Recently Bender, Brody, Jones and Meister found that in the quantum brachistochrone problem the passage time needed for the evolution of certain initial states into specified final states can be made arbitrarily small, when the time-evolution operator is taken to be non-Hermitian but -symmetric. Here we demonstrate that such phenomena can also be obtained for non-Hermitian Hamiltonians for which -symmetry is completely broken, i.e. dissipative systems. We observe that the effect of a tunable passage time can be achieved by projecting between orthogonal eigenstates by means of a time-evolution operator associated with a non-Hermitian Hamiltonian. It is not essential that this Hamiltonian is -symmetric.
Recently, Bender et al. have considered the quantum brachistochrone problem for the non-Hermitian PT -symmetric quantum system and have shown that the optimal time evolution required to transform a given initial state |ψ i into a specific final state |ψ f can be made arbitrarily small.Additionally, it has been shown that finding the shortest possible time requires only the solution of the two-dimensional problem for the quantum system governed by the effective Hamiltonian acting in the subspace spanned by |ψ i and |ψ f .In this paper, we study a similar problem for the generic non-Hermitian Hamiltonian, focusing our attention on the geometric aspects of the problem.
Recently the quantum brachistochrone problem has been considered for non-Hermitian
PT-symmetric quantum system, and it has been shown that the optimal evolution time required to transform a given initial state
∣ψi> into specific final state
∣ψf> can be made arbitrary small. Here we study the similar problem for generic non-Hermitian Hamiltonian and find that the same answer can be obtained. We show that at the exceptional point the distance between
∣ψi> and
∣ψf> becomes virtually vanishing and quantum tunneling occurring at the exceptional point yields the shortest timed required to evolve from the initial state
∣ψi> into the final state
∣ψf>. These results may have applications in holonomic quantum computation and quantum information processing.
A non-Hermitian operator with a real spectrum and a complete set of eigenvectors may serve as the Hamiltonian operator for a unitary quantum system provided that one makes an appropriate choice for the defining the inner product of physical Hilbert state. We study the consequences of such a choice for the representation of states in terms of projection operators and the geometry of the state space. This allows for a careful treatment of the quantum Brachistochrone problem and shows that it is indeed impossible to achieve faster unitary evolutions using PT-symmetric or other non-Hermitian Hamiltonians than those given by Hermitian Hamiltonians.
Fermionic systems differ from bosonic ones in several ways, in particular the time-reversal operator
T is odd, ${\mathcal{T}}^{2}=\ensuremath{-}\mathbb{1}$. For
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
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
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
PT-symmetric fermionic Hamiltonians considered here again uphold the no-signaling principle, but it is not essential that the
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.
Recently, the quantum brachistochrone problem has been discussed in the literature by using non-Hermitian Hamilton operators of different types. Here, it is demonstrated that the passage time is tunable in realistic open quantum systems due to the biorthogonality of the eigenfunctions of the non-Hermitian Hamilton operator. As an example, the numerical results obtained by Bulgakov et al for the transmission through microwave cavities of different shapes are analyzed from the point of view of the brachistochrone problem. The passage time is shortened in the crossover from the weak-coupling to the strong-coupling regime where the resonance states overlap and many branch points (exceptional points) in the complex plane exist. The effect can not be described in the framework of the standard quantum mechanics with the Hermitian Hamilton operator and consideration of S matrix poles.