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Quantum-speed-limit time for multiqubit open systems

Open access
Liu, Chen, Xu, Zhen-Yu, Zhu, Shiqun·2015·Physical Review A
DOI 10.1103/physreva.91.022102
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Abstract

Quantum-speed-limit (QSL) time captures the intrinsic minimal time interval for a quantum system evolving from an initial state to a target state. In single qubit open systems, it was found that the memory (non-Markovian) effect of environment plays an essential role in shortening QSL time or, say, increasing the capacity for potential speedup. In this paper, we investigate the QSL time for multiqubit open systems. We find that for a certain class of states the memory effect still acts as the indispensable requirement for cutting the QSL time down, while for another class of states this takes place even when the environment is of no memory. In particular, when the initial state is in product state $|111\ensuremath{\cdots}1\ensuremath{\rangle}$, there exists a sudden transition from no capacity for potential speedup to potential speedup in a memoryless environment. In addition, we also display evidence for the subtle connection between QSL time and entanglement that weak entanglement may shorten QSL time even more.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper studies quantum-speed-limit (QSL) time in multiqubit open systems and finds that memoryless environments can still allow QSL reduction for certain initial states, unlike the single-qubit case. It also reports a subtle link between entanglement and QSL time, where weak entanglement may shorten QSL time more.

Key findings

  • For a class of multiqubit states such as \(\alpha_1|100\cdots0\rangle+\alpha_2|010\cdots0\rangle+\cdots+\alpha_N|000\cdots1\rangle\), the QSL ratio is the same as in the single-qubit case, so the memory effect remains necessary for QSL reduction.
  • For the product state \(|11\cdots1\rangle\), a memoryless environment can produce potential speedup, with a sudden transition at \(P_\tau=1/2\); for \(n=2\), the minimal QSL time ratio reaches \(3/2\) as \(P_\tau\to0\).
  • In two-qubit Bell-type states, \(|\Psi_1\rangle=\alpha|01\rangle+\sqrt{1-\alpha^2}|10\rangle\) behaves like the single-qubit case, while \(|\Psi_2\rangle=\alpha|11\rangle+\sqrt{1-\alpha^2}|00\rangle\) can show QSL reduction even without memory.
  • Using 20,000 randomly generated pure states under a damping model, the lower bound \(\tau/\tau_{\rm QSL}=1\) is reached by \(|\Psi_1\rangle\), while the upper bound is taken by a subset of \(|\Psi_2\rangle\), indicating weak entanglement may reduce QSL time more.

Methods

  • The system is \(N\) independent two-level atoms, each locally coupled to a leaky vacuum cavity, with dynamics given by a time-dependent master equation and QSL time defined via the Schatten \(p\) norm and Bures angle.
  • The authors derive QSL ratios for typical two-qubit, three-qubit, and \(n\)-qubit states, including Bell-type, W-type, GHZ-type, and product states, under memoryless and memory environments.
  • For the memory environment, a Lorentzian spectral distribution is used with parameters \(\lambda=50\), \(\omega_0=1\), and \(\tau=1\), where \(\gamma_0=\lambda/2\) marks the transition from memoryless to memory; entanglement is studied with 20,000 Monte Carlo random pure states and concurrence.

Limitations

  • The paper states that only non-correlated environments are treated, and that studying initial correlations among composite environment subsystems remains for future work.
  • Evident from the text: exact QSL expressions for a general pure initial state are cumbersome, so the analysis focuses on selected typical states rather than arbitrary multiqubit states.
  • Evident from the text: the entanglement discussion uses numerical Monte Carlo sampling and a specific damping model, so the reported entanglement–QSL connection is model-dependent.

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

Integrity signals

Registry

Crossref · 10.1103/physreva.91.022102

Cites

42 of 42 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 6 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § A. Two-qubit cases · p. 2

    For the initial state |Ψ2(α=1)> = |11> under a memoryless environment, there exists a sudden change point in QSL time at the critical point Pτ = 1/2.

    “Especially, there exists a sudden change point in QSL time with the critical point P τ = 1 / 2 for initial state | Ψ 2 ( α = 1) 〉 = | 11 〉 .”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: A memoryless environment is characterized by |dot P_t| = -dot P.
  • Findingreasoning
    § B. Three-qubit cases · p. 3

    For the three-qubit GHZ-type state |Ψ4(α=1)> = |111> under a memoryless environment, there exists a sudden transition of speedup potential.

    “indicating that there also exists a sudden transition of speedup potential of quantum evolution even the environment is of no memory.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: A memoryless environment is characterized by |dot P_t| = -dot P.
  • Findingreasoning
    § A. Two-qubit cases · p. 2

    For the two-qubit state |Ψ2> = α|11> + sqrt(1-α^2)|00>, QSL reduction can take place even when the environment has no memory.

    “However, a complex but interesting phenomenon appears for initial state | Ψ 2 〉 : QSL reduction can also take place even when the environment is of no memory.”
    Quote in the paperSubstantial claim
  • Findingown data
    § D. Memory effect on QSL time · p. 4

    In the Lorentzian spectral distribution model, more capacity for potential speedup takes place when the environment enters the memory region (γ0 > λ/2).

    “As is clearly shown in Fig. 2, more capacity for potential speedup will take place when the environment enters the memory region.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: The environments coupled to the qubits are non-correlated.
  • Findingreasoning
    § A. Two-qubit cases · p. 3

    For the two-qubit state |Ψ2>, the QSL time ratio satisfies τ/τ_QSL ≥ 1 always.

    “One may check that τ/τ QSL ≥ 1 is always satisfied.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: A memoryless environment is characterized by |dot P_t| = -dot P.
  • May combine two claims. 6
  • The quote only partly states this claim. 1
  • Classified as probably not a substantial claim. 2

  • May combine two claims.

    Findingown data
    § III. DISCUSSION: ENTANGLEMENT AND QSL TIME · p. 4

    For the two-qubit state |Ψ1>, the lower bound τ/τ_QSL = 1 can always be reached, implying that concurrence has nothing to do with τ/τ_QSL under this circumstance.

    “As is clearly displayed in Fig. 3, the lower bound τ/τ QSL = 1 can always be reached by states | Ψ 1 〉 = α | 01 〉 + √ 1 -α 2 | 10 〉 (dark red dots in Fig. 3), implying that the concurrence has nothing to do with τ/τ QSL under this circumstance.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: A memoryless environment is characterized by |dot P_t| = -dot P.

  • May combine two claims.

    Findingown data
    § III. DISCUSSION: ENTANGLEMENT AND QSL TIME · p. 4

    For a subset of states |Ψ2>, the upper bound of τ/τ_QSL is taken, illustrating that weak entanglement may reduce the QSL time more.

    “In addition, the upper bound is taken by a subset of states | Ψ 2 〉 = α | 11 〉 + √ 1 -α 2 | 00 〉 (light green dots in Fig. 3), illustrating that weak entanglement may reduce the QSL time more.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: A memoryless environment is characterized by |dot P_t| = -dot P.

  • May combine two claims.

    Findingreasoning
    § C. N-qubit cases · p. 3

    For an n-qubit open system initially prepared in state α1|100...0> + α2|010...0> + ... + αN|000...1>, the QSL ratio is exactly the same as Eqs. (5) and (10), and the memory effect of the environment is the essential condition for speedup potential to emerge.

    “It is easy to check that if the n-qubit open system is initially prepared in state α 1 | 100 · · · 0 〉 + α 2 | 010 · · · 0 〉 + · · · + α N | 000 · · · 1 〉 , with ∑ N j =1 α 2 j = 1 , the QSL ratio is exactly the same as Eqs. (5) and (10). Therefore, the memory effect of environment becomes the essential condition for the speedup potential emerge in quantum evolution.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § C. N-qubit cases · p. 3

    For the n-qubit initial state |11...1>, the QSL ratio is given by Eq. (13); when the environment is memoryless, Eq. (13) reduces to Eq. (14), and Pτ = 1/2 is the critical point at which the open system experiences a sudden change of QSL time.

    “However, if the initial state is in | 11 · · · 1 〉 , the QSL ratio is given by”
    • Also stated in: § C. N-qubit cases · p. 3 · boosted
      “Clearly, P τ = 1 / 2 is the critical point at which the open system experiences the sudden change of QSL time under a memoryless environment.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: A memoryless environment is characterized by |dot P_t| = -dot P.

  • May combine two claims.

    Findingcitation
    § A. Two-qubit cases · p. 2

    For the two-qubit state |Ψ1> = α|01> + sqrt(1-α^2)|10>, the QSL time ratio τ/τ_QSL is the same as in single-qubit cases, and reduction in QSL time (τ/τ_QSL > 1) occurs only when the environment has memory; otherwise τ/τ_QSL ≡ 1.

    “Obviously, the QSL time ratio τ QSL of state | Ψ 1 〉 is the same as the exact form in single qubit cases discovered in Refs [30, 32], where the reduction in QSL time ( τ/τ QSL > 1) only occurs when the environment is of memory; otherwise the evolution will always be along the QSL time ( τ/τ QSL ≡ 1).”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [30] S. Deffner and E. Lutz, Phys. Rev. Lett. 111 , 010402 (2013). source

      Not checked in the free run.

    • cites [32] Z.-Y. Xu, S. Luo, W. L. Yang, C. Liu, and S. Zhu, Phys. Rev. A 89 , 012307 (2014). source

      Not checked in the free run.

  • May combine two claims. The quote only partly states this claim.

    Findingreasoning
    § C. N-qubit cases · p. 3

    For the n-qubit state |11...1>, there exists a maximal acceleration potential condition when Pτ → 0, and the corresponding minimal QSL time ratio is given by Eq. (15); for n=2, Eq. (15) equals 3/2.

    “Equation (14) also implies that there exists a maximal acceleration potential condition for state | 11 · · · 1 〉 when P τ → 0, and the corresponding minimal QSL time ratio is given by:”
    • Also stated in: § C. N-qubit cases · p. 3 · plain
      “Especially when n = 2, Eq. (15) grows to 3 / 2, which is marked in Fig. 1 as the red circle.”
    Quote in the paperSubstantial claim
    • The authors state this as an assumption: A memoryless environment is characterized by |dot P_t| = -dot P.

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

    Premisecitation
    § I. INTRODUCTION · p. 1

    A QSL time based on the Schatten p norm for an arbitrarily driven open system was presented by Deffner and Lutz.

    “In particular, a QSL time based on the Schatten p norm for an arbitrarily driven open system was presented by Deffner and Lutz [30]”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [30] S. Deffner and E. Lutz, Phys. Rev. Lett. 111 , 010402 (2013). source

      Not checked in the free run.

  • Classified as probably not a substantial claim.

    Premisecitation
    § I. INTRODUCTION · p. 1

    The Bures angle is a reasonable measure for distance between a pure state and a mixed state because the important Riemannian feature is satisfied.

    “It is reasonable to employ the Bures angle above as a measure for distance between an pure state and a mixed state, since the important Riemannian feature is satisfied under such a circumstance [28].”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [28] M. M. Taddei, B. M. Escher, L. Davidovich, and R. L. de Matos-Filho, Phys. Rev. Lett. 110 , 050402 (2013). source

      Not checked in the free run.

8 of 8

  • The authors state this as an assumption.

    A memoryless environment is characterized by |dot P_t| = -dot P.

    “under a memoryless environment, i.e., ∣ ∣ ∣ ∣ · P t ∣ ∣ ∣ ∣ = -· P t”
    “where we have employed the condition of memoryless environment ∣ ∣ ∣ ∣ · P t ∣ ∣ ∣ ∣ = -· P.”

  • The authors state this as an assumption.

    The environments coupled to the qubits are non-correlated.

    “We have only treated non-correlated environments in this paper.”
2 of 2

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?

  • Liu et al found that open multi-qubit systems still exhibit non-Markovian speed-up, and correlations intrinsic to the system can have the same effect.

    “Liu et al [162] found that a open multi-qubit systems still exhibits the non-Markovian speed-up, but that also correlations intrinsic to the system can have the same effect.”

    From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [162]

  • 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 [34]

  • 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 [34]

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.

  • Figure 2 sets its supposedly varying QSL time to one

    Minor

    The text says τ QSL is set to 1 while also describing Figure 2 as plotting τ QSL against coupling strength. With the separately specified τ = 1, that setting would also fix τ/τ QSL at 1, contrary to the stated change in speedup potential.

  • Figure 3 labels the reciprocal of the ratio analyzed in the text

    Minor

    The caption labels the plotted ratio τ QSL /τ, whereas the discussion identifies the sampled ratios as τ/τ QSL. The text's account of an upper bound therefore refers to the reciprocal of the captioned quantity.

  • Figure 1 labels the reciprocal of the ratio discussed in the text

    Minor

    The caption says Figure 1 plots τ QSL /τ, while the accompanying discussion says it plots τ/τ QSL. These are reciprocal quantities, not the same plotted ratio.

  • Figure 2's plotted quantity is identified inconsistently

    Minor

    The caption identifies Figure 2 as QSL time τ QSL, but the text says the plotted quantity is the reciprocal ratio τ/τ QSL. Fixing τ = 1 does not make these quantities equal.

4 of 4

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. L. Mandelstam, 1945, J. Phys. (USSR), 9, 249

Cited by

Sorted by citations

Related

Authors
Liu, C., Xu, Z.-Y., & Zhu, S.
Affiliations
Soochow University (CN)
ORCID
Liu, Chen
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2015-02-02
Volume
91
Issue
2
ISSN
  • 0556-2791 (ISSN-L)
  • 1050-2947
  • 1094-1622
DOI
10.1103/physreva.91.022102
OpenAlex
W2063100204
Semantic Scholar
031371f2c7659e0099b8242e014506615aeb41a7
Cited by
83
References
42
Citations by year
2026: 1 · 2025: 5 · 2024: 5 · 2023: 9 · 2022: 5 · 2021: 8 · 2020: 9 · 2019: 10 · 2018: 3 · 2017: 10 · 2016: 13 · 2015: 5
FWCI
10.14
Type
journal article
Language
en
Open access
Licence
link.aps.org/licenses/aps-default-license (vor)
Locations
  • Physical Review A (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • arXiv (Cornell University)
Funders
National Natural Science Foundation of China (11204196, 11074184)
Topics
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
  • Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
Keywords
quantum speed limit, multi-qubit systems, open quantum systems, non-Markovian effects, speedup, entanglement, product states

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