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SIGMA 3 (2007), 126, 10 pages arXiv:0712.3910
https://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 |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.
Key words:
brachistochrone; PT quantum mechanics; parity; time reversal; time evolution; unitarity.
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