Geometric aspects of noncyclic quantum evolutions
Abstract
The geometric phase is defined for any arbitrary quantum evolution using a ``reference section'' of the bundle covering the curve in the projective Hilbert space. A canonical one-form is defined whose line integral gives the desired geometric phase. It is manifestly gauge, phase, and reparametrization invariant for all quantum evolutions. A simple proof of the vanishing nature of the geometric phase along the geodesic is given. Also, an elementary proof of the nonadditive nature of the geometric phase is given. In the limit of cyclic evolution of a pure quantum state, this phase reduces to the Aharonov and Anandan phase, precisely. It is observed that in addition to the geometric phase, other geometric structures exist, such as the ``length'' and ``distance'' during any arbitrary quantum evolution. The relations among all of these geometric quantities are pointed out. Finally, two simple examples are studied to illustrate the ideas introduced in this paper.
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Giovannetti et al. extended the ML QSL to arbitrary mixed states and showed that entanglement can speed up the dynamical evolution of a closed composite system.
“Giovannetti et al. [18] extended the ML QSL to the case of arbitrary mixed states and also showed that entanglement can speed up the dynamical evolution of a closed composite system.”
From Generalized Geometric Quantum Speed Limits (2016) · cites this paper as [18]
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- Authors
- Pati, A. K.
- Affiliations
- Bhabha Atomic Research Centre (IN)
- Corresponding author
- Pati, Arun Kumar
- ORCID
- Pati, Arun Kumar
- Venue
- Physical Review A
- Publisher
- American Physical Society (APS)
- Published
- 1995-10-01
- Volume
- 52
- Issue
- 4
- Pages
- 2576-2584
- ISSN
- 0556-2791 (ISSN-L)
- 1050-2947
- 1094-1622
- OpenAlex
- W2037607884
- Cited by
- 173
- References
- 24
- Citations by year
- 2026: 1 · 2025: 3 · 2024: 4 · 2023: 4 · 2022: 7 · 2021: 2 · 2020: 4 · 2019: 5 · 2018: 5 · 2017: 7 · 2016: 4 · 2015: 11 · 2013: 6 · 2012: 5
- FWCI
- 1.22
- Type
- journal article
- Language
- en
- Open access
- closed
- Locations
- Physical Review A (publishedVersion)
- PubMed (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Non-Hermitian Physics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Keywords
- Berry phase, projective Hilbert space, reference section, gauge invariance, phase invariance, reparametrization invariance, geodesics, Aharonov-Anandan phase, cyclic evolution, quantum state evolution
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