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Fermionic systems differ from bosonic ones in several ways, in particularthat the time-reversal operator $T$ is odd, $T^2=-1$. For $PT$-symmetricbosonic systems, the no-signaling principle and the quantum brachistochroneproblem have been studied to some degree, both of them controversially. In thispaper, we apply the basic methods proposed for bosonic systems to {\itfermionic} two- and four-dimensional $PT$-symmetric Hamiltonians, and obtainseveral surprising results: We find - in contrast to the bosonic case - thatthe no-signaling principle is upheld for two-dimensional fermionicHamiltonians, however, the $PT$ symmetry is broken. In addition, we find thatthe time required for the evolution from a given initial state, the spin-up, toa given final state, the spin-down, is a constant, independent of theparameters of the Hamiltonian, under the eigenvalue constraint. That is, itcannot - as in the bosonic case - be optimized. We do, however, also find adimensional dependence: four-dimensional $PT$-symmetric fermionic Hamiltoniansconsidered here again uphold the no-signaling principle, but it is notessential that the $PT$ symmetry be broken. The symmetry is, however, broken ifthe measure of entanglement is conserved. In the four-dimensional systems, theevolution time between orthogonal states is dependent on the parameters of theHamiltonian, with the conclusion that it again can be optimized, and approachzero under certain circumstances. However, if we require the conservation ofentanglement, the transformation time between these two states becomes the sameconstant as found in the two-dimensional case, which coincides with the minimumtime for such a transformation to take place in the Hermitian case.

No-signaling principle and quantum brachistochrone problem in $PT$-symmetric fermionic two- and four-dimensional models