Past exam of the mathematics course of the University of Cambridge / 2019 / ii / Paper 2 / 15D (Quantum Information and Computation)

The operator $D$ vanishes on the orthogonal complement of $S=\mathrm{span}\{|{\alpha }_0⟩,|{\alpha }_1⟩\}$, so it has rank at most two. The states are distinct rays, hence they are linearly independent and $S$ has dimension two. In an orthonormal basis of $S$, its determinant isThus its two eigenvalues on $S$ are nonzero and have opposite signs. All remaining eigenvalues are zero.

Since ${\Pi }_1=I-{\Pi }_0$, the success probability is

By the Rayleigh-Ritz variational principle, $\mathrm{tr}({\Pi }_{0}D)$ is maximized by taking ${\Pi }_0$ to project onto the positive-eigenvalue eigenspace of $D$. Put $|⟨{\alpha }_0|{\alpha }_1⟩|=\cos \theta$. On $S$, the characteristic polynomial of $D$ isso its positive eigenvalue isPart (b) then gives the Helstrom-Holevo bound

Let $|{\alpha }_1^{\perp }⟩$ be a unit vector in $S$ orthogonal to $|{\alpha }_1⟩$, and chooseOutcome zero is then impossible when Alice sent $|{\alpha }_1⟩$, so it identifies $|{\alpha }_0⟩$ without error. This measurement succeeds with probabilityThe optimum under the extra requirement is at least the performance of this admissible measurement, so