  Even though the time-reversal is unphysical (it corresponds to the complex
conjugation of the density matrix), for some restricted set of states it can be
achieved unitarily, typically when there is a common de-phasing in a n-level
system. However, in the presence of multiple phases (i. e. a different
de-phasing for each element of an orthogonal basis occurs) the time reversal is
no longer physically possible. In this paper we derive the channel which
optimally approaches in fidelity the time-reversal of multi-phase equatorial
states in arbitrary (finite) dimension. We show that, in contrast to the
customary case of the Universal-NOT on qubits (or the universal conjugation in
arbitrary dimension), the optimal phase covariant time-reversal for equatorial
states is a nonclassical channel, which cannot be achieved via a
measurement/preparation procedure. Unitary realizations of the optimal
time-reversal channel are given with minimal ancillary dimension, exploiting
the simplex structure of the optimal maps.
