# Math 401, Fall 2025: Thesis notes, R4, Superdense coding and Quantum error correcting codes

> Progress: 0/NaN=NaN% (denominator and enumerator may change)

This part may not be a part of "mathematical" research. But that's what I initially begin with.

> [!TIP]
>
> A helpful resource is [The Functional Analysis of Quantum Information Theory](https://arxiv.org/pdf/1410.7188) Section 2.2
>
> Or another way in quantum computing [Quantum Computing and Quantum Information](https://www.cambridge.org/highereducation/books/quantum-computation-and-quantum-information/01E10196D0A682A6AEFFEA52D53BE9AE#overview) Section 2.3

## References to begin with

### Quantum computing and quantum information

Every quantum bit is composed of two orthogonal states, denoted by $|0\rangle$ and $|1\rangle$.

Each state

$$
\varphi=\alpha|0\rangle+\beta|1\rangle
$$

where $\alpha$ and $\beta$ are complex numbers, and $|\alpha|^2+|\beta|^2=1$.

### Logic gates

All the logic gates are unitary operators in $\mathbb{C}^{2\times 2}$.

Example: the NOT gate is represented by the following matrix:

$$
NOT=\begin{pmatrix}
0 & 1 \\
1 & 0
\end{pmatrix}
$$

Hadamard gate is represented by the following matrix:

$$
H=\frac{1}{\sqrt{2}}\begin{pmatrix}
1 & 1 \\
1 & -1
\end{pmatrix}
$$

## Superdense coding


## Quantum error correcting codes

This part is intentionally left blank and may be filled near the end of the semester, by assignments given in CSE5313.
