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This article is cited in 5 scientific papers (total in 5 papers)
Faster than Hermitian Time Evolution
Carl M. Bender Physics Department, Washington University, St. Louis, MO 63130, USA
Abstract:
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.
Keywords:
brachistochrone; PT quantum mechanics; parity; time reversal; time evolution; unitarity.
Received: October 22, 2007; in final form December 22, 2007; Published online December 26, 2007
Citation:
Carl M. Bender, “Faster than Hermitian Time Evolution”, SIGMA, 3 (2007), 126, 10 pp.
Linking options:
https://www.mathnet.ru/eng/sigma252 https://www.mathnet.ru/eng/sigma/v3/p126
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Abstract page: | 203 | Full-text PDF : | 49 | References: | 25 |
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