Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 126, 10 pages      arXiv:0712.3910      http://dx.doi.org/10.3842/SIGMA.2007.126
Contribution to the Proceedings of the Seventh International Conference Symmetry in Nonlinear Mathematical Physics

Faster than Hermitian Time Evolution

Carl M. Bender
Physics Department, Washington University, St. Louis, MO 63130, USA

Received October 22, 2007, in final form December 22, 2007; Published online December 26, 2007

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, 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 |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.

Key words: brachistochrone; PT quantum mechanics; parity; time reversal; time evolution; unitarity.

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