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Cited by 172Quantum Mechanics and ApplicationsQuantum Computing Algorithms and ArchitectureQuantum Information and Cryptography

Geometric derivation of the quantum speed limit

Open access
Jones, Philip J., Kok, Pieter·2010·Physical Review A
DOI 10.1103/physreva.82.022107
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Abstract

The Mandelstam-Tamm and Margolus-Levitin inequalities play an important role in the study of quantum-mechanical processes in nature since they provide general limits on the speed of dynamical evolution. However, to date there has been only one derivation of the Margolus-Levitin inequality. In this paper, alternative geometric derivations for both inequalities are obtained from the statistical distance between quantum states. The inequalities are shown to hold for unitary evolution of pure and mixed states, and a counterexample to the inequalities is given for evolution described by completely positive trace-preserving maps. The counterexample shows that there is no quantum speed limit for nonunitary evolution.

AI snapshot

Written from the paper’s full text.

TL;DR

The paper gives alternative geometric derivations of the Mandelstam-Tamm and Margolus-Levitin quantum speed-limit inequalities from the statistical distance between quantum states. It shows these bounds hold for unitary evolution of pure and mixed states, and constructs a completely positive trace-preserving map counterexample that violates both, implying no general quantum speed limit for non-unitary evolution.

Key findings

  • Both inequalities are derived geometrically: Mandelstam-Tamm follows from the Fisher-information bound, while Margolus-Levitin follows from explicit differentiation of the Wootters distance and the Cauchy-Schwarz inequality.
  • The inequalities are shown to hold for unitary evolution of pure and mixed states; for mixed states, Margolus-Levitin is extended via purification, with the bound interpreted using the system’s average energy by choosing purifications made only of degenerate ground eigenstates.
  • A counterexample using a central qubit plus \(N\) satellite qubits, GHZ entanglement, phase shifts, and simultaneous CZ/Hadamard gates produces orthogonal initial and final states in time \(t < \pi\hbar/E_0\) when \(N\phi=\pi\), with \(E_\phi > E_0\) and \(g\) chosen appropriately; this violates both bounds and shows \(t\) can be made arbitrarily small by making \(E_\phi\) and \(g\) arbitrarily large.
  • The paper notes that mixed states can never attain either equality, and that states attaining the inequalities have the form of superpositions of energy eigenstates.

Methods

  • Theoretical derivation with no experimental data or sample; the work is a mathematical/geometric analysis of quantum speed limits.
  • Uses statistical distance for classical probability distributions and density matrices, Fisher information, Wootters distance, and Uhlmann purification to derive bounds on dynamical speed.
  • The non-unitary counterexample is analyzed with the stabilizer formalism for \(N+1\) qubits, including CZ, Hadamard, and phase-shift gates, and gate-time periodicity arguments.

Limitations

  • The paper states open questions: what general system-environment requirements allow violation of the speed limit; whether the strong interaction can be achieved noiselessly; whether noise may impose a practical speed limit; and whether general bounds can be based on environment properties rather than detailed dynamics.
  • The paper states the counterexample “may be very difficult to implement in practice” and that \(t\) can be made arbitrarily small by making \(E_\phi\) and \(g\) arbitrarily large.
  • Evident from the text: the counterexample is an idealized mathematical construction assuming simultaneous ideal gates and no noise; the mixed-state equality result is cited from prior work, and the derivation of the inequalities assumes unitary evolution.

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Integrity signals

Registry

Crossref · 10.1103/physreva.82.022107

Cites

29 of 29 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 10 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning
    § IV. NON-UNITARY EVOLUTION · p. 6

    The state corresponding to the stabilizer generators after the Hadamard gates is the GHZ state on N+1 qubits.

    “It is straightforward to check that the state state corresponding to these stabilizer generators is the GHZ state ( | 0 , . . . , 0 〉 + | 1 , . . . , 1 〉 ) / √ 2 on the N + 1 qubits.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § IV. NON-UNITARY EVOLUTION · p. 7

    All N CZ gates can be applied simultaneously.

    “we arrive at the conclusion that all N CZ gates can be applied simultaneously.”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § B. Margolus-Levitin inequality · p. 4

    When K is the Hamiltonian and q is time, the derived inequality becomes the Margolus-Levitin inequality.

    “In the case where K is the Hamiltonian and q the time, the inequality becomes the Margolus-Levitin inequality”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § IV. NON-UNITARY EVOLUTION · p. 6

    The central qubit final state is orthogonal to its initial state when Nφ = π.

    “This is orthogonal to the initial state of the central qubit when N f = p .”
    Quote in the paperSubstantial claim
  • Findingreasoning
    § IV. NON-UNITARY EVOLUTION · p. 6

    The minimum time to apply N phase shifts is the same as that for a single phase shift.

    “A similar argument holds for the application of the phase shifts on the satellite qubits, leading to t ( N ) f = t f .”
    Quote in the paperSubstantial claim
  • May combine two claims. 5
  • The quote only partly states this claim. 1
  • Classified as probably not a substantial claim. 4

  • May combine two claims.

    Findingreasoning
    § IV. NON-UNITARY EVOLUTION · p. 6

    The interaction Hamiltonian for the N CZ gates is the sum of the individual Hj because they commute.

    “Since [ Hj , Hk ] = 0 for all j and k , the interaction Hamiltonian for the N CZ gates is H = j Hj .”
    Quote in the paperSubstantial claim

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

    Findingreasoning
    § IV. NON-UNITARY EVOLUTION · p. 6

    After the sequence, the satellite qubits are back to their initial state and the central qubit is in a state with accumulated phase shift Nφ.

    “In other words, the satellite qubits are back to their initial state, and the central qubit is in the state”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Findingnone given
    § A. Statistical distance for classical probability distributions · p. 2

    The statistical distance for classical probability distributions is the Euclidean distance in the space of real probability amplitudes.

    “In other words, the statistical distance in the space of probability distributions is the Euclidean distance in the space of (real) probability amplitudes.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning
    § IV. NON-UNITARY EVOLUTION · p. 7

    The periodicity of Sj(t) implies the periodicity of S0(t), and they must have the same period.

    “The periodicity of Sj ( t ) then implies the periodicity of S 0 ( t ) , and moreover, they must have the same period.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Findingreasoning
    § B. Statistical distance for density matrices · p. 3

    The Wootters distance is the angle between two pure quantum states in Hilbert space.

    “If | y 〉 and | f 〉 are two arbitrary pure quantum states in Hilbert space, then the statistical distance is given by the angle between the two states:”
    Quote in the paperSubstantial claim

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

    Premisecitation
    § B. Statistical distance for density matrices · p. 3

    The Wootters distance is related to the fidelity |〈ψ|φ〉|^2, which is the probability of mistaking |ψ〉 for |φ〉 in a single-shot measurement.

    “Notice how the Wootters distance is related to the fidelity |〈 y | f 〉| 2 , which is the probability of mistaking | y 〉 for | f 〉 in a single-shot measurement.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [8] W. K. Wootters, Phys. Rev. D 23 , 357 (1981). source

      Not checked in the free run.

  • May combine two claims.

    Interpretationown data
    § IV. NON-UNITARY EVOLUTION · p. 7

    The counterexample suggests that the interaction between the system and its environment must be strong, and that the environment must be more energetic than the system (Eq > E0).

    “The counterexample presented here suggests that the interaction between the system and its environment must be strong, and that the environment must be more energetic than the system ( Eq &gt; E 0).”
    Quote in the paperSubstantial claim

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

    Premisecitation
    § B. Margolus-Levitin inequality · p. 5

    The fidelity between two density matrices can be interpreted as the probability of mistaking ρ for σ in a single-shot measurement.

    “which can be interpreted as the probability of mistaking r for s in a single-shot measurement.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [17] R. Jozsa, J. Mod. Opt. 41 , 2315 (1994). source

      Not checked in the free run.

8 of 8

  • The paper relies on this without support.

    To date there has been only one derivation of the Margolus-Levitin inequality.

    “However, to date there has been only one derivation of the Margolus-Levitin inequality.”

  • The paper relies on this without support.

    The purification is merely a mathematical construction, so unrestricted amounts of energy may be added to the system during the process and any extra energy added is physically meaningless.

    “At first glance this seems problematic since the purification is merely a mathematical construction and therefore unrestricted amounts of energy may be added to the system during the process.”

  • The paper relies on this without support.

    The Hadamard gate evolves a state halfway to an orthogonal state.

    “The Hadamard gate evolves a state halfway to an orthogonal state, and the speed limit for this gate can therefore be taken as half the time given in Eq. (45), with E 0 → Eq :”
3 of 3

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?

  • Generalizations of the Mandelstam-Tamm and Margolus-Levitin findings to nonorthogonal states and driven systems have been provided in prior work.

    “Generalizations of the MT and ML findings to nonorthogonal states and to driven systems have been provided in Refs.[14][15][16][17][18][19].”

    From Quantum Speed Limit for Non-Markovian Dynamics (2013) · cites this paper as [15]

  • The statistical distance is the natural choice to define the maximal quantum speed.

    “Since ℓ constitutes the shortest path connecting quantum states, the statistical distance (67) serves as the natural choice to define the maximal quantum speed [60,67,75,76,117].”

    From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [75]

  • The fundamental difference between the minimal time and geometric approaches becomes obvious for parameterized, driven, and time-dependent Liouvillians.

    “The fundamental difference, however, becomes obvious for parameterised, driven, and time-dependent L(ρ t ) [75].”

    From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [75]

  • 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].”

    From Energy–time uncertainty relation for driven quantum systems (2013) · cites this paper as [18]

  • 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].”

    From Energy–time uncertainty relation for driven quantum systems (2013) · cites this paper as [18]

Cites

Flagged first, then by citations

Research works printed but not linked: 2

  1. L. Mandelstam, 1945, J. Phys. (USSR), 9, 249
  2. J. Uffink, 1994, Proceedings of the Third International Workshop on Squeezed States and Uncertainty Relations

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Related

Authors
Jones, P. J., & Kok, P.
Affiliations
University of Sheffield (GB)
ORCID
Kok, Pieter
Venue
Physical Review A
Publisher
American Physical Society (APS)
Published
2010-08-16
Volume
82
Issue
2
ISSN
  • 0556-2791 (ISSN-L)
  • 1050-2947
  • 1094-1622
DOI
10.1103/physreva.82.022107
OpenAlex
W2008738328
Cited by
172
References
29
Citations by year
2026: 6 · 2025: 22 · 2024: 14 · 2023: 12 · 2022: 15 · 2021: 13 · 2020: 17 · 2019: 6 · 2018: 7 · 2017: 7 · 2016: 11 · 2015: 14 · 2014: 12 · 2013: 9 · 2012: 2
FWCI
2.74
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 Computing Algorithms and Architecture
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Information and Cryptography
Keywords
quantum speed limit, statistical distance, quantum states, unitary evolution, mixed states, completely positive trace-preserving maps, non-unitary evolution, counterexample

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