Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 10E / b / Solution

The Helstrom-Holevo bound for equal priors isPerfect cloning twice changes overlap $c$ to $c^2$, so discrimination would succeed with $\frac{1}{2}(1+\sqrt{1-c^4})$, strictly exceeding $\frac{1}{2}(1+\sqrt{1-c^2})$ for $0<c<1$. For $|0⟩,|+⟩$, $c=1/\sqrt{2}$, producing $\frac{1}{2}(1+\sqrt{3}/2)>\frac{1}{2}(1+1/\sqrt{2})$, a contradiction.