updates

content/Math401/Extending_thesis/Math401_R4.md

@@ -4,14 +4,51 @@
 
 This part may not be a part of "mathematical" research. But that's what I initially begin with.
 
-## Superdense coding
-
 > [!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.