Optimal Time Evolution for Hermitian and Non-Hermitian Hamiltonians

Abstract

Consider the set of all Hamiltonians whose largest and smallest energy eigenvalues, E_max and E_min, differ by a fixed energy \omega. Given two quantum states, an initial state |\psi_I> and a final state |\psi_F>, there exist many Hamiltonians H belonging to this set under which |\psi_I> evolves in time into |\psi_F>. Which Hamiltonian transforms the initial state to the final state in the least possible time \tau? For Hermitian Hamiltonians, $\tau$ has a nonzero lower bound. However, among complex non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint, \tau can be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of \tau can be made arbitrarily small because for PT-symmetric Hamiltonians the evolution path from the vector |\psi_I> to the vector |\psi_F>, as measured using the Hilbert-space metric appropriate for this theory, can be made arbitrarily short. The mechanism described here resembles the effect in general relativity in which two space-time points can be made arbitrarily close if they are connected by a wormhole. This result may have applications in quantum computing.