(a) Suppose that Alice and Bob are distantly separated in space and each has one qubit of the 2-qubit state $\left|\phi^{+}\right\rangle=\frac{1}{\sqrt{2}}(|00\rangle+|11\rangle)$. They also have the ability to perform local unitary quantum operations and local computational basis measurements, and to communicate only classically. Alice has a 1-qubit state $|\alpha\rangle$ (whose identity is unknown to her) which she wants to communicate to Bob. Show how this can be achieved using only the operational resources, listed above, that they have available.

Suppose now that a third party, called Charlie, joins Alice and Bob. They are all mutually distantly separated in space and each holds one qubit of the 3-qubit state

$$|\gamma\rangle=\frac{1}{\sqrt{2}}(|000\rangle+|111\rangle) .$$

As previously with Alice and Bob, they are able to communicate with each other only classically, e.g. by telephone, and they can each also perform only local unitary operations and local computational basis measurements. Alice and Bob phone Charlie to say that they want to do some quantum teleportation and they need a shared $\left|\phi^{+}\right\rangle$state (as defined above). Show how Charlie can grant them their wish (with certainty), given their joint possession of $|\gamma\rangle$ and using only their allowed operational resources. [Hint: It may be useful to consider application of an appropriate Hadamard gate action.]

(b) State the quantum no-signalling principle for a bipartite state $|\psi\rangle_{A B}$ of the composite system $A B$.

Suppose we are given an unknown one of the two states

$$\begin{aligned} &\left|\phi^{+}\right\rangle_{A B}=\frac{1}{\sqrt{2}}\left(|00\rangle_{A B}+|11\rangle_{A B}\right) \\ &\left|\phi^{-}\right\rangle_{A B}=\frac{1}{\sqrt{2}}\left(|00\rangle_{A B}-|11\rangle_{A B}\right) \end{aligned}$$

and we wish to identify which state we have. Show that the minimum error probability for this state discrimination task is zero.

Suppose now that we have access only to qubit $B$ of the received state. Show that we can now do no better in the state discrimination task than just making a random guess as to which state we have.