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Cited by 87Quantum Mechanics and ApplicationsQuantum Information and CryptographyQuantum many-body systems

Kind of entanglement that speeds up quantum evolution

Open access
Fröwis, F.·2012·Physical Review A
DOI 10.1103/physreva.85.052127
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Abstract

The ``speed'' of unitary quantum evolution was recently shown to be connected to entanglement in multipartite quantum systems. Here, we discuss a tighter version of the Mandelstam-Tamm uncertainty relation that depends on the Fisher information. The passage time is estimated by a lower bound that depends inversely proportional to the square root of the Fisher information. This leads to a better understanding of the origin of a fast quantum time-evolution of entangled states.

Verified 2026-10-06 · 17 references not checked yet

AI snapshot

Written from the paper’s full text.

TL;DR

This paper derives a tighter Mandelstam-Tamm uncertainty relation for quantum evolution by replacing the Hamiltonian standard deviation with the quantum Fisher information. It argues that only entanglement yielding high Fisher information can speed up unitary evolution relative to nonentangled states.

Key findings

  • The improved bound uses \(\hbar\sqrt{F(\rho,H)}/2 \le (\Delta H)_\rho\), where \(F(\rho,H)\) is the quantum Fisher information; the new relations are identical to the original Mandelstam-Tamm relations for pure states but generically tighter for mixed states.
  • The orthogonalization/passage time is lower bounded inversely by \(\sqrt{F(\rho,H)}\), and the same bound applies to the three survival-probability definitions \(T_\rho(t)\), \(E_\rho(t)\), and \(D_\rho(t)\).
  • For \(k\)-producible states, the paper cites the bound \(F(\rho^{(k)}, H/(2\epsilon)) \le s k^2 + (N-sk)^2\) with \(s=\lfloor N/k\rfloor\), giving an upper bound on evolution speed independent of the specific local Hamiltonian.
  • In the two-qubit example \(\rho=(1-x)|00\rangle\langle00|+x|\psi^+\rangle\langle\psi^+|\) with \(H=\frac12\hbar\Omega(\sigma_x^{(1)}+\sigma_x^{(2)})\), the paper reports \((\Delta H)_\rho=\hbar\Omega\sqrt{(1+x)/2}\) and \(F(\rho,H)=2\Omega^2(1-3x+4x^2)\); the maximal difference occurs at \(x=\sqrt2-1\), where the improved bounds are much tighter than Eq. (1).

Methods

  • Theoretical study of finite-dimensional quantum systems with time-independent Hamiltonians, using the Bures metric, quantum Fisher information, Cramér-Rao bound, and differential inequalities for survival probabilities.
  • Proofs are given for bounds on \(T_\rho(t)\), \(D_\rho(t)\), and \(E_\rho(t)\), plus an extension to arbitrary projective measurements.
  • Illustrative examples include a two-qubit mixed state and a one-qubit projective-measurement case; no experimental data are analyzed.

Limitations

  • The paper states an open question: attainability of the bound (15) when \(F(\rho,H) < 4(\Delta H)^2_\rho/\hbar^2\).
  • The paper notes the new bound is more difficult to calculate because the Fisher information replaces the variance of the energy.
  • Evident from the text: the work is theoretical, with numerical/analytical examples rather than experimental validation; for full-rank \(\rho\), the projection-based survival probability \(E_\rho(t)\) is trivial because \(\Pi_\rho=\mathbb{1}\).

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

Integrity signals

Registry

Crossref · 10.1103/physreva.85.052127

Cites

18 of 35 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 20 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § B. Bound on E ρ ( t ) · p. 3

    The maximal rate with which E_ρ(t) can change is limited by the quantum Fisher information.

    “Equation (12) states that the maximal rate with which E ρ ( t ) can change is limited by the quantum Fisher information.”
    Quote in the paperSubstantial claim
  • Findingnone given
    § B. Explicit example · p. 4

    For the two-qubit state ρ=(1-x)|00〉〈00|+x|ψ+〉〈ψ+| with H=1/2 ħΩ(σ_x^(1)+σ_x^(2)), the standard deviation is (∆H)_ρ = ħΩ√((1+x)/2).

    “The standard deviation reads (∆ H ) ρ = ℏ Ω √ (1 + x ) / 2”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § C. General projective measurement · p. 5

    Similarly, the lower bound (21) holds for all t in [0,(π-2δ′)/√F(ρ,H)] with δ′=arcsin√c.

    “for all t ∈ [ 0 , ( π -2 δ ′ ) / √ F ( ρ, H ) ] with δ ′ = arcsin √ c .”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § IV. CONCLUSION · p. 5

    The bound |〈[H,Π]〉_ρ(t)| ≤ ħ√F(ρ,H)(∆Π)_ρ(t) is tighter than the Heisenberg uncertainty bound |〈[H,Π]〉_ρ(t)| ≤ 2(∆H)_ρ(∆Π)_ρ(t).

    “This is tighter than the Heisenberg uncertainty ∣ ∣ 〈 [ H, Π] 〉 ρ ( t ) ∣ ∣ ≤ 2 (∆ H ) ρ (∆Π) ρ ( t ) .”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § B. Explicit example · p. 4

    For the same two-qubit state, the Fisher information is F(ρ,H)=2Ω^2(1-3x+4x^2).

    “while for the Fisher information (7) we find F ( ρ, H ) = 2Ω 2 ( 1 -3 x +4 x 2 ) .”
    Quote in the paperSubstantial claim
  • May combine two claims. 9
  • The quote only partly states this claim. 2
  • Classified as probably not a substantial claim. 7

  • May combine two claims.

    Findingreasoning
    § B. Bound on E ρ ( t ) · p. 3

    The general inequality (12) constitutes an improvement of Eq. (13), whereas they are identical for pure states.

    “Recalling that F ( ρ, H ) ≤ 4 (∆ H ) 2 ρ / ℏ 2 , one can see that the general inequality (12) constitutes an improvement of Eq. (13) whereas they are identical for pure states.”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingreasoning
    § IV. CONCLUSION · p. 5

    The orthogonalization time for all three definitions of survival probability is lower bounded by the inverse of √F(ρ,H).

    “The orthogonalization (or passage) time is lower bounded by the inverse of √ F ( ρ, H ) .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § A. Survival probability for mixed states · p. 2

    For an incoherent sum of eigenstates of H, T_ρ(t)=E_ρ(t)=1 for all times, whereas bound (2) may suggest a rather quick evolution.

    “An extreme case is an incoherent sum of eigenstates of H . Then T ρ ( t ) = E ρ ( t ) = 1 for all times, whereas the bound (2) may suggest a rather quick evolution.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § II. IMPROVED MANDELSTAM-TAMM BOUND · p. 2

    The new relations are identical to the original relations for pure states but are generically tighter for mixed states.

    “The new relations are identical to the original relations in the case of pure states but are generically tighter for mixed states.”
    • Also stated in: § IV. CONCLUSION · p. 5 · plain
      “The new bound is in general tighter than the original bound (1) for mixed states.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § III. DISCUSSION · p. 4

    In these equality cases, F(ρ,H)=4(∆H)^2_ρ/ħ^2, which is necessary because Eq. (2) cannot be improved further.

    “In these cases, F ( ρ, H ) = 4 (∆ H ) 2 ρ / ℏ 2 , which is necessary, because Eq. (2) cannot be improved further.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given
    § I. INTRODUCTION · p. 1

    In the improved Mandelstam-Tamm relations, the Hamiltonian standard deviation in Eqs. (1) and (2) can be replaced by the Fisher information ħ√F(ρ,H)/2, which is bounded above by the Hamiltonian standard deviation.

    “To this end, we show that (∆ H ) ρ in Eqs. (1) and (2) can be replaced by the Fisher information ℏ √ F ( ρ, H ) / 2 ≤ (∆ H ) ρ .”
    Quote in the paperSubstantial claim

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

    Findingown data
    § C. General projective measurement · p. 5

    In the one-qubit example, T_ρ(t) and D_ρ(t) coincide.

    “They coincide in this example.”
    Quote in the paperNot a substantial claim

  • The quote only partly states this claim.

    Findingreasoning
    § A. Bound on T ρ ( t ) and D ρ ( t ) · p. 3

    For the time interval 0 ≤ √F(ρ,H)|t| ≤ π, the transition probability T_ρ(t) is lower bounded by cos^2(√F(ρ,H)t/2).

    “For the time interval 0 ≤ √ F ( ρ, H ) | t | ≤ π , the transition probability T ρ ( t ) can be bounded from below by”
    • Also stated in: § B. Explicit example · p. 4 · boosted
      “The orange, dashed curve above the shading represents the improved version cos 2 √ F ( ρ, H ) t/ 2 from Eqs. (9), (10) and (14)”
    Quote in the paperSubstantial claim

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

    Findingnone given
    § A. Survival probability for mixed states · p. 2

    For a pure state, the three survival probability definitions coincide with P_ρ(t).

    “For a pure state ρ = | ψ 〉 〈 ψ | , they coincide with P ρ ( t ) .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning
    § III. DISCUSSION · p. 4

    If ρ commutes with H, then F(ρ,H)=0, meaning that there is no finite passage time.

    “Then F ( ρ, H ) = 0 , meaning that there is no finite passage time.”
    Quote in the paperSubstantial claim

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

    Findingown data
    § B. Explicit example · p. 4

    For the two-qubit example, F(ρ,H) coincides with 4(∆H)^2_ρ/ħ^2 for x=0 and x=1.

    “We see that F ( ρ, H ) coincides with 4 (∆ H ) 2 ρ / ℏ 2 for x = 0 , 1 , as it should.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Premisecitation
    § I. INTRODUCTION · p. 1

    Entanglement can speed up time evolution in multipartite systems.

    “Furthermore, it was recognized [5] that entanglement can speed up time evolution in multipartite systems.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [5] V. Giovannetti, S. Lloyd, and L. Maccone, Europhys. Lett. 62 , 615-621 (2003). source

      Not checked in the free run.

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

    Premisecitation
    § I. INTRODUCTION · p. 1

    The expectation value of the Hamiltonian also gives a speed limit on quantum evolution.

    “For instance, it was shown by Margolus and Levitin [3] that the expectation value 〈 H 〉 ρ = Tr( Hρ ) also gives a speed limit on the quantum evolution.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [3] source

      Not checked in the free run.

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

    Premisecitation
    § I. INTRODUCTION · p. 1

    For a pure state, the speed of quantum evolution is limited by the standard deviation of the system Hamiltonian.

    “Already in 1945, Mandelstam and Tamm [1] showed that, for a pure state, the speed of quantum evolution is limited by the standard deviation of the system Hamiltonian.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [1]

      Not checked in the free run.

  • May combine two claims.

    Premisecitation
    § A. Entanglement and quantum speed · p. 4

    For a multipartite qubit state with a Hamiltonian that is a sum of one-particle terms each of norm ε, if F(ρ,H) > 4ε^2 N/ħ^2, then ρ is entangled and is more useful for parameter estimation than any nonentangled state.

    “One can show [17] that if F ( ρ, H ) &gt; 4 ϵ 2 N/ ℏ 2 , then ρ is entangled and is, in principle, more useful for parameter estimation (in the sense of Sec. I B) than any nonentangled state.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [17] L. Pezzé and A. Smerzi,Phys. Rev. Lett. 102 , 100401 (2009).

      Not checked in the free run.

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

    Premisecitation
    § B. Quantum Fisher information · p. 2

    The quantum Fisher information satisfies F(ρ,H) ≤ 4(∆H)^2_ρ/ħ^2, with equality if ρ is pure.

    “Note that F ( ρ, H ) can be further estimated as F ( ρ, H ) ≤ 4 (∆ H ) 2 ρ / ℏ 2 [12, 19]. The equality holds if ρ is pure.”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [12] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72 , 3439 (1994). source

      Not checked in the free run.

    • cites [19] A. Uhlmann, Physics Letters A 161 , 329-331 (1992). source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation
    § B. Quantum Fisher information · p. 2

    When the parameter is time generated by a time-independent Hamiltonian, the Fisher information is time independent and has the form given in Eq. (7).

    “Then one can show [10, 12] that the Fisher information is time independent and it has the form”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [10] C. W. Helstrom, Quantum Detection and Estimation Theory (Academic, New York, 1976). source

      Not checked in the free run.

    • cites [12] S. L. Braunstein and C. M. Caves, Phys. Rev. Lett. 72 , 3439 (1994). source

      Not checked in the free run.

17 of 17

  • The paper relies on this without support.

    Explicit proofs and a clear discussion of the implications for entanglement were missing before this work.

    “However, explicit proofs and a clear discussion on the implications for entanglement were missing.”
1 of 1

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?

  • The resemblance between the Mandelstam-Tamm bound and the Cramer-Rao bound indicates a deep relation between metrology and quantum speed limits based on state distinguishability.

    “the similarity between this Mandelstam-Tamm bound and the Cramer-Rao bound alludes to the deep relation between metrology and quantum speed limits rooted in the fact that both are based on the distinguishability of the states at hand.”

    From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [103]

  • For pure states the quantum Fisher information bound is exactly the Mandelstam-Tamm relation, while for generic mixed states it is strictly tighter.

    “While for pure states this expression is exactly equivalent to the Mandelstam-Tamm relation (32), it turns out to be a strictly tighter bound for generic mixed states.”

    From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [103]

  • For a unitary evolution U = exp(-iHt), to steer a state away from the initial position with a Bures angle D_B = arccos(√f), the evolution time t needs to satisfy t ≥ 2D_B/√F_tt.

    “For a unitary evolution, U = exp(-iHt), to steer a state away from the initial position with a Bures angle D B = arccos( f ) (f is the fidelity defined in equation ( 75)), the evolution time t needs to satisfy [6, 49, 122] t 2D B √ F tt ,(94”

    From Quantum Fisher information matrix and multiparameter estimation (2020) · cites this paper as [122]

Cites

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Research works printed but not linked: 1

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

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Related

Authors
Fröwis, F.
Affiliations
Universität Innsbruck (AT)
Corresponding author
Fröwis, F.
ORCID
Fröwis, F.
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2012-05-29
Volume
85
Issue
5
ISSN
  • 0556-2791 (ISSN-L)
  • 1050-2947
  • 1094-1622
DOI
10.1103/physreva.85.052127
OpenAlex
W1989340955
Cited by
87
References
35
Citations by year
2026: 3 · 2025: 6 · 2024: 6 · 2023: 7 · 2022: 5 · 2021: 9 · 2020: 7 · 2019: 6 · 2018: 5 · 2017: 8 · 2016: 7 · 2015: 5 · 2014: 7 · 2013: 4 · 2012: 2
FWCI
2.98
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)
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 › Atomic and Molecular Physics, and Optics › Quantum many-body systems
Keywords
quantum entanglement, quantum speed limit, Fisher information, unitary evolution, multipartite quantum systems, entangled states

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