**Author(s): ** Carl M. Bender**Journal: ** Symmetry, Integrability and Geometry: Methods and Applications ISSN 1815-0659

**Volume: ** 3;

**Start page: ** 126;

**Date: ** 2007;

Original page**Keywords: ** brachistochrone |

PT quantum mechanics |

parity |

time reversal |

time evolution |

unitarity**ABSTRACT**

For any pair of quantum states, an initial state $|I angle$ and afinal quantum state $|F angle$, in a Hilbert space, there are many Hamiltonians $H$ under which $|I angle$ evolves into $|F angle$. 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 $au$. For Hermitian Hamiltonians, $au$ has a nonzero lower bound. However, amongnon-Hermitian ${cal PT}$-symmetric Hamiltonians satisfying the same energy constraint, $au$ can be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of $au$ can be made arbitrarily small because for ${cal PT}$-symmetric Hamiltonians the path from the vector $|I angle$ to the vector $|F angle$, 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 whichthe 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.

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