7 +11,10 @@
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.
\section{Observable diameters}
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@@ -39,7 +39,14 @@ We can use a symmetric group of permutations of $n$ complex numbers (or $S^2$) t
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}
\section{Space of complex valued functions and pure states}
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\printbibliography[title={References for Chapter 2}]
}
\end{document}

314
chapters/chap1.bib → main.bib
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@@ -1,158 +1,158 @@
@book{parthasarathy1992quantum,
title = {An Introduction to Quantum Stochastic Calculus},
author = {Parthasarathy, K. R.},
series = {Monographs in Mathematics},
volume = {85},
year = {1992},
publisher = {Birkh{\"a}user Basel},
doi = {10.1007/978-3-0348-8641-3},
isbn = {978-3-0348-9711-2},
eisbn = {978-3-0348-8641-3},
pages = {XI, 292},
topics = {Probability Theory and Stochastic Processes}
}
@book{Elizabeth_book,
title ={The Random Matrix Theory of the Classical Compact Groups},
author ={Elizabeth Meckes}
}
@book{parthasarathy2005mathematical,
title = {Mathematical Foundation of Quantum Mechanics},
author = {Parthasarathy, K. R.},
series = {Texts and Readings in Mathematics},
volume = {85},
year = {2005},
publisher = {Hindustan Book Agency},
doi = {10.1007/978-93-86279-28-6},
isbn = {978-93-86279-28-6},
eisbn = {978-93-86279-28-6},
pages = {XI, 292},
topics = {Mathematics, general}
}
@book{Vershynin_book,
title = {High-dimensional probability: an introduction with applications in data science},
author = {Vershynin, Roman},
year = {2018},
publisher = {Cambridge University Press},
doi = {10.1017/9781316278289},
isbn = {9781316278289},
eisbn = {9781316278289},
pages = {X, 368}
}
@inbook{kummer1998elements,
author = {B. Kümmer and H. Maassen},
title = {Elements of quantum probability},
booktitle = {Quantum Probability Communications},
chapter = {},
pages = {73-100},
doi = {10.1142/9789812816054_0003},
url = {https://www.worldscientific.com/doi/abs/10.1142/9789812816054_0003},
abstract = { Abstract This is an introductory article presenting some basic ideas of quantum probability. From a discussion of simple experiments with polarized light and a card game we deduce the necessity of extending the body of classical probability theory. For a class of systems, containing classical systems with finitely many states, a probabilistic model is developed. It can describe, in particular, the polarization experiments. Some examples of “quantum coin tosses” are discussed, closely related to V.F.R. Jones’ approach to braid group representations, to spin relaxation, and to nuclear magnetic resonance. In an appendix we indicate the steps which lead to the full mathematical model of quantum probability. }
}
@misc{Feres,
title = {Math 444 Lecture notes – the mathematics of quantum theory},
url = {https://www.math.wustl.edu/~feres/Math444Spring25/Math444Spring25Syllabus.html},
journal = {Math 444 – the mathematics of quantum theory},
author = {Feres, Renato}
}
@book{romanvershyni,
title = {High-dimensional probability: an introduction with applications in data science},
author = {Roman Vershynin},
year = {2018},
publisher = {Cambridge University Press}
}
@book{MGomolovs,
title = {Metric structures for Riemannian and non-Riemannian spaces},
author = {M. Gromov},
year = {1981},
publisher = {Birkhäuser}
}
@misc{shioya2014metricmeasuregeometry,
title={Metric measure geometry},
author={Takashi Shioya},
year={2014},
eprint={1410.0428},
archivePrefix={arXiv},
primaryClass={math.MG},
url={https://arxiv.org/abs/1410.0428},
}
@inproceedings{Hayden,
title = {Concentration of measure effects in quantum information},
author = {Hayden, Patrick},
booktitle = {Quantum Information Science and Its Contributions to Mathematics},
series = {Proceedings of Symposia in Applied Mathematics},
volume = {68},
pages = {211--260},
year = {2010},
publisher = {American Mathematical Society},
isbn = {978-0-8218-4828-9},
doi = {10.1090/psapm/068}
}
@article{Hayden_2006,
title={Aspects of Generic Entanglement},
volume={265},
ISSN={1432-0916},
url={http://dx.doi.org/10.1007/s00220-006-1535-6},
DOI={10.1007/s00220-006-1535-6},
number={1},
journal={Communications in Mathematical Physics},
publisher={Springer Science and Business Media LLC},
author={Hayden, Patrick and Leung, Debbie W. and Winter, Andreas},
year={2006},
month=mar, pages={95-117}
}
@book{Haar_book,
title = {The random Matrix Theory of the Classical Compact groups},
author = {E. M. S. Meckes},
year = {2013},
publisher = {Princeton University Press}
}
@book{Bengtsson_Życzkowski_2017,
title = {Geometry of Quantum States: An Introduction to Quantum Entanglement},
author = {Bengtsson, Ingemar and Życzkowski, Karol},
year = {2017},
publisher = {Cambridge University Press}
}
@article{Pages_conjecture,
title = {Page's conjecture},
author = {Page, Don N.},
journal = {Physical Review Letters},
}
@article{Pages_conjecture_simple_proof,
title = {Page's conjecture simple proof},
author = {Jorge Sanchez-Ruiz},
journal = {Departament de Fisica Fonamental, Universitat de Barcelona, Diagonal 6/7, 08028 Barcelona, Spain},
year = {1995},
journal = {Physical Review E},
}
@book{Nielsen_Chuang_2010,
place={Cambridge},
title={Quantum Computation and Quantum Information: 10th Anniversary Edition},
publisher={Cambridge University Press},
author={Nielsen, Michael A. and Chuang, Isaac L.},
year={2010}
}
@misc{gupta2015functionalanalysisquantuminformation,
title={The Functional Analysis of Quantum Information Theory},
author={Ved Prakash Gupta and Prabha Mandayam and V. S. Sunder},
year={2015},
eprint={1410.7188},
archivePrefix={arXiv},
primaryClass={quant-ph},
url={https://arxiv.org/abs/1410.7188},
@book{parthasarathy1992quantum,
title = {An Introduction to Quantum Stochastic Calculus},
author = {Parthasarathy, K. R.},
series = {Monographs in Mathematics},
volume = {85},
year = {1992},
publisher = {Birkh{\"a}user Basel},
doi = {10.1007/978-3-0348-8641-3},
isbn = {978-3-0348-9711-2},
eisbn = {978-3-0348-8641-3},
pages = {XI, 292},
topics = {Probability Theory and Stochastic Processes}
}
@book{Elizabeth_book,
title ={The Random Matrix Theory of the Classical Compact Groups},
author ={Elizabeth Meckes}
}
@book{parthasarathy2005mathematical,
title = {Mathematical Foundation of Quantum Mechanics},
author = {Parthasarathy, K. R.},
series = {Texts and Readings in Mathematics},
volume = {85},
year = {2005},
publisher = {Hindustan Book Agency},
doi = {10.1007/978-93-86279-28-6},
isbn = {978-93-86279-28-6},
eisbn = {978-93-86279-28-6},
pages = {XI, 292},
topics = {Mathematics, general}
}
@book{Vershynin_book,
title = {High-dimensional probability: an introduction with applications in data science},
author = {Vershynin, Roman},
year = {2018},
publisher = {Cambridge University Press},
doi = {10.1017/9781316278289},
isbn = {9781316278289},
eisbn = {9781316278289},
pages = {X, 368}
}
@inbook{kummer1998elements,
author = {B. Kümmer and H. Maassen},
title = {Elements of quantum probability},
booktitle = {Quantum Probability Communications},
chapter = {},
pages = {73-100},
doi = {10.1142/9789812816054_0003},
url = {https://www.worldscientific.com/doi/abs/10.1142/9789812816054_0003},
abstract = { Abstract This is an introductory article presenting some basic ideas of quantum probability. From a discussion of simple experiments with polarized light and a card game we deduce the necessity of extending the body of classical probability theory. For a class of systems, containing classical systems with finitely many states, a probabilistic model is developed. It can describe, in particular, the polarization experiments. Some examples of “quantum coin tosses” are discussed, closely related to V.F.R. Jones’ approach to braid group representations, to spin relaxation, and to nuclear magnetic resonance. In an appendix we indicate the steps which lead to the full mathematical model of quantum probability. }
}
@misc{Feres,
title = {Math 444 Lecture notes – the mathematics of quantum theory},
url = {https://www.math.wustl.edu/~feres/Math444Spring25/Math444Spring25Syllabus.html},
journal = {Math 444 – the mathematics of quantum theory},
author = {Feres, Renato}
}
@book{romanvershyni,
title = {High-dimensional probability: an introduction with applications in data science},
author = {Roman Vershynin},
year = {2018},
publisher = {Cambridge University Press}
}
@book{MGomolovs,
title = {Metric structures for Riemannian and non-Riemannian spaces},
author = {M. Gromov},
year = {1981},
publisher = {Birkhäuser}
}
@misc{shioya2014metricmeasuregeometry,
title={Metric measure geometry},
author={Takashi Shioya},
year={2014},
eprint={1410.0428},
archivePrefix={arXiv},
primaryClass={math.MG},
url={https://arxiv.org/abs/1410.0428},
}
@inproceedings{Hayden,
title = {Concentration of measure effects in quantum information},
author = {Hayden, Patrick},
booktitle = {Quantum Information Science and Its Contributions to Mathematics},
series = {Proceedings of Symposia in Applied Mathematics},
volume = {68},
pages = {211--260},
year = {2010},
publisher = {American Mathematical Society},
isbn = {978-0-8218-4828-9},
doi = {10.1090/psapm/068}
}
@article{Hayden_2006,
title={Aspects of Generic Entanglement},
volume={265},
ISSN={1432-0916},
url={http://dx.doi.org/10.1007/s00220-006-1535-6},
DOI={10.1007/s00220-006-1535-6},
number={1},
journal={Communications in Mathematical Physics},
publisher={Springer Science and Business Media LLC},
author={Hayden, Patrick and Leung, Debbie W. and Winter, Andreas},
year={2006},
month=mar, pages={95-117}
}
@book{Haar_book,
title = {The random Matrix Theory of the Classical Compact groups},
author = {E. M. S. Meckes},
year = {2013},
publisher = {Princeton University Press}
}
@book{Bengtsson_Życzkowski_2017,
title = {Geometry of Quantum States: An Introduction to Quantum Entanglement},
author = {Bengtsson, Ingemar and Życzkowski, Karol},
year = {2017},
publisher = {Cambridge University Press}
}
@article{Pages_conjecture,
title = {Page's conjecture},
author = {Page, Don N.},
journal = {Physical Review Letters},
}
@article{Pages_conjecture_simple_proof,
title = {Page's conjecture simple proof},
author = {Jorge Sanchez-Ruiz},
journal = {Departament de Fisica Fonamental, Universitat de Barcelona, Diagonal 6/7, 08028 Barcelona, Spain},
year = {1995},
journal = {Physical Review E},
}
@book{Nielsen_Chuang_2010,
place={Cambridge},
title={Quantum Computation and Quantum Information: 10th Anniversary Edition},
publisher={Cambridge University Press},
author={Nielsen, Michael A. and Chuang, Isaac L.},
year={2010}
}
@misc{gupta2015functionalanalysisquantuminformation,
title={The Functional Analysis of Quantum Information Theory},
author={Ved Prakash Gupta and Prabha Mandayam and V. S. Sunder},
year={2015},
eprint={1410.7188},
archivePrefix={arXiv},
primaryClass={quant-ph},
url={https://arxiv.org/abs/1410.7188},
}

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@@ -14,6 +14,7 @@
% add special notation supports
\usepackage[mathscr]{euscript}
\usepackage{mathtools}
\usepackage{braket}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% add image package and directory
@@ -21,7 +22,9 @@
\usepackage{tikz}
\graphicspath{{./images/}}
% dependency graph
\usetikzlibrary{positioning,arrows.meta}
\usetikzlibrary{trees,positioning,arrows.meta,backgrounds}
% floating graph
\usepackage{float}
% --- Bibliography: biblatex + biber ---
\usepackage[
@@ -33,9 +36,8 @@
% In the assembled book, we load *all* chapter bib files here,
% and print one combined bibliography at the end.
\addbibresource{chapters/chap1.bib}
\addbibresource{chapters/chap2.bib}
\addbibresource{chapters/chap3.bib}
\addbibresource{main.bib}
%%
% Some convenient commands if you need to use integrals
@@ -73,6 +75,7 @@
% Each chapter is in its own file and included as a subfile.
\subfile{preface}
\subfile{chapters/chap0}
\subfile{chapters/chap1}
\subfile{chapters/chap2}
\subfile{chapters/chap3}

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69
preface.tex
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@@ -1,24 +1,44 @@
% preface.tex
\documentclass[main.tex]{subfiles}
\ifSubfilesClassLoaded{
\addbibresource{main.bib}
}
\begin{document}
\chapter*{Preface}
\addcontentsline{toc}{chapter}{Preface}
\section*{Philosophy}
This book is intentionally small and opinionated:
\begin{itemize}
\item Each chapter is a self-contained unit you can compile on its own.
\item Definitions are prioritized over intuition when the two conflict.
\item Every theorem should have an explicit list of dependencies:
what results it uses, and what background it assumes.
\end{itemize}
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}
Think of the graph below as a diagnostic tool.
If something feels like a black box, locate the nearest node you don’t fully own yet,
then check the suggested “symptoms” under that node.
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
@@ -27,7 +47,7 @@ then check the suggested “symptoms” under that node.
\centering
\begin{tikzpicture}[
node distance=10mm and 18mm,
box/.style={draw, rounded corners, align=center, inner sep=4pt},
box/.style={draw, rectangle, fill=white, align=center, inner sep=4pt},
arrow/.style={-Latex}
]
@@ -40,22 +60,27 @@ then check the suggested “symptoms” under that node.
\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] (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);
\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}