Alice wishes to communicate to Bob a 1-bit message $m=0$ or $m=1$ chosen by her with equal prior probabilities $1 / 2$. For $m=0$ (respectively $m=1$ ) she sends Bob the quantum state $\left|a_{0}\right\rangle$ (respectively $\left|a_{1}\right\rangle$ ). On receiving the state, Bob applies quantum operations to it, to try to determine Alice's message. The Helstrom-Holevo theorem asserts that the probability $P_{S}$ for Bob to correctly determine Alice's message is bounded by $P_{S} \leqslant \frac{1}{2}(1+\sin \theta)$, where $\theta=\cos ^{-1}\left|\left\langle a_{0} \mid a_{1}\right\rangle\right|$, and that this bound is achievable.

(a) Suppose that $\left|a_{0}\right\rangle=|0\rangle$ and $\left|a_{1}\right\rangle=\frac{1}{\sqrt{2}}(|0\rangle+|1\rangle)$, and that Bob measures the received state in the basis $\left\{\left|b_{0}\right\rangle,\left|b_{1}\right\rangle\right\}$, where $\left|b_{0}\right\rangle=\cos \beta|0\rangle+\sin \beta|1\rangle$ and $\left|b_{1}\right\rangle=$ $-\sin \beta|0\rangle+\cos \beta|1\rangle$, to produce his output 0 or 1 , respectively. Calculate the probability $P_{S}$ that Bob correctly determines Alice's message, and show that the maximum value of $P_{S}$ over choices of $\beta \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right]$ achieves the Helstrom-Holevo bound.

(b) State the no-cloning theorem as it applies to unitary processes and a set of two non-orthogonal states $\left\{\left|c_{0}\right\rangle,\left|c_{1}\right\rangle\right\}$. Show that the Helstrom-Holevo theorem implies the validity of the no-cloning theorem in this situation.