Perfect NOT and conjugate transformations

We obtain the perfect NOT transformation criteria, two necessary and sufficient conditions for realizing a perfect NOT transformation on a quantum state set S of a qubit. Furthermore, we discuss a probabilistic perfect NOT transformation when there is no perfect NOT transformation on a state set S. We also generalize the perfect NOT transformation to the conjugate transformation in the multi-level quantum system and a lower bound of the best possible efficiencies attained by a probabilistic perfect conjugate transformation are obtained.