= Obstruction to uniform probability lowering by a unitary
{title2=$\operatorname{Tr}(PUI_NU^\dagger/N)=k/N$}
Let $P$ be a rank-$k$ projector with $0<k<N$. No <unitary operator> can map every pure state whose $P$ probability equals $k/N$ to a state with the same fixed smaller probability. Average $\sqrt{k/N}|g\rangle\pm\sqrt{1-k/N}|b\rangle$ over good and bad basis vectors and both signs. The average <density operator> is the <maximally mixed state>, so every unitary preserves its average $P$ probability. This contradicts a strictly smaller probability for all states in the ensemble. A preparation tailored to one known starting state is a different resource and is used in <exact amplitude amplification>.
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