Simon coset state ($(|x⟩+|x\oplus s⟩)/\sqrt{2}$)

In Simon's algorithm, conditioning on a function value leaves the first quantum register in an equal coherent sum of its two preimages. This is a pure state, even though discarding the second quantum register without retaining its outcome generally leaves a mixed state. The label $x$ depends on the measured value; the period $s$ does not.