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Geometric derivation of the quantum speed limit
Open accessAbstract
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.
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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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In the paper
“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.”
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In the paper
“we arrive at the conclusion that all N CZ gates can be applied simultaneously.”
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In the paper
“In the case where K is the Hamiltonian and q the time, the inequality becomes the Margolus-Levitin inequality”
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In the paper
“This is orthogonal to the initial state of the central qubit when N f = p .”
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In the paper
“A similar argument holds for the application of the phase shifts on the satellite qubits, leading to t ( N ) f = t f .”
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In the paper
“The Mandelstam-Tamm and Margolus-Levitin inequalities do not hold for the most general quantum evolutions, described by completely positive, trace preserving maps (CP maps).”
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In the paper
“we need to show that we can choose parameters such that t < p ¯ h / E 0.”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes- Rests on an assumption: The Hadamard gate evolves a state halfway to an orthogonal state.
In the paper
“In the case of pure quantum states this distance simplifies considerably, and we obtain the Wootters distance [8].”
What the sources say
- Not checked yetcites [8] W. K. Wootters, Phys. Rev. D 23 , 357 (1981). source
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In the paper
“a counterexample to the inequalities is given for evolution described by completely positive trace-preserving maps.”
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In the paper
“In the case where q is the time parameter generated by the Hamiltonian H , the inequality reduces to the famous Mandelstam-Tamm inequality”
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- It may state two separate claims as one. 5
- The quote only partly states this claim. 1
- Probably not a substantial claim. 4
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“Since [ Hj , Hk ] = 0 for all j and k , the interaction Hamiltonian for the N CZ gates is H = j Hj .”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In other words, the satellite qubits are back to their initial state, and the central qubit is in the state”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one. The quote only partly states this claim.
The paper states this without a source or data behind it here, so there is nothing to check it against. The quote shows it is in the paper.
In the paper
“In other words, the statistical distance in the space of probability distributions is the Euclidean distance in the space of (real) probability amplitudes.”
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In the paper
“The periodicity of Sj ( t ) then implies the periodicity of S 0 ( t ) , and moreover, they must have the same period.”
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In the paper
“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:”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesProbably not a substantial claim.
In the paper
“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.”
What the sources say
- Not checked yetcites [8] W. K. Wootters, Phys. Rev. D 23 , 357 (1981). source
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it. It may state two separate claims as one.
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In the paper
“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 > E 0).”
Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passesIt may state two separate claims as one.
In the paper
“which can be interpreted as the probability of mistaking r for s in a single-shot measurement.”
What the sources say
- Not checked yetcites [17] R. Jozsa, J. Mod. Opt. 41 , 2315 (1994). source
Strength:plain· not yet calibratedCausal:correlational· not yet calibratedTests✓ Asserted: passes✓ Checkable: passes✗ Load-bearing: fails✓ Based: passesProbably not a substantial claim. The paper’s conclusions may not depend on it.
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 :”
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?
In the citing paper
“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]
In the citing paper
“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]
In the citing paper
“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]
In the citing paper
“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]
In the citing paper
“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]
In the citing paper
“It has lately been recognized that the geometric derivation of the Margolus-Levitin bound put forward in Ref. [18] does not hold true in general [43].”
From Energy–time uncertainty relation for driven quantum systems (2013) · cites this paper as [18]
In the citing paper
“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]).”
From Energy–time uncertainty relation for driven quantum systems (2013) · cites this paper as [18]
In the citing paper
“The corresponding inequality is an extension of the result of Jones and Kok, obtained for constant Hamiltonians and orthogonal states [18].”
From Energy–time uncertainty relation for driven quantum systems (2013) · cites this paper as [18]
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Song, C.-G., & Cai, Q.-Y. (2025) · EPJ Quantum Technology · doi:10.1140/epjqt/s40507-025-00377-8
“Within the realm of quantum sensing, some studies focus on the relationship be-tween quantum speed limits and quantum Fisher information [19, 21, 30–33], while others attempt to derive precision bounds directly from quantum speed limits [34, 35].”
Modulation of Quantum Features of a Driven Two-Level System via a rf Field
Sun, S., Meng, X., & Cai, X.-J. (2025) · Quantum Information and Computation · doi:10.2478/qic-2025-0013
A geometric quantum speed limit: Theoretical insights and photonic implementation
Wang, Q., Wang, B., Wang, J., & Zhang, L. (2025) · Science China Physics Mechanics and Astronomy · doi:10.1007/s11433-025-2691-9
Tight bounds of quantum speed limit for noisy dynamics via maximum rotation angles
Hu, Z., Yuan, H., Zhang, Z., Fung, C. F., & Miao, Z. (2025) · Science China Information Sciences · doi:10.1007/s11432-024-4484-9
“Although various results on unitary dynamics have come out (see e.g. [3, 7, 6, 4, 5, 9, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18]), studies on noisy dynamics and open quantum systems have only been carried out recently.”
Pati, A. K. (2025) · Europhysics Letters (EPL) · doi:10.1209/0295-5075/adfc93
Time-optimal control based on path distance
Du, L.-F., Qin, L., Guo, Q., Xu, X.-X., & Huang, J. (2025) · Physical Review A · doi:10.1103/mv26-qytj
Speed Limit for a Classical Nonlocal Optical SystemPreprint
Wani, S. S., Shi, X., Shabir, A., & Faizal, M. (2025) · SSRN Electronic Journal · doi:10.2139/ssrn.5546538
Propagation-distance limit for a classical nonlocal optical system
Wani, S. S., Shi, X., Al-Kuwari, S. M., Shabir, A., & Faizal, M. (2025) · Physics Letters A · doi:10.1016/j.physleta.2025.131180
“At this dynamical level, we derive and use the MT/ML distance-domain bounds [26].”
InfluentialQuantum speed limit for observables from quantum asymmetry
Budiyono, A., Moody, M. P., Prihadi, H. L., Rahmawati, R., & Deffner, S. (2026) · Physical Review A · doi:10.1103/tfnt-58sw
Unified fidelity-based framework for quantum speed limits in open quantum systems
Osán, T. M., Álvarez, Y., Portesi, M., & Lamberti, P. W. (2026) · Physical Review A · doi:10.1103/nypk-t1qq · Cited by 1
Continuous crossover from quantum-limit-bounded evolution to exponential delay
Zheltikov, A. M. (2025) · Physical Review A · doi:10.1103/8f95-w44f
Some remarks on quantum speed limit for n-dimensional systems
Toan, H. M., Tran, M. C., & Khoi, V. T. (2026) · Quantum Information Processing · doi:10.1007/s11128-026-05079-3
“It is known that many modified Bures distances associated with these fidelities are genuine metrics, i.e., they satisfy the four metric properties listed in Section II of [19].”
Quantum speed limit in open quantum systems: A stochastic approach
Wang, X.-B., & Wu, W. (2026) · Physical Review A · doi:10.1103/gnns-hf69
Dipolar optimal control of entangled current states
Anonymous, Isaule, F., Juliá-Díaz, B., & Guilleumas, M. (2026) · Physical Review Research · doi:10.1103/yt2c-1qh7
Geometric Quantum Speed Limits and Time-Cost Trade-Off in Time-Dependent Quantum ControlPreprint
Tao, H. G., & Jin, Z. (2026) · SSRN Electronic Journal · doi:10.2139/ssrn.7519472
Metadata
- 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
- 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
- green
- arxiv.org
- 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 › Physics and Astronomy › Atomic and Molecular Physics, and Optics › 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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