Faster than Hermitian Time Evolution
For any pair of quantum states, an initial state |Iñ and a final quantum state |Fñ, in a Hilbert space, there are many Hamiltonians H under which |Iñ evolves into |Fñ. Let us impose the constraint that the difference between the largest and smallest eigenvalues of H, Emax and Emin, is held fixed. We...
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| Published in: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Date: | 2007 |
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| Format: | Article |
| Language: | English |
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Інститут математики НАН України
2007
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| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/146898 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Faster than Hermitian Time Evolution / C.M. Bender // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 26 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862695743685394432 |
|---|---|
| author | Bender, C.M. |
| author_facet | Bender, C.M. |
| citation_txt | Faster than Hermitian Time Evolution / C.M. Bender // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 26 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | For any pair of quantum states, an initial state |Iñ and a final quantum state |Fñ, in a Hilbert space, there are many Hamiltonians H under which |Iñ evolves into |Fñ. Let us impose the constraint that the difference between the largest and smallest eigenvalues of H, Emax and Emin, 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 τ. For Hermitian Hamiltonians, τ has a nonzero lower bound. However, among non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint, τ can be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of τ can be made arbitrarily small because for PT-symmetric Hamiltonians the path from the vector |Iñ to the vector |Fñ, 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.
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| first_indexed | 2025-12-07T16:25:57Z |
| format | Article |
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| id | nasplib_isofts_kiev_ua-123456789-146898 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T16:25:57Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Bender, C.M. 2019-02-11T21:16:26Z 2019-02-11T21:16:26Z 2007 Faster than Hermitian Time Evolution / C.M. Bender // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 26 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 81Q10; 81S99 https://nasplib.isofts.kiev.ua/handle/123456789/146898 For any pair of quantum states, an initial state |Iñ and a final quantum state |Fñ, in a Hilbert space, there are many Hamiltonians H under which |Iñ evolves into |Fñ. Let us impose the constraint that the difference between the largest and smallest eigenvalues of H, Emax and Emin, 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 τ. For Hermitian Hamiltonians, τ has a nonzero lower bound. However, among non-Hermitian PT-symmetric Hamiltonians satisfying the same energy constraint, τ can be made arbitrarily small without violating the time-energy uncertainty principle. The minimum value of τ can be made arbitrarily small because for PT-symmetric Hamiltonians the path from the vector |Iñ to the vector |Fñ, 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. This paper is a contribution to the Proceedings of the Seventh International Conference “Symmetry in Nonlinear Mathematical Physics” (June 24–30, 2007, Kyiv, Ukraine). The author receives financial support from the U.S. Department of Energy. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Faster than Hermitian Time Evolution Article published earlier |
| spellingShingle | Faster than Hermitian Time Evolution Bender, C.M. |
| title | Faster than Hermitian Time Evolution |
| title_full | Faster than Hermitian Time Evolution |
| title_fullStr | Faster than Hermitian Time Evolution |
| title_full_unstemmed | Faster than Hermitian Time Evolution |
| title_short | Faster than Hermitian Time Evolution |
| title_sort | faster than hermitian time evolution |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/146898 |
| work_keys_str_mv | AT bendercm fasterthanhermitiantimeevolution |