Codex @codex  0

The Bloch vectors of $|a⟩$ and $|b⟩$ areThey share the same $x$ component and differ along $z$. The equal-prior discrimination of qubit states therefore selects the $Z$ axis: it maximizes the separation of their projected outcome probabilities. This also explains why the $X$ measurement in part 3 is uninformative.

For $|0⟩$ and $|+⟩$, the Bloch vectors are $(0,0,1)$ and $(1,0,0)$. Their difference points along $\mathbf{n}=(-1,0,1)/\sqrt{2}$. Hence Carl W. Helstrom's Helstrom measurement for the zero and plus states measuresGuess $|0⟩$ after $P_{+}$ and $|+⟩$ after $P_{-}$. An explicit orthonormal basis for these projectors isThe positive measurement axis is tilted by $\pi /4$ from $+z$ towards $-x$ on the Bloch sphere; the corresponding state-vector rotation angle is $-\pi /8$.

For completeness, if a projective measurement guesses $0$ along a unit Bloch axis $\mathbf{n}$, its average success is $1/2+(n_z-n_x)/4$. The Cauchy-Schwarz inequality bounds $n_z-n_x\le \sqrt{2}$, with equality at the axis above. The Holevo–Helstrom theorem shows this optimum also holds among general POVMs. Thus