Faster than Hermitian Time Evolution
arXiv:0712.3910 · doi:10.3842/SIGMA.2007.126
Abstract
For any pair of quantum states, an initial state |I> and a final quantum state |F>, in a Hilbert space, there are many Hamiltonians H under which |I> evolves into |F>. Let us impose the constraint that the difference between the largest and smallest eigenvalues of H, E_max and E_min, 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, τhas a nonzero lower bound. However, among non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint, τcan be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of τcan be made arbitrarily small because for PT-symmetric Hamiltonians the path from the vector |I> to the vector |F>, 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.
This is a contribution to the Proc. of the Seventh International Conference ''Symmetry in Nonlinear Mathematical Physics'' (June 24-30, 2007, Kyiv, Ukraine), published in SIGMA (Symmetry, Integrability and Geometry: Methods and Applications) at http://www.emis.de/journals/SIGMA/
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