Past exam of the computer science course of the University of Oxford / 2023 / Quantum information / 1 / b / ii

Write the proposed quantum state ensemble as $\{(|u⟩,p),(|v⟩,1-p)\}$ with normalized pure states, so thatTaking the expectation values in $|u⟩$ and $|v⟩$, and setting $c=|⟨u|v⟩{|}^2$, givesConsequently $p\le 1/2$ and $p\ge 1/2$. Substitution then gives $c=0$. The two pure states are therefore orthogonal, with equal probabilities:Conversely, any orthonormal basis of a qubit has projectors summing to $I$, so such a quantum state ensemble does give $I/2$. This proves the characterization of two-state ensembles of a maximally mixed qubit.