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Cited by 794Quantum Mechanics and ApplicationsQuantum chaos and dynamical systemsQuantum Information and Cryptography

Geometry of quantum evolution

Anandan, J., Aharonov, Y.·1990·Physical Review Letters
DOI 10.1103/physrevlett.65.1697
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Abstract

For an arbitrary quantum evolution, it is shown that the integral of the uncertainty of energy with respect to time is independent of the particular Hamiltonian used to transport the quantum system along a given curve in the projective Hilbert space. It is the distance along this curve measured by the Fubini-Study metric. This gives a new time-energy uncertainty principle. New geometric meanings to time as measured by a clock and the transition probability during a quantum measurement are also obtained.

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Crossref · 10.1103/physrevlett.65.1697

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2 of 2 references verified0 citation errors · 0 not checkable

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  • Another bound, based on the average energy E, was established by Margolus and Levitin.

    “Another bound, based on the average energy E , was established by Margolus and Levitin.”

    From Fundamental Limit on the Rate of Quantum Dynamics: The Unified Bound Is Tight (2009) · cites this paper as [3]

  • The bounds coincide, and can be exactly attained by certain initial states if ΔE = E.

    “The bounds coincide, and can be exactly attained by certain initial states if ∆ E = E .”

    From Fundamental Limit on the Rate of Quantum Dynamics: The Unified Bound Is Tight (2009) · cites this paper as [3]

  • The Anandan-Aharonov relation states that the speed of evolution of a quantum state is 2ħ^{-1} ΔH.

    “This relation is indeed natural if we recall the Anandan-Aharonov relation [12] which states that the 'speed' of the evolution of a given quantum state is given by 2h -1 H .”

    From Elementary derivation for passage times (2003) · cites this paper as [12]

  • The above expression generalizes in a unified manner previous results derived in Refs. [7-22].

    “The above expression generalizes in an unified manner previous results derived in Refs. [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22].”

    From Energy–time uncertainty relation for driven quantum systems (2013) · cites this paper as [7]

  • The correct interpretation of the time-energy Heisenberg uncertainty principle is that a quantum state with spread in energy ΔE takes time at least Δt = π hbar/2ΔE to evolve to an orthogonal state.

    “In particular, the correct interpretation of the time-energy Heisenberg uncertainty principle ∆ E ∆ t ≥ ¯ h is not that it takes time ∆ t to measure energy to an accuracy ∆ E (a fallacy that was put to rest by Aharonov and Bohm 23 -24 ) but rather that that a quantum state with spread in energy ∆ E takes time at least ∆ t = π ¯ h/ 2∆ E to evolve to an orthogonal (and hence distinguishable) state 2”

    From Ultimate physical limits to computation (2000) · cites this paper as [25]

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Authors
Anandan, J., & Aharonov, Y.
Affiliations
  • University of Cambridge (GB)
  • University of South Carolina (US)
Venue
Physical Review Letters
Publisher
American Physical Society (APS)
Published
1990-10-01
Volume
65
Issue
14
Pages
1697-1700
ISSN
0031-9007 (ISSN-L)
DOI
10.1103/physrevlett.65.1697
OpenAlex
W2017028422
Cited by
794
References
2
Citations by year
2026: 46 · 2025: 59 · 2024: 50 · 2023: 51 · 2022: 53 · 2021: 49 · 2020: 48 · 2019: 32 · 2018: 34 · 2017: 24 · 2016: 28 · 2015: 36 · 2014: 25 · 2013: 29 · 2012: 22
FWCI
5.30
Type
journal article
Language
en
Open access
closed
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review Letters (publishedVersion)
  • PubMed (publishedVersion)
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Quantum chaos and dynamical systems
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
Keywords
time-energy uncertainty relation, quantum evolution, projective Hilbert space, Fubini-Study metric, quantum clocks, quantum measurement, transition probabilities, quantum geometry

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