Optimal Evolution Time Generated by Pseudo-Hermitian Hamiltonian

If an initial state $|\psi_{\scriptscriptstyle\mathrm I}\rangle$ and a final state $|\psi_{\scriptscriptstyle\mathrm{F}}\rangle$ are given, then there exist many Hamiltonians under whose action $|\psi_{\scriptscriptstyle\mathrm I}\rangle$ evolves into $|\psi_{\scriptscriptstyle\mathrm F}\rangle$. In this case, the problem of the transition of $|\psi_{\scriptscriptstyle\mathrm I}\rangle$ to $|\psi_{\scriptscriptstyle \mathrm F}\rangle$ in the least time is very interesting. It was previously shown that for a Hermitian Hamiltonian, there is an optimum evolution time if $|\psi_{\scriptscriptstyle\mathrm I}\rangle$ and $|\psi_{\scriptscriptstyle\mathrm F}\rangle$ 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.