Summary: For any pair of quantum states, an initial state |$I$ñ and a final quantum state |$F$ñ, in a Hilbert space, there are many Hamiltonians $H$ under which |$I$ñ evolves into |$F$ñ. 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 $\tau $. For Hermitian Hamiltonians, $\tau $ has a nonzero lower bound. However, among non-Hermitian $P$T-symmetric Hamiltonians satisfying the same energy constraint, $\tau $ can be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of $\tau $ can be made arbitrarily small because for $P$T-symmetric Hamiltonians the path from the vector |$I$ñ to the vector |$F$ñ, 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.

81Q10, 81S99

brachistochrone, PT quantum mechanics, parity, time reversal, time evolution, unitarity