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Generalized Geometric Quantum Speed Limits

Open access

Pires, Diego Paiva, Cianciaruso, Marco, Céleri, Lucas C., Adesso, Gerardo, Soares-Pinto, Diogo O.·2016·Physical Review X

DOI 10.1103/physrevx.6.021031

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Abstract

Understanding the speed with which a quantum system can evolve between distinguishable states has applications in quantum technology. A new theoretical study demonstrates a general family of quantum speed limits that can be applied to any physical process.

Checked 2026-10-06 · 49 references not checked yet

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TL;DR

The paper derives an infinite family of geometric quantum speed limits (QSLs) in one-to-one correspondence with contractive Riemannian metrics on quantum state space, valid for unitary and nonunitary evolutions. It shows that the conventional quantum Fisher information metric is only one choice and that the Wigner-Yanase skew information metric can yield tighter bounds for open-system dynamics.

Key findings

The central result is the family of bounds \(L_f(\rho_0,\rho_\tau) \le l^f_\gamma(\rho_0,\rho_\tau)\), where each contractive Riemannian metric, characterized by a Morozova-Čencov function \(f\), gives a different QSL; the quantum Fisher information metric recovers the QSL of Taddei et al.
For single-qubit unitary dynamics, the paper proves that the quantum Fisher information metric gives a tighter QSL than the Wigner-Yanase information metric, and conjectures it is tightest among all contractive Riemannian metrics for any one-qubit unitary dynamics.
For open-system examples, the Wigner-Yanase metric can be tighter: in parallel dephasing it is generally tighter when the frequency \(\omega_0\) is sufficiently small, e.g. for \(\beta=0\) and \(\theta_0=\pi/4\), while quantum Fisher information is slightly tighter for \(\beta=8\).
For amplitude damping, the Wigner-Yanase QSL is nearly saturated with \(\delta^{\mathrm{WY}}_\gamma \simeq 0\) and tighter than quantum Fisher information in almost the whole parameter space, except a small region near the north pole; for transversal dephasing with a plus state, Wigner-Yanase is tighter at short times for small enough \(\Gamma\) and \(\omega_0\).

Methods

Theoretical information-geometric derivation: the authors use the Morozova-Čencov-Petz characterization of contractive Riemannian metrics on density operators and the inequality that geodesic length is a lower bound to path length.
They decompose the metric tensor into population and coherence contributions, \(g^f_{\mu\nu}=F_{\mu\nu}+Q^f_{\mu\nu}\), and define a tightness indicator \(\delta^f_\gamma\) to compare bounds.
They illustrate the formalism with single-qubit unitary dynamics and nonunitary dephasing and amplitude-damping channels, comparing quantum Fisher information, Wigner-Yanase skew information, and the minimal Morozova-Čencov metric.

Limitations

The paper states that finding the tightest bound over all Morozova-Čencov functions is a formidable problem, and that only the quantum Fisher information and Wigner-Yanase information metrics have analytically known geodesic lengths for general dynamics.
The optimization is therefore restricted primarily to those two metrics; the authors leave open whether quantum Fisher information remains tightest for higher-dimensional systems or for other contractive Riemannian metrics.
The paper states that the family is within MT-like bounds and that extending the geometric approach to ML-like speed limits is left for further study; the experimental investigation is also stated to be reported elsewhere.
Evident from the text: the concrete examples are single-qubit dynamics, and the comparison is analytical/numerical rather than experimental.

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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
“Specifically, we establish an infinite family of quantum speed limits valid for unitary and nonunitary evolutions, based on an elegant information geometric formalism.”
- Also stated in: § III. GENERALIZED GEOMETRIC QUANTUM SPEED LIMITS · plain
  “We are now ready to present our main result, that is, a family of geometric QSLs that hold for any physical process and are in one-to-one correspondence with the contractive Riemannian metrics defined on the set of quantum states.”
§ Generalized Geometric Quantum Speed Limits > Abstract
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Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes
- The authors state this as an assumption: A Riemannian metric must be contractive under physical maps to be a bona fide measure of distinguishability between quantum states.
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
“L f ðρ 0 ; ρ τ Þ ≤ l f γ ðρ 0 ; ρ τ Þ;ð17Þ”
- Also stated in: § III. GENERALIZED GEOMETRIC QUANTUM SPEED LIMITS · boosted
  “Equation ( 17) represents the anticipated infinite family of generalized geometric QSLs and is the central result of this paper.”
§ Generalized Geometric Quantum Speed Limits > III. GENERALIZED GEOMETRIC QUANTUM SPEED LIMITS
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- The authors state this as an assumption: A Riemannian metric must be contractive under physical maps to be a bona fide measure of distinguishability between quantum states.
- The authors state this as an assumption: The dynamical evolution ρλ depends analytically on each parameter λμ.
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
“τ ≥ ℏ ΔE L QF ðρ 0 ; ρ τ Þ:ð27Þ”
§ Generalized Geometric Quantum Speed Limits > 1. Quantum Fisher information metric
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Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes
- The authors state this as an assumption: A Riemannian metric must be contractive under physical maps to be a bona fide measure of distinguishability between quantum states.
- The authors state this as an assumption: The dynamical evolution ρλ depends analytically on each parameter λμ.
None of the sources cited for this claim have been checked yet.
In the paper
“We are now ready to present our main result, that is, a family of geometric QSLs that hold for any physical process and are in one-to-one correspondence with the contractive Riemannian metrics defined on the set of quantum states.”
- Also stated in: § I. INTRODUCTION · plain
  “which is in one-to-one correspondence with the family of contractive Riemannian metrics characterized by the Morozova, Čencov, and Petz theorem.”
§ Generalized Geometric Quantum Speed Limits > III. GENERALIZED GEOMETRIC QUANTUM SPEED LIMITS
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- The authors state this as an assumption: A Riemannian metric must be contractive under physical maps to be a bona fide measure of distinguishability between quantum states.
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
“g QF μν ≤ 1 2ℏ 2 X j;l ðp j þ p l ÞhjjΔH λ μ jlihljΔH λ ν jji ¼ ℏ -2 C ðΔH λ μ ; ΔH λ ν Þ;ð24Þ”
§ Generalized Geometric Quantum Speed Limits > 1. Quantum Fisher information metric
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Tests✓ Asserted: passes✓ Checkable: passes✓ Load-bearing: passes✓ Based: passes
- The authors state this as an assumption: A Riemannian metric must be contractive under physical maps to be a bona fide measure of distinguishability between quantum states.
- The authors state this as an assumption: The dynamical evolution ρλ depends analytically on each parameter λμ.
In the paper
“CðΔH λ μ ; ΔH λ ν Þ ≔ -ð1=2ÞTrð½ ffiffiffiffi ffi ρ λ p ;ΔH λ μ ½ ffiffiffiffi ffi ρ λ p ;ΔH λ ν Þ reduces to the skew information between the evolved state and [57].”
§ Generalized Geometric Quantum Speed Limits > 2. Wigner-Yanase information metric
What the sources say
- Not checked yetcites [57] source
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The quote only partly states this claim. 5
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It may state two separate claims as one. 41

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
“By substituting the inequality (24) into Eq.( 17), we get the new bound”
- Also stated in: § 1. Quantum Fisher information metric · plain
  “L QF ðρ 0 ; ρ τ Þ ≤ 1 ℏ Z τ 0 dt ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi X r μ;ν¼1 C ðΔH λ μ ; ΔH λ ν Þ dλ μ dt dλ ν dt v u u t :ð25Þ”
§ Generalized Geometric Quantum Speed Limits > 1. Quantum Fisher information metric
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The quote only partly states this claim.
- The authors state this as an assumption: A Riemannian metric must be contractive under physical maps to be a bona fide measure of distinguishability between quantum states.
- The authors state this as an assumption: The dynamical evolution ρλ depends analytically on each parameter λμ.
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
“However, when restricting to the relevant case of an initial plus state (which is an optimal probe state for frequency estimation), i.e., ρ 0 ¼ jþihþj with jþi ¼ ðj0i þ j1iÞ= ffiffi ffi 2 p , one obtains the following simple expressions:”
§ Generalized Geometric Quantum Speed Limits > 1. Parallel and transversal dephasing channels
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Probably not a substantial claim. It may state two separate claims as one. The quote only partly states this claim.
- Rests on an assumption: The initial plus state is an optimal probe state for frequency estimation.
A result the paper reports from its own data — 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 QSL constructed with the Wigner-Yanase skew information is nearly globally optimal (as δ WY γ ≃ 0) and tighter than the one constructed with the quantum Fisher information (as indicated by Δδ γ ≥ 0) in almost the whole parameter space, except for a small region in the bottom-right corner (large r 0 , small θ 0 , i.e., around the state j0i) in which the quantum Fisher information bound is margi”
- Also stated in: § Additional information · boosted
  “What is more, in Fig. 7 one can see that for almost all initial states, except for a small neighborhood of the north pole (which is the asymptotic state of the amplitude damping channel), δ QF ≥ δ WY and δ WY ≃ 0; i.e., the Wigner-Yanase information metric provides us with a definitely tighter (and nearly saturated) QSL than the quantum Fisher information metric.”
§ Generalized Geometric Quantum Speed Limits > APPENDIX B: PARALLEL AND TRANSVERSAL DEPHASING
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In the paper
“Our work unifies and generalizes existing results on quantum speed limits and provides instances of novel bounds that are tighter than any established one based on the conventional quantum Fisher information.”
- Also stated in: § II. GEOMETRIC MEASURES OF DISTINGUISHABILITY · plain
  “Here, we establish a general family of geometric quantum speed limits with respect to an infinite hierarchy of contractive Riemannian metrics on the space of quantum states, unifying and extending previous results under an information geometry framework.”
§ Generalized Geometric Quantum Speed Limits > Abstract
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It may state two separate claims as one.
- The authors state this as an assumption: A Riemannian metric must be contractive under physical maps to be a bona fide measure of distinguishability between quantum states.
In the paper
“As pointed out by Kubo and Ando [55], among the MC functions, there exists a minimal one, f min ðtÞ ¼ 2t=ð1 þ tÞ, and a maximal one, f max ðtÞ ¼ ð1 þ tÞ=2, such that a generic MC function fðtÞ must satisfy f min ðtÞ ≤ fðtÞ ≤ f max ðtÞ.”
§ Generalized Geometric Quantum Speed Limits > II. GEOMETRIC MEASURES OF DISTINGUISHABILITY
What the sources say
- Not checked yetcites [55] source
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It may state two separate claims as one.
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
“Quite remarkably, this restriction will be enough to reveal how the choice of the quantum Fisher information metric, though ubiquitous in the existing literature, is only a special case that does not always provide the tightest lower bound.”
§ Generalized Geometric Quantum Speed Limits > III. GENERALIZED GEOMETRIC QUANTUM SPEED LIMITS
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It may state two separate claims as one.
- The authors state this as an assumption: A Riemannian metric must be contractive under physical maps to be a bona fide measure of distinguishability between quantum states.
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 quantum Fisher information metric corresponds to the MC function fðtÞ ¼ ð1 þ tÞ=2, so c f ðx; yÞ ¼ 2=ðx þ yÞ and Eq. ( 21) becomes”
§ Generalized Geometric Quantum Speed Limits > 1. Quantum Fisher information metric
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Probably not a substantial claim. It may state two separate claims as one.
- The authors state this as an assumption: A Riemannian metric must be contractive under physical maps to be a bona fide measure of distinguishability between quantum states.
In the paper
“Moreover, we can immediately see that it exactly coincides with the one reported in Ref. [9] and reduces to a MT-like bound, when further restricting to the case of a time-independent generator of the dynamics.”
§ Generalized Geometric Quantum Speed Limits > 1. Quantum Fisher information metric
What the sources say
- Not checked yetcites [9] source
Strength:plain· not yet calibratedCausal:direct· not yet calibrated
Tests✓ Asserted: passes✓ Checkable: passes? Load-bearing: unclear✓ Based: passes
It may state two separate claims as one.
- The authors state this as an assumption: A Riemannian metric must be contractive under physical maps to be a bona fide measure of distinguishability between quantum states.
- The authors state this as an assumption: The dynamical evolution ρλ depends analytically on each parameter λμ.
In the paper
“Giovannetti et al. [18] extended the ML QSL to the case of arbitrary mixed states and also showed that entanglement can speed up the dynamical evolution of a closed composite system.”
§ Generalized Geometric Quantum Speed Limits > I. INTRODUCTION
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
“More precisely, transversal dephasing noise stands as the relevant scenario whereby one can attain a precision in the estimation of the parameter ω 0 that scales superclassically with the number of qubits, even if such noise applies independently to each qubit (while any superclassical advantage is lost in the case of parallel dephasing noise).”
§ Generalized Geometric Quantum Speed Limits > 1. Parallel and transversal dephasing channels
What the sources say
- Not checked yetcites [59] source
- Not checked yetcites [60] source
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In the paper
“Then, Deffner and Lutz [33] extended the ML bound to open quantum systems by adopting again a geometric approach using the Bures angle.These authors have also introduced a new sort of bound, which is tighter than both the ML and MT ones, and shown that non-Markovianity can speed up the quantum evolution.”
§ Generalized Geometric Quantum Speed Limits > I. INTRODUCTION
What the sources say
- Not checked yetcites [33] source
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In the paper
“Specifically, the present study allows one to certify that, in a particular implementation, quantum states have been driven at the ultimate speed limit [42] and their evolution cannot be sped up further: This occurs whenever saturation of one of our bounds is demonstrated.”
§ Generalized Geometric Quantum Speed Limits > V. CONCLUSIONS
What the sources say
- Not checked yetcites [42] source
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In the paper
“where the analytical formulas of the fidelity Fðρ 0 ; ρ τ Þ and the affinity Aðρ 0 ; ρ τ Þ for any pair of one-qubit states ρ 0 and ρ τ are the following [29,73,74]:”
§ Generalized Geometric Quantum Speed Limits > V. CONCLUSIONS
What the sources say
- Not checked yetcites [29] source
- Not checked yetcites [73] source
- Not checked yetcites [74] source
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In the paper
“In fact, dephasing and amplitude damping are naturally occurring sources of decoherence in such an implementation, and our results can be accessed by means of spin ensemble measurements, which constitute the conventional types of detection in such a technique [71,72].”
§ Generalized Geometric Quantum Speed Limits > V. CONCLUSIONS
What the sources say
- Not checked yetcites [71]
- Not checked yetcites [72] source
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55 of 55

The authors state this as an assumption.
A Riemannian metric must be contractive under physical maps to be a bona fide measure of distinguishability between quantum states.
“However, it is also natural to assume that for a Riemannian metric to be bona fide in quantifying the distinguishability between two states, it must be contractive under the physical maps that represent the mathematical counterpart of noise, i.e., stochastic maps in the classical settings and completely positive trace-preserving maps in quantum.”
The authors state this as an assumption.
The dynamical evolution ρλ depends analytically on each parameter λμ.
“in such a way that ρ λ depends analytically on each parameter λ μ (μ ¼ 1; …; r).”
The paper relies on this without support.
The noisy evolution of the single qubit is governed by a master equation of Lindblad form with the specified Hamiltonian and Liouvillian.
“Let us now consider a noisy evolution of this state governed by a master equation of Lindblad form”
“We further consider the Hamiltonian H ¼ ðω 0 =2Þσ 3 , where ω 0 is the unitary frequency, and the Liouvillian”
The paper relies on this without support.
The initial plus state is an optimal probe state for frequency estimation.
“However, when restricting to the relevant case of an initial plus state (which is an optimal probe state for frequency estimation), i.e., ρ 0 ¼ jþihþj with jþi ¼ ðj0i þ j1iÞ= ffiffi ffi 2 p , one obtains the following simple expressions:”
The paper relies on this without support.
The quantum Fisher information and the Wigner-Yanase skew information are the only two metrics admitting known geodesics.
“While such a problem can only be solved in restricted form at present, due to the fact that the quantum Fisher information and the Wigner-Yanase skew information are the only two metrics admitting known geodesics”

5 of 5

Claims other papers make about this one

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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.

Transversal-dephasing eigenvalues do not sum to one
Minor
The displayed formula assigns the same value to both eigenvalues. Their sum is 1 þ r 0 ξt, rather than the trace-one value required for a density operator, except when r 0 ξt is zero.
Fidelity is not equal to affinity for commuting states
Minor
For commuting states, the paper's definitions give F=A2, not F=A. For example, a pure state and the maximally mixed qubit state commute but have F=1/2 and A=1/√2. The stated equality of the angles nevertheless follows from √F=A.

2 of 2

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Authors: Pires, D. P., Cianciaruso, M., Céleri, L. C., Adesso, G., & Soares-Pinto, D. O.
Affiliations: Universidade de São Paulo (BR)
University of Salerno (IT)
Universidade Federal de Goiás (BR)
University of Nottingham (GB)
Corresponding author: Pires, Diego Paiva
ORCID: Pires, Diego Paiva
Céleri, Lucas C.
Adesso, Gerardo
Soares-Pinto, Diogo O.
Venue: Physical Review X
Publisher: American Physical Society (APS)
Published: 2016-06-02
Volume: 6
Issue: 2
ISSN: 2160-3308 (ISSN-L)
DOI: 10.1103/physrevx.6.021031
OpenAlex: W2264466972
Semantic Scholar: e579c01d9ce8b70d2ff460ab76e23d822d547c56
Cited by: 265
References: 92
Citations by year: 2026: 18 · 2025: 35 · 2024: 22 · 2023: 28 · 2022: 23 · 2021: 41 · 2020: 26 · 2019: 21 · 2018: 22 · 2017: 24 · 2016: 4
FWCI: 19.49
Type: journal article
Language: en
Open access: gold
doi.org
Licence: cc-by
creativecommons.org/licenses/by/3.0/ (vor)
Locations: Physical Review X (publishedVersion, cc-by)
arXiv (Cornell University) (publishedVersion)
Repository@Nottingham (University of Nottingham) (publishedVersion, cc-by)
Repository@Nottingham (University of Nottingham) (submittedVersion, cc-by)
DOAJ (DOAJ: Directory of Open Access Journals) (submittedVersion, cc-by-sa)
Nottingham ePrints (University of Nottingham) (submittedVersion, cc-by)
arXiv (Cornell University)
Greater South Information System
Funders: Conselho Nacional de Desenvolvimento Científico e Tecnológico (445516/2014-3, 401230/2014-7, 305086/2013-8, 304955/2013-2, 443828/2014-8)
Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (108/2012)
European Research Council (637352)
Brazilian National Institute of Science and Technology
Topics: 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
Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Mechanics and Applications
Keywords: quantum speed limit, quantum state evolution, quantum dynamics, quantum technologies

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Generalized Geometric Quantum Speed Limits

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