Эта публикация цитируется в
5 статьях
Faster than Hermitian Time Evolution
Carl M. Bender Physics Department, Washington University, St. Louis, MO 63130, USA
Аннотация:
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.
Ключевые слова:
brachistochrone; PT quantum mechanics; parity; time reversal; time evolution; unitarity.
MSC: 81Q10;
81S99 Поступила: 22 октября 2007 г.; в окончательном варианте
22 декабря 2007 г.; опубликована
26 декабря 2007 г.
Язык публикации: английский
DOI:
10.3842/SIGMA.2007.126