Let $\left|\alpha_{0}\right\rangle \neq\left|\alpha_{1}\right\rangle$ be two quantum states and let $p_{0}$ and $p_{1}$ be associated probabilities with $p_{0}+p_{1}=1, p_{0} \neq 0, p_{1} \neq 0$ and $p_{0} \geqslant p_{1}$. Alice chooses state $\left|\alpha_{i}\right\rangle$ with probability $p_{i}$ and sends it to Bob. Upon receiving it, Bob performs a 2-outcome measurement $\mathcal{M}$ with outcomes labelled 0 and 1 , in an attempt to identify which state Alice sent.

(a) By using the extremal property of eigenvalues, or otherwise, show that the operator $D=p_{0}\left|\alpha_{0}\right\rangle\left\langle\alpha_{0}\left|-p_{1}\right| \alpha_{1}\right\rangle\left\langle\alpha_{1}\right|$ has exactly two nonzero eigenvalues, one of which is positive and the other negative.

(b) Let $P_{S}$ denote the probability that Bob correctly identifies Alice's sent state. If the measurement $\mathcal{M}$ comprises orthogonal projectors $\left\{\Pi_{0}, \Pi_{1}\right\}$ (corresponding to outcomes 0 and 1 respectively) give an expression for $P_{S}$ in terms of $p_{1}, \Pi_{0}$ and $D$.

(c) Show that the optimal success probability $P_{S}^{\text {opt }}$, i.e. the maximum attainable value of $P_{S}$, is

$$P_{S}^{\mathrm{opt}}=\frac{1+\sqrt{1-4 p_{0} p_{1} \cos ^{2} \theta}}{2}$$

where $\cos \theta=\left|\left\langle\alpha_{0} \mid \alpha_{1}\right\rangle\right|$.

(d) Suppose we now place the following extra requirement on Bob's discrimination process: whenever Bob obtains output 0 then the state sent by Alice was definitely $\left|\alpha_{0}\right\rangle$. Show that Bob's $P_{S}^{\mathrm{opt}}$ now satisfies $P_{S}^{\mathrm{opt}} \geqslant 1-p_{0} \cos ^{2} \theta$.