OPTIMAL TIME EVOLUTION FOR PSEUDO-HERMITIAN HAMILTONIANS

If an initial state vertical bar psi(I)> and a final state vertical bar psi(F)> are given, then there exist many Hamiltonians under whose action vertical bar psi(I)> evolves into vertical bar psi(F)>. In this case, the problem of the transition of vertical bar psi(I)> to vertical bar psi(F)> in the least time is very interesting. It was previously shown that for a Hermitian Hamiltonian, there is an optimum evolution time if vertical bar psi(I)> and vertical bar psi(F)> are orthogonal. But for a PT-symmetric Hamiltonian, this time can be arbitrarily small, which seems amazing. We discuss the optimum time evolution for pseudo-Hermitian Hamiltonians and obtain a lower bound for the evolution time under the condition that the Hamiltonian is bounded. The optimum evolution time can be attained in the case where two quantum states are orthogonal with respect to some inner product. The results in the Hermitian and pseudo-Hermitian cases coincide if the evolution is unitary with some well-defined inner product. We also analyze two previously studied examples and find that they are consistent with our theory. In addition, we give some explanations of our results with two examples.