CoChat
Sign Up
Cited by 199Quantum Mechanics and ApplicationsQuantum Information and CryptographyAdvanced Thermodynamics and Statistical Mechanics

Energy–time uncertainty relation for driven quantum systems

Open access
Deffner, Sebastian, Lutz, Eric·2013·Journal of Physics A: Mathematical and Theoretical
DOI 10.1088/1751-8113/46/33/335302
Open this paper in CoChat — free accountOpen PDF

Abstract

We derive generalizations of the energy-time uncertainty relation for driven quantum systems. Using a geometric approach based on the Bures length between mixed quantum states, we obtain explicit expressions for the quantum speed limit time, valid for arbitrary initial and final quantum states and arbitrary driving protocols. Our results establish the fundamental limit on the rate of evolution of generic quantum systems.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper derives generalized energy–time uncertainty relations and quantum speed limit times for driven closed quantum systems using a geometric Bures-length approach. It obtains Mandelstam–Tamm and Margolus–Levitin type bounds valid for arbitrary unitary driving, with mixed-state extensions via purification.

Key findings

  • A Mandelstam–Tamm-type relation is derived for arbitrary initial and final mixed quantum states and arbitrary time-dependent Hamiltonians, with the time-averaged energy variance \(\Delta E_\tau = (1/\tau)\int_0^\tau dt\,(\langle H_t^2\rangle-\langle H_t\rangle^2)^{1/2}\).
  • A Margolus–Levitin-type bound is derived first for pure initial states with ground-state energy set to zero, then extended to mixed states by purification; for arbitrary driving the generalized bound uses the time average of nondiagonal matrix elements \(|\langle \psi_0|H_t U_t|\psi_0\rangle|\).
  • The Margolus–Levitin-type bound has prefactor \(4/\pi^2\), which the authors note is smaller than the prefactor \(2/\pi\) obtained numerically in Giovannetti, Lloyd, and Maccone.
  • The quantum speed limit time depends strongly on the driving: it is inversely proportional to mean energy or energy spread, can be reduced exponentially if mean energy increases exponentially, can increase if driving decreases energy, and vanishes as \(\hbar \to 0\).

Methods

  • Theoretical derivation using the Bures length/angle between density operators, quantum fidelity, the Braunstein–Caves Bures metric, Fisher information, the von Neumann equation, and a triangle-type inequality.
  • The Mandelstam–Tamm bound is derived for arbitrary mixed states and arbitrary time-dependent Hamiltonians; the Margolus–Levitin bound assumes an initial pure state and \(E_g=0\), then uses purification for mixed states.
  • The appendix treats time-independent and time-dependent Hamiltonians, including the condition \([H_t,H_0]=0\) for one form of the time-dependent bound; no experimental data are analyzed.

Limitations

  • Stated: the Margolus–Levitin derivation assumes an initial pure state for simplicity, handles mixed states via purification, and sets the ground-state energy to zero.
  • Stated: for time-dependent Hamiltonians, the time-independent-form Margolus–Levitin inequality remains valid only if \([H_t,H_0]=0\) for all \(t\in[0,\tau]\); otherwise a more general expression with nondiagonal matrix elements is required.
  • Stated/evident: the derivative of the Bures length may change sign later, requiring care in deriving inequalities, and the paper corrects prior geometric Margolus–Levitin derivations; the work is purely theoretical with no experimental validation reported.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1088/1751-8113/46/33/335302

Cites

49 of 49 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 29 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § Appendix · p. 8

    For time-dependent Hamiltonians, the inequality Eq. (32) holds true if [H_t, H_0] = 0 for all t in [0, τ].

    “holds true if [ H t , H 0 ] = 0 , ∀ t ∈ [0 , τ ].”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
  • Findingnone given
    § 3. Discussion · p. 5

    The quantum speed limit time τ_QSL expresses the generalized energy-time uncertainty relation for arbitrary quantum unitary processes in the generic form Eq. (15).

    “Summing up, we have obtained the quantum speed limit time τ QSL , which expresses the generalized energy-time uncertainty relation for arbitrary quantum unitary processes, in the generic form,”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
  • Findingreasoning
    § 1. Mandelstam-Tamm bound · p. 4

    Equation (8) implies that the Fisher information f_t is bounded from above by the variance of the energy of the system.

    “Note that Eq. (8) also implies that the Fisher information f t is bounded from above by the variance of the energy of the system.”
    Quote in the paperNot a substantial claim
  • Findingcitation
    § 1. Mandelstam-Tamm bound · p. 4

    Equation (11) is a Mandelstam-Tamm uncertainty relation valid for arbitrary initial and final mixed quantum states and arbitrary time-dependent Hamiltonians.

    “Equation (11) is a Mandelstam-Tamm uncertainty relation valid for arbitrary initial and final mixed quantum states and arbitrary time-dependent Hamiltonians.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
  • Findingreasoning
    § Appendix · p. 8

    Expression (37) is a generalized Margolus-Levitin uncertainty relation valid for arbitrary angles between initial and final pure states and arbitrary external driving.

    “Expression (37) is a generalized Margolus-Levitin uncertainty relation valid for arbitrary angles between initial and final pure states and arbitrary external driving.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
    • The authors state this as an assumption: For the Margolus-Levitin bound, the driven quantum system starts in a pure state.
  • May combine two claims. 8
  • The quote only partly states this claim. 1
  • Classified as probably not a substantial claim. 7

  • May combine two claims.

    Findingnone given
    § 2. Margolus-Levitin bound · p. 5

    The generalization of the Margolus-Levitin bound to arbitrary mixed quantum states can be done invoking purification, and the mean energy becomes Eτ = (1/τ)∫ dt |tr{ρ0 Ht}|.

    “The generalization to arbitrary mixed quantum states can be done invoking purification and the mean energy becomes E τ = (1 /τ ) ∫ τ 0 dt | tr { ρ 0 H t }| , cf. also appendix.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.

  • May combine two claims.

    Findingcitation
    § Appendix · p. 8

    The derivation can be extended to arbitrary mixed states by interpreting pure states as purifications of mixed states in an enlarged Hilbert space; by choosing the purification that maximizes the fidelity, the Margolus-Levitin inequality also applies to the purifications.

    “The above derivation can be extended to arbitrary mixed states by interpreting pure states as purifications of mixed states in an enlarged Hilbert space [35]. Specifically, by choosing the purification that maximizes the fidelity between two density operators, the Margolus-Levitin inequality also applies to the purifications [18].”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [35] source

      Not checked in the free run.

    • cites [18] source

      Not checked in the free run.

    • The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.

  • May combine two claims.

    Findingreasoning
    § Appendix · p. 8

    Equation (30) is the Margolus-Levitin bound for quantum systems evolving under time-independent Hamiltonians, valid for arbitrary angles between initial and final states; it corrects previously published versions.

    “Equation (30) is the Margolus-Levitin bound for quantum systems evolving under time-independent Hamiltonians, valid for arbitrary angles between initial and final states; it corrects previously published versions [18, 43].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [18] source

      Not checked in the free run.

    • cites [43] source

      Not checked in the free run.

    • The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
    • The authors state this as an assumption: The ground state energy is set to zero, E_g = 0.

  • May combine two claims.

    Findingcitation
    § Appendix · p. 9

    Equation (38) generalizes the Margolus-Levitin-type bound introduced by Giovannetti, Lloyd and Maccone for time-independent systems, with numerical prefactor 4/π2 smaller than the prefactor 2/π obtained numerically in Ref. [15].

    “Equation (38) generalizes the Margolus-Levitin-type bound introduced by Giovannetti, Lloyd and Maccone for time-independent systems [15] (note that the numerical prefactor 4 /π 2 in Eq. (38) is smaller than the prefactor 2 /π obtained numerically in Ref. [15]).”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [15] source

      Not checked in the free run.

    • The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.

  • The quote only partly states this claim.

    Findingreasoning
    § 2. Margolus-Levitin bound · p. 5

    The minimal quantum evolution time from initial to final state can be estimated from below by Eq. (16), where Eτ is the time-averaged mean energy with respect to the initial state.

    “As a consequence, the minimal quantum evolution time from initial to final state can be estimated from below by”
    • Also stated in: § 2. Margolus-Levitin bound · p. 5 · plain
      “where E τ = (1 /τ ) ∫ τ 0 dt |〈 ψ 0 | H t | ψ 0 〉| is the time-averaged mean energy with respect to the initial state.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
    • The authors state this as an assumption: For the Margolus-Levitin bound, the driven quantum system starts in a pure state.
    • The authors state this as an assumption: The ground state energy is set to zero, E_g = 0.

  • May combine two claims.

    Findingreasoning
    § Appendix · p. 8

    In general, the instantaneous eigenbasis of H_t is time-dependent; as a result, the estimations in Eqs. (25)-(27) do not necessarily hold true.

    “In general, the instantaneous eigenbasis of H t is time-dependent. As a result, the estimations in Eqs. (25)-(27) do not necessarily hold true.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § Appendix · p. 6

    The derivative d_t L may change sign later on, and special care should be taken when deriving inequalities; this fact seems not to have been noticed by Jones and Kok.

    “Note that d t L is positive at short times, since L is initially zero (0 ≤ L ≤ π/ 2). However, the derivative may change sign later on, and special care should be taken when deriving inequalities (this fact seems not have been noticed by Jones and Kok [18]).”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [18] source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § Energy-time uncertainty relation for driven quantum systems · p. 2

    The quantum speed limit time gives the fundamental limit on the rate of evolution of quantum systems.

    “The quantum speed limit time gives the fundamental limit on the rate of evolution of quantum systems [20, 21, 22].”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [20] source

      Not checked in the free run.

    • cites [21] source

      Not checked in the free run.

    • cites [22] source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § 1. Mandelstam-Tamm bound · p. 3

    The Bures metric for two infinitesimally close density operators is given by the Braunstein-Caves expression.

    “According to Braunstein and Caves, the Bures metric for two infinitesimally close density operators, ρ ′ = ρ + dρ , is [37] (see also Ref. [38]),”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [37] source

      Not checked in the free run.

    • cites [38] source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning
    § 3. Discussion · p. 5

    The quantum speed limit time strongly depends on the driving protocol: the more energy is pumped into a system and the more it is distributed among different states, the faster the system can evolve.

    “The quantum speed limit time, therefore, strongly depends on the driving protocol: the more energy is pumped into a system and the more it is distributed among different states, the faster the system can evolve.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § Energy-time uncertainty relation for driven quantum systems · p. 2

    Wootters' statistical distance is the only Riemannian metric, up to a constant factor, that is invariant under all unitary transformations.

    “Wootters' statistical distance is the only Riemannian metric, up to a constant factor, which is invariant under all unitary transformations.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [31] source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § Energy-time uncertainty relation for driven quantum systems · p. 3

    The Bures length is a natural generalization of Wootters' statistical distance for arbitrary mixed quantum states.

    “The Bures length (3) has been shown to be a natural generalization of Wootters' statistical distance for arbitrary mixed quantum states [37].”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [37] source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation
    § 3. Discussion · p. 6

    If a classical system is driven so rapidly that the driving frequency becomes comparable to the typical transition frequency between quantum states, quantum effects can no longer be neglected and have hence to be taken into account.

    “However, as discussed by e.g. Landau and Lifshitz [42], if a classical system is driven so rapidly that the driving frequency becomes comparable to the typical transition frequency between quantum states, quantum effects can no longer be neglected and have hence to be taken into account.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [42] Landau L D, 1980, Statistical Mechanics

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationreasoning
    § 3. Discussion · p. 5

    The quantum speed limit time is reduced when the Bures length between initial and final states is smaller.

    “We first remark, as one might expect, that the quantum speed limit time is reduced when the Bures length between initial and final states is smaller, indicating that the closer the two states, the less time is needed to go from one state to the other.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning
    § 3. Discussion · p. 5

    Whereas τ_QSL is determined by the initial energy in the time-independent case, it is set by the time-averaged mean energy or energy variance in the case of driven quantum systems.

    “Whereas τ QSL is determined by the initial energy in the time-independent case, it is set by the time-averaged mean energy or energy variance in the case of driven quantum systems.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Interpretationreasoning
    § 2. Margolus-Levitin bound · p. 4

    Due to the unitary dynamics, a system that starts in a pure state remains in a pure state at all times.

    “Due to the unitary dynamics the system then remains in a pure state at all times.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
16 of 16

  • The authors state this as an assumption.

    The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.

    “Specifically, we derive the quantum speed limit time for generic mixed states with arbitrary angle and for arbitrary timedependent unitary driving.”

  • The authors state this as an assumption.

    For the Margolus-Levitin bound, the driven quantum system starts in a pure state.

    “For simplicity, and without loss of generality, we assume that the driven quantum system starts in a pure state ρ 0 = | ψ 0 〉〈 ψ 0 | .”

  • The authors state this as an assumption.

    The ground state energy is set to zero, E_g = 0.

    “We additionally set the ground state energy to zero, E g = 0 [14].”

  • The authors state this as an assumption.

    The time evolution of the quantum system obeys the time-dependent Schrödinger equation.

    “The time evolution of the quantum system obeys the time-dependent Schr¨ odinger equation,”

  • The authors state this as an assumption.

    The system's Hamiltonian is driven by arbitrary time-dependent parameters during time τ.

    “In the following, we consider quantum systems whose Hamiltonians H t are driven by arbitrary timedependent parameters during time τ .”
5 of 5

Claims other papers make about this one

Each is checked against this paper’s own text: does it say what the citing paper says it does?

Self-check

Percentages, sums and reported differences re-computed from the paper’s own numbers.

Each contradiction between two passages listed here was confirmed by an independent check.

  • Energy variance is defined with opposite signs

    Minor

    The introduction defines the energy spread using 〈H〉2 − 〈H2〉, while the later time-averaged definition uses 〈H2〉 − 〈H〉2. The introductory definition reverses the variance used in the paper's Mandelstam-Tamm bound.

1 of 1

Cites

Flagged first, then by citations

Research works printed but not linked: 2

  1. Mandelstam L, 1945, J. Phys. (USSR), 9, 249
  2. Bures D J C, 1969, Trans. Am. Math. Soc., 135, 199

Cited by

Sorted by citations

Related

Authors
Deffner, S., & Lutz, E.
Affiliations
University of Augsburg (DE)
Corresponding author
Deffner, Sebastian
ORCID
Venue
Journal of Physics A: Mathematical and Theoretical
Publisher
IOP Publishing
Published
2013-07-29
Volume
46
Issue
33
Pages
335302
ISSN
  • 1751-8113 (ISSN-L)
  • 1751-8121
DOI
10.1088/1751-8113/46/33/335302
OpenAlex
W2161551823
Semantic Scholar
856d66591e01e1d5d7aeb0c7c06e0cfdec753e28
Cited by
199
References
49
Citations by year
2026: 8 · 2025: 18 · 2024: 16 · 2023: 16 · 2022: 20 · 2021: 14 · 2020: 21 · 2019: 13 · 2018: 14 · 2017: 12 · 2016: 15 · 2015: 19 · 2014: 9 · 2013: 4
FWCI
8.03
Type
journal article
Language
en
Open access
Licence
Locations
  • Journal of Physics A Mathematical and Theoretical (publishedVersion)
  • arXiv (Cornell University) (publishedVersion)
  • Maryland Shared Open Access Repository (USMAI Consortium) (submittedVersion)
  • arXiv (Cornell University)
  • Digital Repository at the University of Maryland (University of Maryland College Park)
Funders
  • Deutscher Akademischer Austauschdienst
  • Deutsche Forschungsgemeinschaft
Topics
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
Keywords
time-energy uncertainty relation, driven quantum systems, quantum speed limit, mixed quantum states, closed quantum systems, quantum state evolution

About these checks

Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.

Claims read by deepseek-v4.1-flash under definition version 3 (extract-v3) on 2026-10-06.

Classified by jev-1.13-20260917 (gold-guideline-v0.2) on 2026-10-06.

Statistics and arithmetic are recomputed, not judged.

Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.

What counts as a claim ›How we measure confidence ›How we check this ›

Are you an author, or do you see a problem with this page? Request a review