Faster than Hermitian Time Evolution

For any pair of quantum states, an initial state $|I\rangle$ and a final quantum state $|F\rangle$, in a Hilbert space, there are many Hamiltonians $H$ under which $|I\rangle$ evolves into $|F\rangle$. 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 $\mathcal{PT}$-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 $\mathcal{PT}$-symmetric Hamiltonians the path from the vector $|I\rangle$ to the vector $|F\rangle$, 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.