Faster than Hermitian Time Evolution

Abstract

For any pair of quantum states, an initial state |Iñ and a final quantum state |Fñ, in a Hilbert space, there are many Hamiltonians H under which |Iñ evolves into |Fñ. Let us impose the constraint that the difference between the largest and smallest eigenvalues of H, Emax and Emin, is held fixed. We can then determine the Hamiltonian H that satisfies this constraint and achieves the transformation from the initial state to the final state in the least possible time τ. For Hermitian Hamiltonians, τ has a nonzero lower bound. However, among non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint, τ can be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of τ can be made arbitrarily small because for PT-symmetric Hamiltonians the path from the vector |Iñ to the vector |Fñ, as measured using the Hilbert-space metric appropriate for this theory, can be made arbitrarily short. The mechanism described here is similar to that in general relativity in which the distance between two space-time points can be made small if they are connected by a wormhole. This result may have applications in quantum computing.