# HonorThesis

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.

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, 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 $d_B\geq d_A\geq 3$.

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.

> [!TIPS]
>
> To compile the bibliography for each chapter, run
>
> ```bash
biber chapters/chap1
pdflatex chapers/chap1.tex
> ```