$$\left|\beta_{x z}\right\rangle=\left(Z^{z} X^{x}\right) \otimes I\left(\frac{|00\rangle+|11\rangle}{\sqrt{2}}\right)$$

where $X$ and $Z$ are the standard qubit Pauli operations and $x, z \in\{0,1\}$.

(a) For any 1-qubit state $|\alpha\rangle$ show that the 3 -qubit state $|\alpha\rangle_{C}\left|\beta_{00}\right\rangle_{A B}$ of system $C A B$ can be expressed as

$$|\alpha\rangle_{C}\left|\beta_{00}\right\rangle_{A B}=\frac{1}{2} \sum_{x, z=0}^{1}\left|\beta_{x z}\right\rangle_{C A}\left|\mu_{x z}\right\rangle_{B}$$

where the 1 -qubit states $\left|\mu_{x z}\right\rangle$ are uniquely determined. Show that $\left|\mu_{10}\right\rangle=X|\alpha\rangle$.

(b) In addition to $\left|\mu_{10}\right\rangle=X|\alpha\rangle$ you may now assume that $\left|\mu_{x z}\right\rangle=X^{x} Z^{z}|\alpha\rangle$. Alice and Bob are separated distantly in space and share a $\left|\beta_{00}\right\rangle_{A B}$ state with $A$ and $B$ labelling qubits held by Alice and Bob respectively. Alice also has a qubit $C$ in state $|\alpha\rangle$ whose identity is unknown to her. Using the results of part (a) show how she can transfer the state of $C$ to Bob using only local operations and classical communication, i.e. the sending of quantum states across space is not allowed.

(c) Suppose that in part (b), while sharing the $\left|\beta_{00}\right\rangle_{A B}$ state, Alice and Bob are also unable to engage in any classical communication, i.e. they are able only to perform local operations. Can Alice now, perhaps by a modified process, transfer the state of $C$ to Bob? Give a reason for your answer.