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Choose an orthonormal eigenbasis of $V$, with eigenphases ${\varphi }_0,{\varphi }_1$, counting multiplicity. The maximally mixed state is basis independent, soLinearity means that the experiment is the equal mixture of the two eigenstate experiments. After the inverse quantum Fourier transform, the joint state is $\frac{1}{2}{\sum }_{j=0}^1|{\chi }_j⟩⟨{\chi }_j|\otimes |v_j⟩⟨v_j|$, where $|{\chi }_j⟩={\sum }_m{a}_m({\varphi }_j)|m⟩$ andThus the exact output distribution for quantum phase estimation with a maximally mixed target isFor a nonzero phase mismatch, the squared probability amplitude is ${\sin }^2[L(\varphi -2\pi m/L)/2]/\{L^2{\sin }^2[(\varphi -2\pi m/L)/2]\}$, with value one at an exact match. If both eigenphases are exactly representable on the $L$-point grid, their two labels occur with probability $1/2$ each; coincident labels have their probabilities added. For general eigenphases there can be nonzero probabilities at other labels, so it would not be exact to list only the two best approximations with probability $1/2$ each. Each eigenphase is sampled with spectral weight $1/2$, and its nearest-label event has conditional probability at least $4/{\pi }^2$. In particular a nearest label for one eigenphase has unconditional probability at least $2/{\pi }^2$, with additional contributions possible from the other phase.