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Generalized Geometric Quantum Speed Limits

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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

Overview

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Written from the paper’s full text.

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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Crossref · 10.1103/physrevx.6.021031

Cites

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Figures

FIG. 1 Illustration of geometric quantum speed limits. The dashed blue curve is the path γ in the quantum-state space representing a generic evolution between an initial state ρ 0 and a final state ρτ , parametrized by time t ∈ ½ 0 ; τ Š . Given a metric on the quantum-state space, the length of this path is denoted by l γ ð ρ 0 ; ρτ Þ . The solid red curve denotes the geodesic connecting ρ 0 to ρτ , whose length is given by L ð ρ 0 ; ρτ Þ . Quantum speed limits originate from the fact that the geodesic amounts to the path of shortest length among all physical evolutions between the given initial and final states: L ð ρ 0 ; ρτ Þ ≤ l γ ð ρ 0 ; ρτ Þ ∀ γ . Such inequality can be interpreted as follows. For any given physical evolution γ from ρ 0 to ρτ , and according to any valid metric, the maximum distance between the initial ρ 0 and the final state ρτ is the length of the path l γ ð ρ 0 ; ρτ Þ followed by the system. The ensuing minimal time necessary for this distance to reach a chosen value is the time at which the path length reaches this value. This interpretation provides a neat criterion for the saturation of the lower bound on the evolution time, that is, when the dynamical evolution coincides with a geodesic of the considered metric. 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.
FIG. 1 Illustration of geometric quantum speed limits. The dashed blue curve is the path γ in the quantum-state space representing a generic evolution between an initial state ρ 0 and a final state ρτ , parametrized by time t ∈ ½ 0 ; τ Š . Given a metric on the quantum-state space, the length of this path is denoted by l γ ð ρ 0 ; ρτ Þ . The solid red curve denotes the geodesic connecting ρ 0 to ρτ , whose length is given by L ð ρ 0 ; ρτ Þ . Quantum speed limits originate from the fact that the geodesic amounts to the path of shortest length among all physical evolutions between the given initial and final states: L ð ρ 0 ; ρτ Þ ≤ l γ ð ρ 0 ; ρτ Þ ∀ γ . Such inequality can be interpreted as follows. For any given physical evolution γ from ρ 0 to ρτ , and according to any valid metric, the maximum distance between the initial ρ 0 and the final state ρτ is the length of the path l γ ð ρ 0 ; ρτ Þ followed by the system. The ensuing minimal time necessary for this distance to reach a chosen value is the time at which the path length reaches this value. This interpretation provides a neat criterion for the saturation of the lower bound on the evolution time, that is, when the dynamical evolution coincides with a geodesic of the considered metric. 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.CC-BY · From the PDF
FIG. 2 The family of MC functions is upper bounded by a maximal one, f max ð t Þ ¼ ð 1 þ t Þ = 2 , corresponding to the quantum Fisher information metric (red solid line), and lower bounded by a minimal one, f min ð t Þ ¼ 2 t= ð 1 þ t Þ (purple dotted line). Any MCfunction f ð t Þ satisfies f min ð t Þ ≤ f ð t Þ ≤ f max ð t Þ ; i.e., its graph falls into the shaded area, as shown in the particular case of the MC function f WY ð t Þ ¼ ð 1 = 4 Þð ffi ffi t p þ 1 Þ 2 corresponding to the Wigner-Yanase information metric (blue dashed line). All the MC functions collapse to 1 at t ¼ 1 . This reflects the fact that the corresponding Riemannian metrics are regular on the maximally mixed state.
FIG. 2 The family of MC functions is upper bounded by a maximal one, f max ð t Þ ¼ ð 1 þ t Þ = 2 , corresponding to the quantum Fisher information metric (red solid line), and lower bounded by a minimal one, f min ð t Þ ¼ 2 t= ð 1 þ t Þ (purple dotted line). Any MCfunction f ð t Þ satisfies f min ð t Þ ≤ f ð t Þ ≤ f max ð t Þ ; i.e., its graph falls into the shaded area, as shown in the particular case of the MC function f WY ð t Þ ¼ ð 1 = 4 Þð ffi ffi t p þ 1 Þ 2 corresponding to the Wigner-Yanase information metric (blue dashed line). All the MC functions collapse to 1 at t ¼ 1 . This reflects the fact that the corresponding Riemannian metrics are regular on the maximally mixed state.CC-BY · From the PDF
FIG. 3 Evolution path lengths l f γ for parallel dephasing processes by considering the contractive Riemannian metrics corresponding to the following MC functions: f QF ð t Þ ¼ f max ð t Þ (red solid line), f WY ð t Þ (blue dashed line), and f min ð t Þ (purple dotted line) for (a) r 0 ¼ 1 = 4 , θ 0 ¼ π = 4 , β ¼ 8 , (b) r 0 ¼ 3 = 4 , θ 0 ¼ π = 4 , β ¼ 8 , (c) r 0 ¼ 1 = 2 , θ 0 ¼ π = 4 , β ¼ 0 , and (d) r 0 ¼ 1 = 2 , θ 0 ¼ π = 2 , β ¼ 0 . The insets in each panel show the relative difference δ f γ , Eq. (19), for parallel dephasing processes by considering the quantum Fisher information metric (red solid line) and the Wigner-Yanase information metric (blue dashed line); such a relative difference can be regarded as an indicator of the tightness of the bounds (the smaller δ f γ , the tighter the bounds).
FIG. 3 Evolution path lengths l f γ for parallel dephasing processes by considering the contractive Riemannian metrics corresponding to the following MC functions: f QF ð t Þ ¼ f max ð t Þ (red solid line), f WY ð t Þ (blue dashed line), and f min ð t Þ (purple dotted line) for (a) r 0 ¼ 1 = 4 , θ 0 ¼ π = 4 , β ¼ 8 , (b) r 0 ¼ 3 = 4 , θ 0 ¼ π = 4 , β ¼ 8 , (c) r 0 ¼ 1 = 2 , θ 0 ¼ π = 4 , β ¼ 0 , and (d) r 0 ¼ 1 = 2 , θ 0 ¼ π = 2 , β ¼ 0 . The insets in each panel show the relative difference δ f γ , Eq. (19), for parallel dephasing processes by considering the quantum Fisher information metric (red solid line) and the Wigner-Yanase information metric (blue dashed line); such a relative difference can be regarded as an indicator of the tightness of the bounds (the smaller δ f γ , the tighter the bounds).CC-BY · From the PDF
FIG. 4 Contour plot of the difference Δ δγ ≡ δ QF γ -δ WY γ between the tightness parameter corresponding to the quantum Fisher information metric and the one corresponding to the Wigner-Yanase information metric, for the parallel dephasing process as a function of r 0 and θ 0 , for (a) Γ τ ¼ 10 , ω 0 ¼ 10 and (b) Γ τ ¼ 10 , ω 0 ¼ 0 . 1 . The QSL constructed with the WignerYanase skew information is tighter than (respectively, looser than) the one constructed with the quantum Fisher information when Δ δ γ > 0 (resp. Δ δ γ < 0 ), as in panel (b) [resp. (a)].
FIG. 4 Contour plot of the difference Δ δγ ≡ δ QF γ -δ WY γ between the tightness parameter corresponding to the quantum Fisher information metric and the one corresponding to the Wigner-Yanase information metric, for the parallel dephasing process as a function of r 0 and θ 0 , for (a) Γ τ ¼ 10 , ω 0 ¼ 10 and (b) Γ τ ¼ 10 , ω 0 ¼ 0 . 1 . The QSL constructed with the WignerYanase skew information is tighter than (respectively, looser than) the one constructed with the quantum Fisher information when Δ δ γ > 0 (resp. Δ δ γ < 0 ), as in panel (b) [resp. (a)].CC-BY · From the PDF
FIG. 5 Plot of the relative difference δ f γ , Eq. (19), indicating the tightness of the QSLs (the smaller δ f γ , the tighter the bounds) corresponding, respectively, to the quantum Fisher information (red solid line) and the Wigner-Yanase skew information (blue dashed line), for a qubit initially in the plus state and undergoing a transversal dephasing process with (a) Γ ¼ 0 . 1 and ω 0 ¼ 0 . 01 , (b) Γ ¼ 0 . 25 and ω 0 ¼ 0 . 01 , (c) Γ ¼ 0 . 25 and ω 0 ¼ 0 . 033 , and (d) Γ ¼ 1 and ω 0 ¼ 1 .
FIG. 5 Plot of the relative difference δ f γ , Eq. (19), indicating the tightness of the QSLs (the smaller δ f γ , the tighter the bounds) corresponding, respectively, to the quantum Fisher information (red solid line) and the Wigner-Yanase skew information (blue dashed line), for a qubit initially in the plus state and undergoing a transversal dephasing process with (a) Γ ¼ 0 . 1 and ω 0 ¼ 0 . 01 , (b) Γ ¼ 0 . 25 and ω 0 ¼ 0 . 01 , (c) Γ ¼ 0 . 25 and ω 0 ¼ 0 . 033 , and (d) Γ ¼ 1 and ω 0 ¼ 1 .CC-BY · From the PDF
FIG. 6 Evolution path lengths l f γ for amplitude damping processes related to the contractive Riemannian metrics corresponding to the following MC functions: f QF ð t Þ ¼ f max ð t Þ (red solid line), f WY ð t Þ (blue dashed line), and f min ð t Þ (purple dotted line) for (a) r 0 ¼ 1 = 2 , θ 0 ¼ π = 2 , (b) r 0 ¼ 3 = 4 , θ 0 ¼ π = 2 , (c) r 0 ¼ 1 = 4 , θ 0 ¼ π = 4 , (d) r 0 ¼ 1 = 4 , θ 0 ¼ π = 2 , (e) r 0 ¼ 1 = 4 , θ 0 ¼ 0 , and (f) r 0 ¼ 1 = 2 , θ 0 ¼ 0 . The insets in each panel show the relative difference δ f γ , Eq. (19), for amplitude damping processes by considering the quantum Fisher information metric (red solid line) and the Wigner-Yanase information metric (blue dashed line); such a relative difference can be regarded as an indicator of the tightness of the bounds (the smaller δ f γ , the tighter the bounds).
FIG. 6 Evolution path lengths l f γ for amplitude damping processes related to the contractive Riemannian metrics corresponding to the following MC functions: f QF ð t Þ ¼ f max ð t Þ (red solid line), f WY ð t Þ (blue dashed line), and f min ð t Þ (purple dotted line) for (a) r 0 ¼ 1 = 2 , θ 0 ¼ π = 2 , (b) r 0 ¼ 3 = 4 , θ 0 ¼ π = 2 , (c) r 0 ¼ 1 = 4 , θ 0 ¼ π = 4 , (d) r 0 ¼ 1 = 4 , θ 0 ¼ π = 2 , (e) r 0 ¼ 1 = 4 , θ 0 ¼ 0 , and (f) r 0 ¼ 1 = 2 , θ 0 ¼ 0 . The insets in each panel show the relative difference δ f γ , Eq. (19), for amplitude damping processes by considering the quantum Fisher information metric (red solid line) and the Wigner-Yanase information metric (blue dashed line); such a relative difference can be regarded as an indicator of the tightness of the bounds (the smaller δ f γ , the tighter the bounds).CC-BY · From the PDF
FIG. 7 Contour plot of (a) the tightness indicator δ WY γ of the bound specified by the Wigner-Yanase information metric and (b) the difference Δ δ γ ≡ δ QF γ -δ WY γ between the tightness parameter corresponding to the quantum Fisher information metric and the one corresponding to the Wigner-Yanase information metric, for the amplitude damping process as a function of r 0 and θ 0 , for Γ τ ¼ 10 . 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 Δ δ 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 j 0 i ) in which the quantum Fisher information bound is marginally tighter.
FIG. 7 Contour plot of (a) the tightness indicator δ WY γ of the bound specified by the Wigner-Yanase information metric and (b) the difference Δ δ γ ≡ δ QF γ -δ WY γ between the tightness parameter corresponding to the quantum Fisher information metric and the one corresponding to the Wigner-Yanase information metric, for the amplitude damping process as a function of r 0 and θ 0 , for Γ τ ¼ 10 . 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 Δ δ 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 j 0 i ) in which the quantum Fisher information bound is marginally tighter.CC-BY · From the PDF
FIG. 8 The difference between the right-hand side and lefthand side of Eq. (A18) as a function of r 0 and of the relative angle φ 0 ; τ .
FIG. 8 The difference between the right-hand side and lefthand side of Eq. (A18) as a function of r 0 and of the relative angle φ 0 ; τ .CC-BY · From the PDF

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  • The quote only partly states this claim. 5
  • Probably not a substantial claim. 21
  • It may state two separate claims as one. 41
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”
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Citations

Cites

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Research works printed but not linked: 3

  1. W. Pauli, 1926, Handbuch der Physik
  2. W. Pauli, 1933, Handbuch der Physik
  3. L. Mandelstam, 1945, J. Phys. USSR, 9, 249

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Metadata

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
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
Licence
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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