Faster than Hermitian Quantum Mechanics

Given an initial quantum state |ψI⟩ and a final quantum state |ψF⟩, there exist Hamiltonians H under which |ψI⟩ evolves into |ψF⟩. Consider the following quantum brachistochrone problem: subject to the constraint that the difference between the largest and smallest eigenvalues of H is held fixed, which H achieves this transformation in the least 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. This is because for such Hamiltonians the path from |ψI⟩ to |ψF⟩ can be made 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.