= Quantum phase estimation with a maximally mixed target
{title2=$p(m)=d^{-1}\sum_j|a_m(\phi_j)|^2$}
For a $d$-dimensional <unitary operator> $V=\sum_{j=1}^de^{i\phi_j}|v_j\rangle\langle v_j|$, a <maximally mixed state> $I/d$ on its target is the equal mixture of these <orthonormal> <eigenvectors>. Therefore <quantum phase estimation> with register size $L$ produces label $m$ with <probability>
$$
p(m)=\frac1d\sum_{j=1}^d|a_m(\phi_j)|^2,\qquad a_m(\phi)=\frac1L\sum_{x=0}^{L-1}e^{ix(\phi-2\pi m/L)}.
$$
This follows by <linearity> of <unitary time evolution> and the <Born rule> on the spectral mixture. If every <eigenphase> has an exact $L$-grid label, that label has <probability> equal to its multiplicity divided by $d$. Otherwise the exact distribution has tails, and the nearest labels are guaranteed only the <nearest-integer success bound for quantum phase estimation> weighted by the corresponding spectral <probabilities>.
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