By considering the behaviour of two indistinguishable and identical quantum particles under an exchange operation, show that the particles must fall into one of two classes of particles, where these two classes correspond to those particles with symmetric or anti-symmetric wavefunctions. Finally, state which of these particle classes represent the bosons and which represents the fermions.

Consider the degenerate Hamiltonian

$H_{0}^{1}=\begin{bmatrix} 0&0&0 \\ 0&0&0 \\ 0&0&1 \end{bmatrix},$

on which we can apply transformations of the form $\mathbf{U(z)}=U_{1}(z_{1})U_{2}(z_{2})$ with $U_{\alpha}(z_{\alpha})=\text{exp}(z_{\alpha}\left|\alpha\right> \left<\tilde{2}\right|-\bar{z_{\alpha}}\left|\tilde{2}\right> \left<\alpha\right|)$ for $\alpha=1,2$ and $z_{\alpha}=\theta_{\alpha}\text{exp}(i\phi_{\alpha})$. Find the Hamiltonian $H$ given by $H=\mathbf{U(z)}H_{0}^{1}\mathbf{U^{\dagger}(z)}$, which is the correct Hamiltonian given in the solutions for the first question of the chapter 2 exercises. For more information on the context of this question and how to answer it see the research paper Quantum Computation by Geometrical Means by Jiannis K. Pachos.

In question 4 of chapter 3 in the book you were asked to find the operations that would transform the $\left|\psi\right>=\left|00\right>$ state into the maximally entangled state, $\left|\psi\right>=\frac{1}{\sqrt{2}}(\left|00\right>+\left|11\right>)$. The solution to which was given in Dirac notation, but this question can also be solved using matrices. Solve the same problem using matrices (or using Dirac notation if you used matrices originally).