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Faster than Hermitian quantum mechanics. (English) Zbl 1228.81027
Summary: Given an initial quantum state $|{\psi }_{I}〉$ and a final quantum state $|{\psi }_{F}〉$, there exist Hamiltonians $H$ under which $|{\psi }_{I}〉$ evolves into $|{\psi }_{F}〉$. Consider the following quantum brachistochrone problem: subject to the constraint that the difference between the largest and smallest eigenvalues of $H$ is held fixed, which $H$ achieves this transformation in the least time $\tau$? For Hermitian Hamiltonians $\tau$ has a nonzero lower bound. However, among non-Hermitian $𝒫𝒯$-symmetric Hamiltonians satisfying the same energy constraint, $\tau$ can be made arbitrarily small without violating the time-energy uncertainty principle. This is because for such Hamiltonians the path from $|{\psi }_{I}〉$ to $|{\psi }_{F}〉$ can be made 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 the points are connected by a wormhole. This result may have applications in quantum computing.
##### MSC:
 81P05 General and philosophical topics in quantum theory 81P68 Quantum computation