Unitary and anti-unitary quantum description of the classical not gate

Two possible quantum descriptions of the classical Not gate are investigated in the framework of the Hilbert space C: the unitary and the anti–unitary operator realizations. The two cases are distinguished interpreting the unitary Not as a quantum realization of the classical gate which on a fixed orthogonal pair of unit vectors, realizing once for all the classical bits 0 and 1, produces the required transformations 0 → 1 and 1 → 0 (i.e., logical quantum Not). The anti–unitary Not is a quantum realization of a gate which acts as a classical Not on any pair of mutually orthogonal vectors, each of which is a potential realization of the classical bits (i.e., universal quantum Not). Finally, we consider the unitary and the anti–unitary operator realizations of two important genuine quantum gates that transform elements of the computational basis of C into superpositions: the square root of the identity and the square root of the negation.