Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 58 / 3 / d / Solution

Continue with equal prior probabilities as in part (c). The two Bloch vectors areTheir density-matrix difference is $\Delta =\sin \theta \cos \phi {\sigma }_x$. If a binary measurement effect $E$ declares the first state, its success isDiagonalize $\Delta$. Because $0\le E\le I$, the trace is maximized by taking $E$ to be its positive-eigenvalue projector. Thus perform a Pauli measurement of ${\sigma }_x$, equivalently measure in $(|0⟩\pm |1⟩)/\sqrt{2}$. If $\cos \phi >0$, identify the first state on the positive outcome and the second on the negative; reverse the labels if $\cos \phi <0$. The optimum isFor $\cos \phi =0$ the states coincide, so the best probability is $1/2$. This explicit positive-eigenspace optimization proves the Helstrom measurement for two pure states result in the present case, rather than only quoting its name.