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

Fermionic systems differ from bosonic ones in several ways, in particular the time-reversal operator $\mathcal{T}$ is odd, ${\mathcal{T}}^2=-\mathbb{1}$. For $\mathcal{PT}$-symmetric bosonic systems, the no-signaling principle and the quantum brachistochrone problem have been studied to some degree, both of them controversially. In this paper, we apply the basic methods proposed for bosonic systems [Y. Lee et al., Phys. Rev. Lett. 112, 130404 (2014); C. M. Bender et al., Phys. Rev. Lett. 98, 040403 (2007)] to fermionic two- and four-dimensional $\mathcal{PT}$-symmetric Hamiltonians and obtain several surprising results: We find, in contrast to the bosonic case, that the no-signaling principle is upheld for two-dimensional fermionic Hamiltonians; however, the $\mathcal{PT}$ symmetry is broken. In addition, we find that the time required for the evolution from a given initial state, the spin up, to a given final state, the spin down, is a constant, independent of the parameters of the Hamiltonian, under the eigenvalue constraint. That is, it cannot, as in the bosonic case, be optimized. We do, however, also find a dimensional dependence: Four-dimensional $\mathcal{PT}$-symmetric fermionic Hamiltonians considered here again uphold the no-signaling principle, but it is not essential that the $\mathcal{PT}$ symmetry be broken. The symmetry is, however, broken if the measure of entanglement is conserved. In the four-dimensional systems, the evolution time between orthogonal states is dependent on the parameters of the Hamiltonian, with the conclusion that it again can be optimized and approach zero under certain circumstances. However, if we require the conservation of entanglement, the transformation time between these two states becomes the same constant as that found in the two-dimensional case, which coincides with the minimum time for such a transformation to take place in the Hermitian case.