Math 401, Fall 2025: Overview of thesis

This is a note base on first discussion with Prof. Feres on 2025-09-02

Due to time constraint, our goal for this semester is to extend the study of concentration of measure effects described by Hayden’s paper to Majorana stellar representation of quantum states.

That is, we want to build connection between the system described by follows:

Bounding the entropy of the state via Levy’s concentration theorem and Page’s lemma

Recall that the bipartite quantum states of $\mathcal{P}(A\otimes B)$. Assume $A$ has $\dim A=d_A$, $\dim B=d_B$, the the system is isomorphic to the complex projective space $\mathbb{C}P^{d_A d_B-1}$.

Then over partial trace operations over $B$, we can obtain a mixed quantum state denoted by $S_A$ on the Hilbert space $A$.

Then we measure the von Neumann entropy of $S_A$ to get the entropy of the state.

From the Hayden’s work, using analysis of Levy’s concentration theorem, and Page’s lemma, we can find that the entropy of the state is concentrated around a certain value which is close to maximally entangled state.

This project is incomplete due to several critical missing parts that I don’t have comprehensive knowledge to fill in.

One goal for this section of study is to fully investigate the missing parts and fill in the gaps. It is irrelevant to any one except me for trivial reasons. But I don’t want to speak anything that I don’t have a good understanding of.

To achieve this goal, I will set up few side project that continue to investigate the missing parts, and the notes will start with letter R, for recollections.

To make these sections self-contained. Some materials will be borrowed from other notes.

Bounding the entropy of the state via exploring Majorana stellar representation of quantum states

As Professor Feres mentioned, we can further explore the Majorana stellar representation of quantum states to bound the entropy of the state.

The new topics discovered will be noted with letter S. for stellar representation.