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Energy–time uncertainty relation for driven quantum systems
Open accessAbstract
We derive generalizations of the energy-time uncertainty relation for driven quantum systems. Using a geometric approach based on the Bures length between mixed quantum states, we obtain explicit expressions for the quantum speed limit time, valid for arbitrary initial and final quantum states and arbitrary driving protocols. Our results establish the fundamental limit on the rate of evolution of generic quantum systems.
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TL;DR
The paper derives generalized energy–time uncertainty relations and quantum speed limit times for driven closed quantum systems using a geometric Bures-length approach. It obtains Mandelstam–Tamm and Margolus–Levitin type bounds valid for arbitrary unitary driving, with mixed-state extensions via purification.
Key findings
- A Mandelstam–Tamm-type relation is derived for arbitrary initial and final mixed quantum states and arbitrary time-dependent Hamiltonians, with the time-averaged energy variance \(\Delta E_\tau = (1/\tau)\int_0^\tau dt\,(\langle H_t^2\rangle-\langle H_t\rangle^2)^{1/2}\).
- A Margolus–Levitin-type bound is derived first for pure initial states with ground-state energy set to zero, then extended to mixed states by purification; for arbitrary driving the generalized bound uses the time average of nondiagonal matrix elements \(|\langle \psi_0|H_t U_t|\psi_0\rangle|\).
- The Margolus–Levitin-type bound has prefactor \(4/\pi^2\), which the authors note is smaller than the prefactor \(2/\pi\) obtained numerically in Giovannetti, Lloyd, and Maccone.
- The quantum speed limit time depends strongly on the driving: it is inversely proportional to mean energy or energy spread, can be reduced exponentially if mean energy increases exponentially, can increase if driving decreases energy, and vanishes as \(\hbar \to 0\).
Methods
- Theoretical derivation using the Bures length/angle between density operators, quantum fidelity, the Braunstein–Caves Bures metric, Fisher information, the von Neumann equation, and a triangle-type inequality.
- The Mandelstam–Tamm bound is derived for arbitrary mixed states and arbitrary time-dependent Hamiltonians; the Margolus–Levitin bound assumes an initial pure state and \(E_g=0\), then uses purification for mixed states.
- The appendix treats time-independent and time-dependent Hamiltonians, including the condition \([H_t,H_0]=0\) for one form of the time-dependent bound; no experimental data are analyzed.
Limitations
- Stated: the Margolus–Levitin derivation assumes an initial pure state for simplicity, handles mixed states via purification, and sets the ground-state energy to zero.
- Stated: for time-dependent Hamiltonians, the time-independent-form Margolus–Levitin inequality remains valid only if \([H_t,H_0]=0\) for all \(t\in[0,\tau]\); otherwise a more general expression with nondiagonal matrix elements is required.
- Stated/evident: the derivative of the Bures length may change sign later, requiring care in deriving inequalities, and the paper corrects prior geometric Margolus–Levitin derivations; the work is purely theoretical with no experimental validation reported.
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Claims the paper makes
13 Not checked yet · 9 Own interpretation · 8 No basis given · 1 Supported · 1 Partly supported · accuracy not yet measured
Every source the paper cites for this claim was checked and supports it.
In the paper
“They showed that the Krotov algorithm of quantum optimal control theory fails to converge when the duration of the process is chosen below the quantum speed limit time.”
What the sources say
- Supportedcites [28] source
The paper states on p. 1 that, when the process duration is fixed below the quantum speed limit, optimal control fails to converge. Its Landau–Zener results on p. 2 report that the Krotov algorithm’s infidelity does not converge to zero below that limit. (LLM, conf=0.94)
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- Supported
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In the paper
“On the other hand, we note that the quantum speed limit time can also be increased when the energy of the system is decreased by the driving [41].”
What the sources say
- Partly supportedcites [41] sourceExperimental — accuracy not yet measured
The Discussion describes a driven harmonic oscillator being cooled as energy is extracted from it. It does not discuss quantum speed limits or say that extracting energy increases a quantum speed-limit time (searched: quantum speed limit, evolution time, decreased energy). (LLM, conf=0.82)
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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
“holds true if [ H t , H 0 ] = 0 , ∀ t ∈ [0 , τ ].”
plainconditional- Asserted: passes — The authors put it forward as true.
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- The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
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
“Summing up, we have obtained the quantum speed limit time τ QSL , which expresses the generalized energy-time uncertainty relation for arbitrary quantum unitary processes, in the generic form,”
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- The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
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
“Note that Eq. (8) also implies that the Fisher information f t is bounded from above by the variance of the energy of the system.”
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In the paper
“Equation (11) is a Mandelstam-Tamm uncertainty relation valid for arbitrary initial and final mixed quantum states and arbitrary time-dependent Hamiltonians.”
plain- Asserted: passes — The authors put it forward as true.
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- The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
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
“Expression (37) is a generalized Margolus-Levitin uncertainty relation valid for arbitrary angles between initial and final pure states and arbitrary external driving.”
plain- Asserted: passes — The authors put it forward as true.
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- The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
- The authors state this as an assumption: For the Margolus-Levitin bound, the driven quantum system starts in a pure state.
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 contrast to Eq. (16), the quantum speed limit time (38) depends on the square of the Bures angle.”
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In the paper
“Our results establish the fundamental limit on the rate of evolution of closed quantum systems.”
- Also stated in: § 3. Discussion · p. 6 · plain
“Our result provides the fundamental limit on the evolution rate of closed quantum systems, in particular quantum computation and information processing devices.”
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- The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
- Also stated in: § 3. Discussion · p. 6 · plain
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
“More importantly, the quantum speed limit time is inversely proportional to the energy of the system (mean energy or energy spread).”
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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
“The generalization to arbitrary mixed quantum states can be done invoking purification and the mean energy becomes E τ = (1 /τ ) ∫ τ 0 dt | tr { ρ 0 H t }| , cf. also appendix.”
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- The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
None of the sources cited for this claim have been checked yet.
In the 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].”
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- The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
None of the sources cited for this claim have been checked yet.
In the 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].”
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It may state two separate claims as one.
- The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
- The authors state this as an assumption: The ground state energy is set to zero, E_g = 0.
In the paper
“Equation (38) generalizes the Margolus-Levitin-type bound introduced by Giovannetti, Lloyd and Maccone for time-independent systems [15] (note that the numerical prefactor 4 /π 2 in Eq. (38) is smaller than the prefactor 2 /π obtained numerically in Ref. [15]).”
What the sources say
- Not checked yetcites [15] source
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It may state two separate claims as one.
- The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
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
“As a consequence, the minimal quantum evolution time from initial to final state can be estimated from below by”
- Also stated in: § 2. Margolus-Levitin bound · p. 5 · plain
“where E τ = (1 /τ ) ∫ τ 0 dt |〈 ψ 0 | H t | ψ 0 〉| is the time-averaged mean energy with respect to the initial state.”
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The quote only partly states this claim.
- The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
- The authors state this as an assumption: For the Margolus-Levitin bound, the driven quantum system starts in a pure state.
- The authors state this as an assumption: The ground state energy is set to zero, E_g = 0.
- Also stated in: § 2. Margolus-Levitin bound · p. 5 · plain
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 general, the instantaneous eigenbasis of H t is time-dependent. As a result, the estimations in Eqs. (25)-(27) do not necessarily hold true.”
hedgeddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
In the 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]).”
What the sources say
- Not checked yetcites [18] source
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None of the sources cited for this claim have been checked yet.
In the paper
“The quantum speed limit time gives the fundamental limit on the rate of evolution of quantum systems [20, 21, 22].”
What the sources say
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
None of the sources cited for this claim have been checked yet.
In the paper
“According to Braunstein and Caves, the Bures metric for two infinitesimally close density operators, ρ ′ = ρ + dρ , is [37] (see also Ref. [38]),”
plain- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
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
“The quantum speed limit time, therefore, strongly depends on the driving protocol: the more energy is pumped into a system and the more it is distributed among different states, the faster the system can evolve.”
boosteddirect- Asserted: passes — The authors put it forward as true.
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It may state two separate claims as one.
In the paper
“Wootters' statistical distance is the only Riemannian metric, up to a constant factor, which is invariant under all unitary transformations.”
What the sources say
- Not checked yetcites [31] source
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Probably not a substantial claim. The paper’s conclusions may not depend on it.
In the paper
“The Bures length (3) has been shown to be a natural generalization of Wootters' statistical distance for arbitrary mixed quantum states [37].”
What the sources say
- Not checked yetcites [37] source
boostedcorrelational- 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.
In the paper
“However, as discussed by e.g. Landau and Lifshitz [42], if a classical system is driven so rapidly that the driving frequency becomes comparable to the typical transition frequency between quantum states, quantum effects can no longer be neglected and have hence to be taken into account.”
What the sources say
- Not checked yetcites [42] Landau L D, 1980, Statistical Mechanics
plaindirect- 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.
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In the paper
“We first remark, as one might expect, that the quantum speed limit time is reduced when the Bures length between initial and final states is smaller, indicating that the closer the two states, the less time is needed to go from one state to the other.”
hedgeddirect- 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.
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
“Whereas τ QSL is determined by the initial energy in the time-independent case, it is set by the time-averaged mean energy or energy variance in the case of driven quantum systems.”
plaindirect- Asserted: passes — The authors put it forward as true.
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In the paper
“Due to the unitary dynamics the system then remains in a pure state at all times.”
plaindirect- Asserted: passes — The authors put it forward as true.
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Probably not a substantial claim.
- The authors state this as an assumption: The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
The authors state this as an assumption.
The quantum system is closed and evolves unitarily under arbitrary time-dependent driving.
“Specifically, we derive the quantum speed limit time for generic mixed states with arbitrary angle and for arbitrary timedependent unitary driving.”
The authors state this as an assumption.
For the Margolus-Levitin bound, the driven quantum system starts in a pure state.
“For simplicity, and without loss of generality, we assume that the driven quantum system starts in a pure state ρ 0 = | ψ 0 〉〈 ψ 0 | .”
The authors state this as an assumption.
The ground state energy is set to zero, E_g = 0.
“We additionally set the ground state energy to zero, E g = 0 [14].”
The authors state this as an assumption.
The time evolution of the quantum system obeys the time-dependent Schrödinger equation.
“The time evolution of the quantum system obeys the time-dependent Schr¨ odinger equation,”
The authors state this as an assumption.
The system's Hamiltonian is driven by arbitrary time-dependent parameters during time τ.
“In the following, we consider quantum systems whose Hamiltonians H t are driven by arbitrary timedependent parameters during time τ .”
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
“The usual QSL is tight for unitary dynamics of pure states [3].However, it becomes rather loose for mixed states.”
From Tight, robust, and feasible quantum speed limits for open dynamics (2019) · cites this paper as [3]
In the citing paper
“The distance term in many QSLs requires the square root of the initial and final states, thus the solution of the eigenvalue problem for the initial state ρ and final state σ [3,30,33,48].”
From Tight, robust, and feasible quantum speed limits for open dynamics (2019) · cites this paper as [3]
In the citing paper
“For isolated systems it quickly became clear that the Bures angle (25) would do the job [66,67]”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [67]
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 [67]
In the citing paper
“Hence, the relation to the quantum speed limit becomes apparent, since loosely speaking the quantum speed limit time sets a lower bound on the action E τ QSL /2 [67].”
From Quantum speed limits: from Heisenberg’s uncertainty principle to optimal quantum control (2017) · cites this paper as [67]
In the citing paper
“Such infinitely powerful driving, however, would be in conflict with so-called quantum speed limits which bound the evolution time by the inverse mean energy (Margolus and Levitin 1998), or, alternatively, its variance (Mandelstam and Tamm 1945) and apply to time-dependent unitary evolution between any two quantum states (Deffner and Lutz 2013).”
From Quantacell: powerful charging of quantum batteries (2015) · cites this paper as Deffner and Lutz 2013
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.
Rated minor: it would not touch a result — a value no result rests on, a p-value that stays on the same side of the significance level, or a conflict that is clerical, resolved elsewhere in the paper, or outside its own findings.
In the paper
§ Energy-time uncertainty relation for driven quantum systems · p. 2
“∆ E = ( 〈 H 〉 2 -〈 H 2 〉 ) 1 / 2”
§ 1. Mandelstam-Tamm bound · p. 4
“∆ E τ = (1 /τ ) ∫ τ 0 dt ( 〈 H 2 t 〉 -〈 H t 〉 2 ) 1 / 2”
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Metadata
- Authors
- Deffner, S., & Lutz, E.
- Affiliations
- University of Augsburg (DE)
- Corresponding author
- Deffner, Sebastian
- Venue
- Journal of Physics A: Mathematical and Theoretical
- Publisher
- IOP Publishing
- Published
- 2013-07-29
- Volume
- 46
- Issue
- 33
- Pages
- 335302
- ISSN
- 1751-8113 (ISSN-L)
- 1751-8121
- OpenAlex
- W2161551823
- Semantic Scholar
- 856d66591e01e1d5d7aeb0c7c06e0cfdec753e28
- Cited by
- 199
- References
- 49
- Citations by year
- 2026: 8 · 2025: 18 · 2024: 16 · 2023: 16 · 2022: 20 · 2021: 14 · 2020: 21 · 2019: 13 · 2018: 14 · 2017: 12 · 2016: 15 · 2015: 19 · 2014: 9 · 2013: 4
- FWCI
- 8.03
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- Journal of Physics A Mathematical and Theoretical (publishedVersion)
- arXiv (Cornell University) (publishedVersion)
- Maryland Shared Open Access Repository (USMAI Consortium) (submittedVersion)
- arXiv (Cornell University)
- Digital Repository at the University of Maryland (University of Maryland College Park)
- Funders
- Deutscher Akademischer Austauschdienst
- Deutsche Forschungsgemeinschaft
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Information and Cryptography
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Advanced Thermodynamics and Statistical Mechanics
- Keywords
- time-energy uncertainty relation, driven quantum systems, quantum speed limit, mixed quantum states, closed quantum systems, quantum state evolution
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.
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Statistics and arithmetic are recomputed, not judged.
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