Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 10E (Quantum Information and Computation)

The quantum no-signalling principle says Alice's uncommunicated local choice cannot alter Bob's reduced state. Expanding the two tensor products cancels the cross terms and givesAlice obtains either result with probability $1/2$; conditionally Bob has probability ${\cos }^2\theta$ or ${\sin }^2\theta$ of $0$, respectively. Averaging,independent of $\theta$, exactly as no-signalling requires.

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.