From d4df6b37c9c76f602e95896c99f1063d926bd08b Mon Sep 17 00:00:00 2001 From: Zheyuan Wu <60459821+Trance-0@users.noreply.github.com> Date: Mon, 19 Jan 2026 21:00:36 -0600 Subject: [PATCH] update readme --- README.md | 23 +++++++++++++++++++++++ chapters/chap1.pdf | Bin 572623 -> 572227 bytes 2 files changed, 23 insertions(+) diff --git a/README.md b/README.md index 7615456..c1c24ed 100644 --- a/README.md +++ b/README.md @@ -1,2 +1,25 @@ # 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. 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