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Cited by 176Quantum Mechanics and ApplicationsQuantum Information and CryptographyAdvanced Thermodynamics and Statistical Mechanics

Quantum speed limit for arbitrary initial states

Open access
Zhang, Ying-Jie, Han, Wei, Xia, Yun-Jie, Cao, Jun-Peng, Fan, Heng·2014·Scientific Reports
DOI 10.1038/srep04890PMC PMC4013937
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Abstract

The minimal time a system needs to evolve from an initial state to its one orthogonal state is defined as the quantum speed limit time, which can be used to characterize the maximal speed of evolution of a quantum system. This is a fundamental question of quantum physics. We investigate the generic bound on the minimal evolution time of the open dynamical quantum system. This quantum speed limit time is applicable to both mixed and pure initial states. We then apply this result to the damped Jaynes-Cummings model and the Ohimc-like dephasing model starting from a general time-evolution state. The bound of this time-dependent state at any point in time can be found. For the damped Jaynes-Cummings model, when the system starts from the excited state, the corresponding bound first decreases and then increases in the Markovian dynamics. While in the non-Markovian regime, the speed limit time shows an interesting periodic oscillatory behavior. For the case of Ohimc-like dephasing model, this bound would be gradually trapped to a fixed value. In addition, the roles of the relativistic effects on the speed limit time for the observer in non-inertial frames are discussed.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper derives a quantum speed limit time (QSLT) for arbitrary initial states, including mixed states, using relative purity as the distance measure. It applies the unified bound to the damped Jaynes-Cummings model, Ohmic-like dephasing, and non-inertial-frame relativistic settings.

Key findings

  • A unified QSLT expression is obtained for arbitrary initially mixed states in open systems; for a pure initial state it recovers the unified bound of Ref. 18.
  • In the damped Jaynes-Cummings model starting from the excited state, the QSLT first decreases and then increases to the driving time \(\tau_D\) in the Markovian regime, while in the non-Markovian regime it shows periodic oscillatory behavior.
  • For Ohmic-like dephasing, Ohmic and sub-Ohmic environments give a speeded-up evolution; for super-Ohmic environments the QSLT is gradually trapped to a fixed value, leading to uniform evolution speed.
  • Relativistic effects increase the speed of evolution in purely Ohmic-like dephasing channels, while in the damped Jaynes-Cummings model the paper states they slow the evolution and lead to a smaller QSLT.

Methods

  • Theoretical derivation: relative purity is used as the distance measure; the von Neumann trace inequality and Cauchy-Schwarz inequality yield ML-type and MT-type bounds, combined into a unified QSLT for arbitrary initial mixed states.
  • Applications: a qubit is studied in the exactly solvable damped Jaynes-Cummings model and the Ohmic-like dephasing model, with the QSLT calculated from an arbitrary time-dependent mixed state \(\rho_\tau\) over a driving time \(\tau_D\).
  • Non-inertial-frame analysis: the relativistic effect is treated by considering a uniformly accelerated observer, with the Minkowski vacuum and excited states expressed in Rindler regions and the parameter \(r\) ranging from \(0\) to \(\pi/4\).

Limitations

  • The paper states no explicit limitations section.
  • The text notes that using fidelity for mixed states would require optimization over all purifications, which is “a challenging task,” so relative purity is used instead.
  • Evident from the text: the analysis is restricted to specific exactly solvable models (damped Jaynes-Cummings and Ohmic-like dephasing), a qubit system, zero-temperature vacuum environments, and particular parameter regimes such as \(\lambda > 2\gamma_0\) and \(\lambda < 2\gamma_0\).

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

Integrity signals

Registry

Crossref · 10.1038/srep04890

Cites

37 of 37 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 9 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning

    The quantum speed limit time for the qubit to evolve from ρτ to in the Ohmic-like dephasing model can be obtained as above.

    “With regard to the QSLT for the qubit to evolve from ρτ to in the Ohimc-like dephasing model, we can obtain as the above derivation .”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The environment in the spin-boson dephasing model is initially in its vacuum state at zero temperature.
    • The authors state this as an assumption: The spectral density of the environmental modes is Ohmic-like.
  • Findingnone given

    The open system executes a speeded-up dynamical evolution process in the Ohmic and sub-Ohmic dephasing models.

    “The open system executes a speeded-up dynamical evolution process in the Ohmic and sub-Ohmic dephasing models.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The environment in the spin-boson dephasing model is initially in its vacuum state at zero temperature.
    • The authors state this as an assumption: The spectral density of the environmental modes is Ohmic-like.
  • Findingreasoning

    Equation (4) expresses a Margolus-Levitin type bound on the speed of quantum evolution valid for mixed initial states.

    “Regarding general nonunitary open system dynamics, Eq. (4) expresses a ML type bound on the speed of quantum evolution valid for mixed initial states.”
    Quote in the paperSubstantial claim
  • Findingown data

    In the Ohmic-like dephasing model, the quantum speed limit time of a qubit can be described by the Margolus-Levitin type bound.

    “In this Ohmic-like dephasing model, the QSLT of a qubit can also be described by ML type bound.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The environment in the spin-boson dephasing model is initially in its vacuum state at zero temperature.
    • The authors state this as an assumption: The spectral density of the environmental modes is Ohmic-like.
  • Findingreasoning

    For a generally mixed state of a qubit, the Margolus-Levitin type bound on the quantum speed limit time is tight for the open system.

    “For the generally mixed state ρτ of a qubit, is always less than , so we reach the result that the ML type bound on the QSLT is tight for the open system.”
    • Also stated in: § Quantum speed limit time in open system dynamics · plain
      “So we can obtain that is always less than , and reach the result that the ML type bound on the QSLT is tight for the open system.”
    Quote in the paperSubstantial claim
  • May combine two claims. 6
  • Classified as probably not a substantial claim. 5

  • May combine two claims.

    Findingown data

    For s > 1, the qubit dephasing qτ stops after a finite time, leading to coherence trapping.

    “On the other hand, for s > 1 the qubit dephasing qτ will stop after a finite time, therefore leading to coherence trapping, as shown by the inset of Fig. 3.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The environment in the spin-boson dephasing model is initially in its vacuum state at zero temperature.
    • The authors state this as an assumption: The spectral density of the environmental modes is Ohmic-like.

  • May combine two claims.

    Findingreasoning

    τQSL is independent of vz and relates to the dephasing rate of the Ohmic-like environment and to the coherence of the initial state ρ0 under a given driving time τD.

    “With this, it is easy to show that τQSL is independent of vz, and not only it relates to the dephasing rate of the Ohmic-like environment but also to the coherence of the initial state ρ0 under a given driving time τD.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The environment in the spin-boson dephasing model is initially in its vacuum state at zero temperature.
    • The authors state this as an assumption: The spectral density of the environmental modes is Ohmic-like.

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Findingnone given

    For the Ohmic-like dephasing model, the quantum speed limit time under the relativistic effect can be calculated.

    “While for the case of the Ohimc-like dephasing model, the QSLT can be calculated .”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: Bob moves with a uniform acceleration and carries a detector for the qubit.
    • The authors state this as an assumption: The single-mode approximation is used for the Minkowski vacuum and excited states in the accelerated frame.

  • May combine two claims.

    Findingown data

    For the same driving time τD = 1, larger coherence of the initial state decreases the speed of evolution of a quantum system and demands a longer quantum speed limit time.

    “We observe that, for the same driving time τD = 1, the lager coherence of the initial state can decrease the speed of evolution of a quantum system, and thus demand the longer QSLT.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The environment in the spin-boson dephasing model is initially in its vacuum state at zero temperature.
    • The authors state this as an assumption: The spectral density of the environmental modes is Ohmic-like.

  • Classified as probably not a substantial claim.

    Findingreasoning

    Equation (6) is a Mandelstam-Tamm type bound for nonunitary dynamics process.

    “Integrating Eq. (5) over time leads to the following MT type bound for nonunitary dynamics process, where means the time-averaged variance of the energy.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    For the super-Ohmic dephasing model, the qubit first exhibits a speeded-up dynamical process before a finite time, and then complies with a uniform evolution speed after this finite time.

    “But for the super-Ohmic dephasing model the qubit firstly exhibits a speeded-up dynamical process before a finite time, and then complies with an uniform evolution speed after this finite time.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The environment in the spin-boson dephasing model is initially in its vacuum state at zero temperature.
    • The authors state this as an assumption: The spectral density of the environmental modes is Ohmic-like.

  • Classified as probably not a substantial claim.

    Premisecitation

    A unified bound of quantum speed limit time including both Mandelstam-Tamm and Margolus-Levitin types for non-Markovian dynamics can be formulated.

    “It is shown that a unified bound of QSLT including both MT and ML types for non-Markovian dynamics can be formulated18.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [18] Deffner, S. & Lutz, E. Quantum speed limit for non-Markovian dynamics. Phys. Rev. Lett. 111, 010402 (2013). source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Interpretationnone given

    Due to the presence of a Rindler horizon, Bob is forced to trace over a causally disconnected region-II of space-time.

    “Due to the presence of a Rindler horizon, Bob is forced to trace over a causally disconnected region-II of space-time.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: Bob moves with a uniform acceleration and carries a detector for the qubit.
    • The authors state this as an assumption: The single-mode approximation is used for the Minkowski vacuum and excited states in the accelerated frame.

  • May combine two claims.

    Interpretationown data

    In the Markovian regime, the evolution of the qubit first exhibits a speeded-up process for τ < τc and then shows a gradual deceleration process for τ > τc.

    “That is to say, in the Markovian regime, the evolution of the qubit first exhibits a speeded-up process for τ < τc and then shows gradual deceleration process for τ > τc.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The environment in the damped Jaynes-Cummings model is initially in a vacuum state.

  • May combine two claims.

    Premisecitation

    The unified bound for non-Markovian dynamics is applicable for a given driving time for pure initial states but is not feasible for mixed initial states.

    “However, while this unified bound is applicable for a given driving time for the pure initial states, it is not feasible for mixed initial states.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [18] Deffner, S. & Lutz, E. Quantum speed limit for non-Markovian dynamics. Phys. Rev. Lett. 111, 010402 (2013). source

      Not checked in the free run.

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

    Premisenone given

    The parameter r ranges from 0 to π/4 for 0≤a<∞.

    “Here r ranges from 0 to π/4 for 0≤ a < ∞.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: Bob moves with a uniform acceleration and carries a detector for the qubit.
    • The authors state this as an assumption: The single-mode approximation is used for the Minkowski vacuum and excited states in the accelerated frame.
11 of 11

  • The paper relies on this without support.

    Decoherence and inaccurate operations are indispensable and may result in mixed initial states.

    “As we all know that decoherence and inaccurate operations are indispensable which may result in mixed initial states.”

  • The authors state this as an assumption.

    The environment in the damped Jaynes-Cummings model is initially in a vacuum state.

    “The environment is supposed to be initially in a vacuum state.”

  • The authors state this as an assumption.

    There are no correlations between the system and the environment at t = 0 in the spin-boson dephasing model.

    “There exists no correlations between the system and the environment at t = 0.”

  • The authors state this as an assumption.

    The environment in the spin-boson dephasing model is initially in its vacuum state at zero temperature.

    “Furthermore, the environment is initially in its vacuum state at zero temperature.”

  • The authors state this as an assumption.

    The spectral density of the environmental modes is Ohmic-like.

    “Here, we suppose that the spectral density of the environmental modes is Ohmic-like , with ωc being the cutoff frequency and η a dimensionless coupling constant.”

  • The authors state this as an assumption.

    Bob moves with a uniform acceleration and carries a detector for the qubit.

    “Now we assume that Bob moves with a uniform acceleration and takes with him a detector for the qubit.”

  • The authors state this as an assumption.

    The single-mode approximation is used for the Minkowski vacuum and excited states in the accelerated frame.

    “From the perspective of Bob, under the single-mode approximation, the Minkowski vacuum state |0〉M and the Minkowski only excited state |1〉M, in a uniformly accelerated frame with acceleration a are expressed as30”
7 of 7

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?

  • Non-Markovianity in the dynamics process and the associated information backflow from the reservoir can lead to faster quantum evolution and hence to smaller QSLT.

    “As noted in Refs. [25-27], the non-Markovianity in the dynamics process ( ρ 0 to ρ τ D ), and the associated information backflow from the reservoir, can lead to faster quantum evolution, and hence to smaller QSLT.”

    From Classical-driving-assisted quantum speed-up (2015) · cites this paper as [26]

  • Non-Markovian effect characterized by non-Markovianity can speed up quantum evolution.

    “Deffner and Lutz [25] showed that non-Markovian effect characterized by non-Markovianity [29-32] can speed up quantum evolution, which has been verified in different settings [26, 28, 33, 34].”

    From Mechanism for quantum speedup in open quantum systems (2016) · cites this paper as [26]

  • Our mechanism cannot be captured by the previous treatment with Lorentzian spectrum, where the lower limit of the frequency integration is artificially extended from 0 to -∞, and this approximation loses the availability of the bound state.

    “Note that our mechanism cannot be captured by the previous treatment with Lorentzian spectrum [25, 26, 28, 33, 34], where the lower limit of the frequency integration in f ( t -τ ) is artificially extended from 0 to -∞ . This approximation is mathematically convenient but loses the availability of the bound state [40].”

    From Mechanism for quantum speedup in open quantum systems (2016) · cites this paper as [26]

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.

  • Slower evolution is said to give a smaller QSLT

    Major

    The paper elsewhere states that decreased evolution speed requires a longer QSLT. Its conclusion for the damped Jaynes-Cummings model instead says that slowing the evolution leads to a smaller QSLT.

1 of 1

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. von Neumann, J. Some matrix-inequalities and metrization of matric-space. Tomsk Univ. Rev. 1, 286–300 (1937).

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Related

Authors
Zhang, Y.-J., Han, W., Xia, Y.-J., Cao, J.-P., & Fan, H.
Affiliations
  • Qufu Normal University (CN)
  • Chinese Academy of Sciences (CN)
Corresponding author
Fan, Heng
ORCID
Venue
Scientific Reports
Publisher
Springer Science and Business Media LLC
Published
2014-05-08
Volume
4
Issue
1
Pages
4890-4890
ISSN
2045-2322 (ISSN-L)
DOI
10.1038/srep04890
PMC
PMC4013937
OpenAlex
W1980954462
Semantic Scholar
f1b8e39d3a8ccfe9669508f910f83eea3f60c636
Cited by
176
References
36
Citations by year
2026: 3 · 2025: 19 · 2024: 14 · 2023: 16 · 2022: 12 · 2021: 17 · 2020: 22 · 2019: 13 · 2018: 13 · 2017: 16 · 2016: 16 · 2015: 14 · 2014: 1
FWCI
7.80
Type
journal article
Language
en
Open access
Licence
Locations
  • Scientific Reports (publishedVersion, cc-by)
  • Europe PMC (PubMed Central) (publishedVersion, cc-by)
  • PubMed Central (publishedVersion, cc-by)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
  • National Natural Science Foundation of China (973 Program, 61178012, 11204156, 11104162, 11247240, 11175248, 11304179, 2010CB922904)
  • Natural Science Foundation of Shandong Province
  • National Key Research and Development Program of China (2010CB922904, “973” Program)
  • Specialized Research Fund for the Doctoral Program of Higher Education of China (20123705120002, 20133705110001)
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
quantum speed limit, open quantum systems, mixed states, non-Markovian dynamics, Markovian dynamics, time-dependent states, relativistic effects, non-inertial frames

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