= Solution
Choose an <orthonormal> eigenbasis of $V$, with <eigenphases> $\phi_0,\phi_1$, counting multiplicity. The <maximally mixed state> is <basis> independent, so
$$
\rho=\frac12|v_0\rangle\langle v_0|+\frac12|v_1\rangle\langle v_1|.
$$
<Linearity> means that the experiment is the equal mixture of the two <eigenstate> experiments. After the inverse <quantum Fourier transform>, the joint state is $\frac12\sum_{j=0}^1|\chi_j\rangle\langle\chi_j|\otimes|v_j\rangle\langle v_j|$, where $|\chi_j\rangle=\sum_m a_m(\phi_j)|m\rangle$ and
$$
a_m(\phi)=\frac1L\sum_{x=0}^{L-1}e^{ix(\phi-2\pi m/L)}.
$$
Thus the exact output distribution for <quantum phase estimation with a maximally mixed target> is
$$
\boxed{\Pr(m)=\frac12|a_m(\phi_0)|^2+\frac12|a_m(\phi_1)|^2,\qquad m=0,\ldots,L-1.}
$$
For a nonzero phase mismatch, the squared <probability amplitude> is $\sin^2[L(\phi-2\pi m/L)/2]/\{L^2\sin^2[(\phi-2\pi m/L)/2]\}$, 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.
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