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Zbl 1228.81027
Bender, Carl M.; Brody, Dorje C.; Jones, Hugh F.; Meister, Bernhard K.
Faster than Hermitian quantum mechanics.
(English)
[J] Phys. Rev. Lett. 98, No. 4, Article ID 040403, 4 p. (2007). ISSN 0031-9007; ISSN 1079-7114/e

Summary: Given an initial quantum state $|\psi_I\rangle$ and a final quantum state $|\psi_F\rangle$, there exist Hamiltonians $H$ under which $|\psi_I\rangle$ evolves into $|\psi_F\rangle$. 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 $\cal P\cal T$-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 $|\psi_I\rangle$ to $|\psi_F\rangle$ 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 2000:
*81P05 General and philosophical topics in quantum theory
81P68 Quantum computation and quantum cryptography

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