 Solution

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