The Pauli matrices $\boldsymbol{\sigma}=\left(\sigma_{x}, \sigma_{y}, \sigma_{z}\right)=\left(\sigma_{1}, \sigma_{2}, \sigma_{3}\right)$, with

$$\sigma_{1}=\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}\right), \quad \sigma_{2}=\left(\begin{array}{cc} 0 & -i \\ i & 0 \end{array}\right), \quad \sigma_{3}=\left(\begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array}\right)$$

are used to represent angular momentum operators with respect to basis states $|\uparrow\rangle$ and $|\downarrow\rangle$ corresponding to spin up and spin down along the $z$-axis. They satisfy

$$\sigma_{i} \sigma_{j}=\delta_{i j}+i \epsilon_{i j k} \sigma_{k}$$

(i) How are $|\uparrow\rangle$ and $|\downarrow\rangle$ represented? How is the spin operator s related to $\sigma$ and $\hbar$ ? Check that the commutation relations between the spin operators are as desired. Check that $\mathbf{s}^{2}$ acting on a spin one-half state has the correct eigenvalue.

What are the states obtained by applying $s_{x}, s_{y}$ to the eigenstates $|\uparrow\rangle$ and $|\downarrow\rangle$ of $s_{z}$ ?

(ii) Let $V$ be the space of states for a spin one-half system. Consider a combination of three such systems with states belonging to $V^{(1)} \otimes V^{(2)} \otimes V^{(3)}$ and spin operators acting on each subsystem denoted by $s_{x}^{(i)}, s_{y}^{(i)}$ with $i=1,2,3$. Find the eigenvalues of the operators

$$s_{x}^{(1)} s_{y}^{(2)} s_{y}^{(3)}, \quad s_{y}^{(1)} s_{x}^{(2)} s_{y}^{(3)}, \quad s_{y}^{(1)} s_{y}^{(2)} s_{x}^{(3)} \quad \text { and } \quad s_{x}^{(1)} s_{x}^{(2)} s_{x}^{(3)}$$

of the state

$$|\Psi\rangle=\frac{1}{\sqrt{2}}\left[|\uparrow\rangle_{1}|\uparrow\rangle_{2}|\uparrow\rangle_{3}-|\downarrow\rangle_{1}|\downarrow\rangle_{2}|\downarrow\rangle_{3}\right]$$

(iii) Consider now whether these outcomes for measurements of particular combinations of the operators $s_{x}^{(i)}, s_{y}^{(i)}$ in the state $|\Psi\rangle$ could be reproduced by replacing the spin operators with classical variables $\tilde{s}_{x}^{(i)}, \tilde{s}_{y}^{(i)}$ which take values $\pm \hbar / 2$ according to some probabilities. Assume that these variables are identical to the quantum measurements of $s_{x}^{(1)} s_{y}^{(2)} s_{y}^{(3)}, s_{y}^{(1)} s_{x}^{(2)} s_{y}^{(3)}, s_{y}^{(1)} s_{y}^{(2)} s_{x}^{(3)}$ on $|\Psi\rangle$. Show that classically this implies a unique possibility for

$$\tilde{s}_{x}^{(1)} \tilde{s}_{x}^{(2)} \tilde{s}_{x}^{(3)},$$

and find its value.

State briefly how this result could be used to experimentally test quantum mechanics.