Alice and Bob are separated in space and can communicate only over a noiseless public classical channel, i.e. they can exchange bit string messages perfectly, but the messages can be read by anyone. An eavesdropper Eve constantly monitors the channel, but cannot alter any passing messages. Alice wishes to communicate an $m$-bit string message to Bob whilst keeping it secret from Eve.

(a) Explain how Alice can do this by the one-time pad method, specifying clearly any additional resource that Alice and Bob need. Explain why in this method, Alice's message does, in fact, remain secure against eavesdropping.

(b) Suppose now that Alice and Bob do not possess the additional resource needed in part (a) for the one-time pad, but that they instead possess $n$ pairs of qubits, where $n \gg 1$, with each pair being in the state

$$|\psi\rangle_{A B}=t|00\rangle_{A B}+s|11\rangle_{A B},$$

where the real parameters $(t, s)$ are known to Alice and Bob and obey $t>s>0$ and $t^{2}+s^{2}=1$. For each qubit pair in state $|\psi\rangle_{A B}$, Alice possesses qubit $A$ and Bob possesses qubit $B$. They each also have available a supply of ancilla qubits, each in state $|0\rangle$, and they can each perform local quantum operations on qubits in their possession.

Show how Alice, using only local quantum operations, can convert each $|\psi\rangle_{A B}$ state into $\left|\phi^{+}\right\rangle_{A B}=\frac{1}{\sqrt{2}}\left(|00\rangle_{A B}+|11\rangle_{A B}\right)$ by a process that succeeds with non-zero probability. [Hint: It may be useful for Alice to start by adjoining an ancilla qubit $|0\rangle_{A^{\prime}}$ and work locally on her two qubits in $\left.|0\rangle_{A^{\prime}}|\psi\rangle_{A B \cdot}\right]$

Hence, or otherwise, show how Alice can communicate a bit string of expected length $\left(2 s^{2}\right) n$ to Bob in a way that keeps it secure against eavesdropping by Eve.