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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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15 No basis given · 13 Own interpretation · 8 Not checked yet · 2 Own result
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“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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“we arrive at the conclusion that all N CZ gates can be applied simultaneously.”
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“In the case where K is the Hamiltonian and q the time, the inequality becomes the Margolus-Levitin inequality”
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“This is orthogonal to the initial state of the central qubit when N f = p .”
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“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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“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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“we need to show that we can choose parameters such that t < p ¯ h / E 0.”
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- 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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“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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“We see from Eq. (16) that the Fisher information is directly related to the derivative of the statistical distance, as expected.”
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“We show that orthogonal initial and final states can be created in arbitrary short times.”
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“Since all Hadamard gates are applied simultaneously to different qubits, the minimum time to apply N Hadamard gates is the same as that for a single Hadamard gate.”
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In the paper
“According to Uhlmann's theorem [18], the square root of the fidelity is given by”
What the sources say
- Not checked yetcites [18] A. Uhlmann, Rep. Math. Phys 9 , 273 (1976). source
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“consequently there is no general bound that is valid for all possible (unitary and non-unitary) quantum evolution.”
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“In fact, t can be made arbitrarily small by making Eq and g arbitrarily large”
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“The inequalities are shown to hold for unitary evolution of pure and mixed states”
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In the paper
“Note that the Mandelstam-Tamm inequality in Eq. (25) was derived for unitary evolution of arbitrary mixed states .”
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“It demonstrates that the bounds are valid only for unitary evolution.”
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“The expectation value 〈 KS 〉 + 〈 KR 〉 is then the average energy of the system S and the purification system R taken together.”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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“any extra energy added is physically meaningless and therefore the true bound on the evolution occurs for 〈 H 〉 = 〈 HS 〉 .”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- Rests on an assumption: 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.
None of the sources cited for this claim have been checked yet.
In the paper
“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.”
What the sources say
- Not checked yetcites [12] L. Mandelstam and I. Tamm, J. Phys. (USSR) 9 , 249 (1945). source
- Not checked yetcites [13] N. Margolus and L. B. Levitin, Phys. D 120 , 188 (1998).
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In the paper
“So far, we have shown that the Margolus-Levitin inequality holds for the unitary evolution of pure states.”
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“The counterexample show that there is no quantum speed limit for non-unitary evolution.”
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“In other words, the amount of information about q in any measurement is bounded by the variance in its generator K [16].”
What the sources say
- Not checked yetcites [16] S. L. Braunstein, C. M. Caves, and G. J. Milburn, Ann. Phys. 247 , 135 (1996). source
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“It was show in Ref. [19] that mixed states can never attain either equality.”
What the sources say
- Not checked yetcites [19] source
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In the paper
“In addition, any state that does attain the inequalities has the form [19, 20]”
What the sources say
- Not checked yetcites [19] source
- Not checked yetcites [20] J. Soderholm, G. Björk, T. Tsegaye, and A. Trifonov, Phys. Rev. A 59 , 1788 (1999).
plaincorrelational- Asserted: passes — The authors put it forward as true.
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In the paper
“This means that the evolution of mixed states is also bounded by Eq. (38).”
plaincorrelational- Asserted: passes — The authors put it forward as true.
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- Rests on an assumption: 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.
A conclusion the paper draws from its own results — it cites no other source for it, so there is none to check. The quote shows it is in the paper.
In the paper
“In this paper, alternative geometric derivations for both inequalities are obtained from the statistical distance between quantum states.”
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“For mixed states, this interpretation makes sense only when the states have the same purity Tr ( r 2 ) = Tr ( s 2 ) , and the evolution is unitary.”
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“Since [ Hj , Hk ] = 0 for all j and k , the interaction Hamiltonian for the N CZ gates is H = j Hj .”
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“In other words, the satellite qubits are back to their initial state, and the central qubit is in the state”
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“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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“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:”
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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
plaincorrelational- Asserted: passes — The authors put it forward as true.
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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).”
hedgeddirectSingle case- Asserted: passes — The authors put it forward as true.
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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
plaincorrelational- Asserted: passes — The authors put it forward as true.
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Probably 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]
Citations
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Law, J. (2001) · ACM SIGSOFT Software Engineering Notes · doi:10.1145/505482.505499 · Cited by 20,450
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Jaynes, E. T., & Bretthorst, G. L. (2003) · Cited by 4,600
Probability Theory: The Logic of Science
Jaynes, E. T. (2003) · doi:10.1017/cbo9780511790423 · Cited by 4,045
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Helstrom, C. W. (1969) · Journal of Statistical Physics · doi:10.1007/bf01007479 · Cited by 3,776
Quantum-mechanical noise in an interferometer
Caves, C. M. (1981) · Physical Review D · doi:10.1103/physrevd.23.1693 · Cited by 3,202
Statistical distance and the geometry of quantum states
Braunstein, S. L., & Caves, C. M. (1994) · Physical Review Letters · doi:10.1103/physrevlett.72.3439 · Cited by 3,176
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(1976) · Mathematics in Science and Engineering/Mathematics in science and engineering · doi:10.1016/s0076-5392(08)x6017-5 · Cited by 2,712
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Holevo, A. (2011) · doi:10.1007/978-88-7642-378-9 · Cited by 2,582
Giovannetti, V., Lloyd, S., & Maccone, L. (2006) · Physical Review Letters · doi:10.1103/physrevlett.96.010401 · Cited by 2,440
Research works printed but not linked: 2
- L. Mandelstam, 1945, J. Phys. (USSR), 9, 249
- J. Uffink, 1994, Proceedings of the Third International Workshop on Squeezed States and Uncertainty Relations
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“The fundamental difference, however, becomes obvious for parameterised, driven, and time-dependent L(ρt) [75].”
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del Campo, A., Egusquiza, I. L., Plenio, M. B., & Huelga, S. F. (2013) · Physical Review Letters · doi:10.1103/physrevlett.110.050403 · Cited by 493
Quantum speed limits: from Heisenberg's uncertainty principle to optimal quantum control
Deffner, S., & Campbell, S. (2017) · Maryland Shared Open Access Repository (USMAI Consortium) · doi:10.13016/m2heya-i4ys · Cited by 485
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
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
A language model drew the claims from the paper on 2026-10-06, under version 3 of the claim definition (model deepseek-v4.1-flash, prompt extract-v3).
A second model labelled each claim on 2026-10-06 (model jev-1.13-20260917, definition revision gold-guideline-v0.2).
Statistics and arithmetic are recomputed, not judged.
Confidence bands are shown only for fields calibrated on our annotated test set. None is calibrated yet, so no band is shown.
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