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Cited by 265Quantum Computing Algorithms and ArchitectureQuantum Information and CryptographyQuantum Mechanics and Applications

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.

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

AI snapshot

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.

Model-written summary. Check it against the paper before relying on it.

Integrity signals

Registry

Crossref · 10.1103/physrevx.6.021031

Cites

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Claims the paper makes

0 of 31 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning

    The paper establishes an infinite family of quantum speed limits valid for unitary and nonunitary evolutions, based on an information geometric formalism.

    “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.”
    Quote in the paperSubstantial 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.
  • Findingreasoning

    For any contractive Riemannian metric g_f, the geodesic distance L_f(ρ0,ρτ) between initial and final states is a lower bound to the length l_fγ of the path followed by the evolved state.

    “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.”
    Quote in the paperSubstantial 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.
  • Findingreasoning

    In the one-parameter case, τ ≥ ħ/ΔE L_QF(ρ0,ρτ).

    “τ ≥ ℏ ΔE L QF ðρ 0 ; ρ τ Þ:ð27Þ”
    Quote in the paperSubstantial 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.
  • Findingcitation

    The family of geometric QSLs is in one-to-one correspondence with the contractive Riemannian metrics defined on the set of quantum states.

    “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.”
    Quote in the paperSubstantial 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.
  • Findingreasoning

    For the quantum Fisher information metric, g_QFμν ≤ ħ^-2 C(ΔHλμ, ΔHλν).

    “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”
    Quote in the paperSubstantial 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 quote only partly states this claim. 5
  • Classified as probably not a substantial claim. 21
  • May combine two claims. 41

  • The quote only partly states this claim.

    Findingreasoning

    The new bound L_QF(ρ0,ρτ) ≤ (1/ħ)∫0^τ dt sqrt(Σ C(ΔHλμ,ΔHλν) dλμ/dt dλν/dt) holds for r-parameter unitary dynamics.

    “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”
    Quote in the paperSubstantial 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.

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingown data

    The parallel dephasing channel is unital; its Kraus operators satisfy both Σ K†K=I and Σ KK†=I.

    “Notice that the Kraus operators satisfy not only P 1 j¼0 K † j K j ¼ I but also P 1 j¼0 K j K † j ¼ I, as such a channel is unital.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given

    Qf does not depend on the initial azimuthal angle φ0, even though the eigenstates of the evolved state do depend on φ0.

    “Interestingly, Q f does not depend on the initial azimuthal angle ϕ 0 as well, even though the eigenstates of the evolved state do depend on ϕ 0 .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    For amplitude damping, the evolved state has spectral decomposition Eq. (53) with p± and eigenstates Eq. (54).

    “In this case, it is easy to verify that the evolved state ρ t ¼ P j Kj ρ 0 K † j ¼ ð1=2ÞðI þ rt • σÞ has the following spectral decomposition:”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingnone given

    The transversal dephasing channel is unital; it leaves the maximally mixed state invariant.

    “It is worthwhile noticing that the transversal dephasing channel is also unital; i.e., it leaves the maximally mixed state invariant.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim. May combine two claims.

    Findingreasoning

    Putting the spectral decomposition into Eqs. (15) and (16) yields F Eq. (38) and Qf Eq. (39).

    “By putting the above equations into Eqs.( 15) and ( 16), one obtains, respectively,”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingnone given

    The amplitude damping channel is not unital; its Kraus operators satisfy only Σ K†K=I.

    “satisfying only P j K † j Kj ¼ I since this channel is not unital.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    Fig. 3(d) shows that according to Eq. (39), Qf is zero for θ0=π/2 and ω0=0 (β=0); thus g_f is described solely by F Eq. (38), arising only from populations.

    “Indeed, according to Eq. ( 39), the purely quantum contribution Q f to the metric is equal to zero for θ 0 ¼ π=2 and ω 0 ¼ 0 (β ¼ 0).Thus, the speed of evolution g f is described solely by the term F given in Eq. ( 38) and arising only from the populations of the evolved state.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim. May combine two claims. The quote only partly states this claim.

    Findingreasoning

    For transversal dephasing with initial plus state, F is given by Eq. (49) and Qf by Eq. (50).

    “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:”
    Quote in the paperSubstantial claim
    • Rests on an assumption: The initial plus state is an optimal probe state for frequency estimation.

  • May combine two claims.

    Findingreasoning

    For transversal dephasing, the evolved state has spectral decomposition Eq. (45) with p± and eigenstates Eq. (46).

    “By writing the spectral decomposition of the density operator ρ t , we get”
    Quote in the paperSubstantial claim

  • May combine two claims. The quote only partly states this claim.

    Findingcitation

    Consequently, sqrt(I) ≤ ΔE and the QSL satisfies Eq. (32).

    “so ffiffiffi ffi I p ≤ ΔE and”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingreasoning

    The bound in Eq. (29) turns into Eq. (30) in the one-parameter case.

    “Therefore, the bound in Eq. ( 29) turns into”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given

    The effect of amplitude damping consists in shrinking the Bloch sphere towards the north pole, or the state |0>.

    “The effect of amplitude damping consists in shrinking the Bloch sphere towards the north pole, or the state j0i.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    For parallel dephasing with β=8, the dynamics does not saturate the bound for either QFI or WY metrics, although QFI provides in general a slightly tighter QSL.

    “We can see that for β ¼ 8, the dynamics does not saturate the bound for either of the two metrics, although the quantum Fisher information metric provides, in general, a slightly tighter QSL.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    For parallel dephasing, the evolved state has spectral decomposition Eq. (36) with p± and eigenstates Eq. (37).

    “One can easily see that the evolved state ρ t has the following spectral decomposition:”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Findingreasoning

    The QSL becomes Eq. (31).

    “The QSL thus becomes”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Findingreasoning

    Putting Eq. (28) into the bound in Eq. (17) gives Eq. (29).

    “By putting Eq. ( 28) into the bound in Eq. ( 17), we get”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingnone given

    The family of speed limits provided in the paper are within the class of MT-like bounds.

    “It is important to remark that the family of speed limits provided in this paper are within the class of MT-like bounds.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingnone given

    The effect of parallel dephasing is the same as phase flip and consists in shrinking the Bloch sphere onto the z axis; states diagonal in the computational basis are left invariant.

    “The effect of parallel dephasing is exactly the same as the one of phase flip and consists in shrinking the Bloch sphere onto the z axis of states diagonal in the computational basis, which are instead left invariant.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    The choice of quantum Fisher information is only a special case and does not always provide the tightest lower bound in open system dynamics.

    “By investigating paradigmatic examples of unitary and noisy physical processes and of contractive Riemannian metrics, we have seen, in fact, how the choice of the quantum Fisher information, corresponding to an extremal metric and being ubiquitous in the existing literature, is only a special case, which does not always provide the tightest lower bound in the realistic case of open system dynamics”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingnone given

    The Wigner-Yanase information metric corresponds to the MC function f(t)=(1/4)(sqrt(t)+1)^2, so c_f(x,y)=4/(sqrt(x)+sqrt(y))^2 and Eq. (21) becomes Eq. (28).

    “The Wigner-Yanase information metric corresponds to the MC function fðtÞ ¼ ð1=4Þð ffiffi t p þ 1Þ 2 , so c f ðx; yÞ ¼ 4=ð ffiffi ffi x p þ ffiffi ffi y p Þ 2 and Eq. ( 21) becomes”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    For single-qubit evolutions, the QFI bound is the tightest for idealized unitary dynamics, while the Wigner-Yanase skew information bound can be significantly tighter in common open dynamics, effectively yielding the optimal bound for amplitude damping dynamics, as certified by a nearly vanishing tightness indicator.

    “For single-qubit evolutions, we showed that the latter is in fact the tightest for the idealized case of unitary dynamics, while our novel bound based on the Wigner-Yanase skew information can instead be significantly tighter in the most common instances of open dynamics, effectively yielding the optimal bound (even among all the other unverifiable Riemannian metrics) for amplitude damping”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Findingreasoning

    Putting the spectral decomposition into Eqs. (15) and (16) yields F Eq. (58) and Qf Eq. (59).

    “By putting the above equations into Eqs.(15) and ( 16) one obtains, respectively,”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    F does not depend on the initial azimuthal angle φ0 because the eigenvalues p_j of the evolved state do not depend on φ0.

    “Although F is a function of the initial purity r 0 and of time, it does not depend on the initial azimuthal angle ϕ 0 since the eigenvalues of the evolved state p j do not depend on ϕ 0 .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    For amplitude damping, Qf vanishes only for θ0=0 or π, whereas F vanishes in neither of these cases nor for θ0=π/2, because only the north pole, not the entire z axis, is left invariant by the dynamics.

    “However, contrarily to the parallel dephasing channel case, here the purely quantum contribution Q f vanishes only for θ 0 ¼ 0; π, whereas the term F vanishes in neither of these cases nor for θ 0 ¼ π=2, as expected due to the fact that now only the north pole, and not the entire z axis of the Bloch sphere, is left invariant by the dynamics.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    Eq. (39), the quantum contribution to speed, depends on the MC function and is identically zero for incoherent initial states with θ0=0 or π.

    “Equation (39), which instead describes the truly quantum contribution to the speed of evolution g f and depends on the specific choice of the MC function f, is identically zero for all the incoherent initial states such that θ 0 is either 0 or π.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    According to Eq. (59), Qf vanishes identically for θ0=0, and the speed of evolution g and corresponding evolution path length in Eq. (17) become independent of the choice of MC function f.

    “Moreover, according to Eq. ( 59), the purely quantum contribution Q f vanishes identically for θ 0 ¼ 0, and the speed of evolution g and corresponding evolution path length in Eq. ( 17) become independent of the choice of the MC function f.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    For amplitude damping, neither F nor Qf depends on the initial azimuthal angle φ0.

    “As in the case of the parallel dephasing channel, neither contribution F or Q f to the speed of evolution g f depends on the initial azimuthal angle ϕ 0 .”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    The generalization in Eq. (A20) is only true in the one-qubit unitary case; for systems of dimension higher than 2, the coefficients c_f cannot be extracted from the path integral as done above.

    “However, we also note that this is only true in the onequbit unitary case.If the dimensionality of the system is higher than 2, then Q f becomes a nontrivial sum, as defined in Eq. ( 16), so that the various time-independent coefficients c f ðp j ; p l Þ cannot be extracted from the path integral as we have just done above.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    For amplitude damping, the Wigner-Yanase skew information QSL is nearly globally optimal (δ_WY ≃ 0) and tighter than the quantum Fisher information QSL (Δδ_γ ≥ 0) in almost the whole parameter space, except for a small region in the bottom-right corner (large r0, small θ0, around |0>) where the quantum Fisher information bound is marginally tighter.

    “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.”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Findingreasoning

    For transversal dephasing, the geodesic distances corresponding to the quantum Fisher information and the Wigner-Yanase information metric can be analytically computed using Eqs. (5), (6), (A12), and (A13) with Eq. (B9).

    “On the other hand, regarding transversal dephasing noise, we are able to analytically compute both the geodesic distances corresponding to the quantum Fisher information and the Wigner-Yanase information metric by using Eqs.(5), ( 6), (A12), and (A13) with”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). The quote only partly states this claim.

    Findingnone given

    For amplitude damping, the path length for the Wigner-Yanase information metric is evaluated as Eq. (C4).

    “For the Wigner-Yanase information metric, the path length is evaluated as”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingown data

    For parallel dephasing, the QSL constructed with the Wigner-Yanase skew information is tighter than the one constructed with the quantum Fisher information when Δδ_γ > 0, and looser when Δδ_γ < 0.

    “The QSL constructed with the Wigner-Yanase skew information is tighter than (respectively, looser than) the one constructed with the quantum Fisher information when Δδ γ > 0 (resp.Δδ γ < 0), as in panel (b) [resp.(a)].”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim.

    Findingnone given

    Equation (A10) can be generalized to any pair of contractive Riemannian metrics g_f and g_h as Eq. (A20).

    “On the one hand, Eq. (A10) can be trivially generalized to any pair of contractive Riemannian metrics g f and g h as follows:”
    Quote in the paperSubstantial claim

  • The quote only partly states this claim.

    Findingreasoning

    For amplitude damping, the path length for the quantum Fisher information metric is given by Eq. (C2).

    “For the quantum Fisher information metric, the analytical expression of the path length is given by”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    For a unitarily evolving one-qubit state, the eigenvalues p1,2=(1±r0)/2 are time independent, so F=0.

    “Since the eigenvalues p 1;2 ¼ ð1 Æ r 0 Þ=2 of the unitarily evolving one-qubit state are time independent, it immediately follows that F ¼ 0;”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The paper's work unifies and generalizes existing results on quantum speed limits.

    “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.”
    Quote in the paperSubstantial 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.

  • May combine two claims.

    Premisecitation

    Among MC functions, there is a minimal f_min(t)=2t/(1+t) and maximal f_max(t)=(1+t)/2, and any MC function satisfies f_min ≤ f ≤ f_max.

    “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.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [55] source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning

    The choice of the quantum Fisher information metric, though ubiquitous, is only a special case that does not always provide the tightest lower bound.

    “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.”
    Quote in the paperSubstantial 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.

  • Classified as probably not a substantial claim. May combine two claims.

    Premisenone given

    The quantum Fisher information metric corresponds to the MC function f(t)=(1+t)/2, so c_f(x,y)=2/(x+y), and Eq. (21) becomes Eq. (22).

    “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”
    Quote in the paperSubstantial 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.

  • May combine two claims.

    Interpretationcitation

    Eq. (27) exactly coincides with the one reported in Ref. [9] and reduces to a MT-like bound when further restricting to a time-independent generator.

    “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.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [9] source

      Not checked in the free run.

    • 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.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given

    An experimental investigation of the main results for closed and open system dynamics can be achieved using a highly controllable nuclear magnetic resonance setup, with no need for complete quantum state tomography.

    “In this respect, we would like to point out that an experimental investigation of the main results presented here, for both closed and open system dynamics, can be achieved, in particular, using a highly controllable nuclear magnetic resonance setup, with no need for a complete quantum state tomography.”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing). May combine two claims.

    Premisecitation

    Giovannetti et al. extended the ML QSL to arbitrary mixed states and showed that entanglement can speed up the dynamical evolution of a closed composite system.

    “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.”
    Quote in the paperNot a substantial claim0 of 1 sources checked
    • cites [18] source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation

    Transversal dephasing noise is the relevant scenario whereby one can attain precision in estimation of ω0 scaling superclassically with the number of qubits, even if noise applies independently to each qubit; any superclassical advantage is lost for parallel dephasing.

    “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).”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [59] source

      Not checked in the free run.

    • cites [60] source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing).

    Interpretationown data

    This highlights the power of the general approach to reach beyond the state of the art.

    “This highlights the power of our general approach to reach beyond the state of the art.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationreasoning

    Each bona fide geometric measure of distinguishability gives rise to a different QSL tailored to initial mixed states and separates contributions of populations and coherences.

    “Specifically, each bona fide geometric measure of distinguishability gives rise to a different quantum speed limit which is particularly tailored to the case of initial mixed states and such that the contributions of the populations of the evolved state and of the coherences of its time variation are clearly separated.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    When initial and final states commute, fidelity and affinity coincide, the Bures angle equals the Hellinger angle, and the Wigner-Yanase bound is less tight than the QFI bound by a factor 1/sqrt(2).

    “However, when the initial and final states commute, we have that the corresponding fidelity and affinity coincide, Fðρ 0 ; ρ τ Þ ¼ Aðρ 0 ; ρ τ Þ, and so the Bures angle is equal to the Hellinger angle, which implies that in this case the bound emerging from the Wigner-Yanase metric is less tight than the one corresponding to the quantum Fisher information by a factor of 1= ffiffi ffi 2 p .”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationown data

    The present analysis shows that the approach applies straightforwardly to obtain novel, tighter bounds in dynamical cases of interest for quantum technologies, as corroborated for transversal dephasing noise.

    “Once more, the present analysis shows that our approach applies straightforwardly to obtain novel, tighter bounds in dynamical cases of interest for quantum technologies, as corroborated here, in particular, for the metrologically relevant case of transversal dephasing noise.”
    Quote in the paperSubstantial claim

  • Classified as probably not a substantial claim. May combine two claims.

    Interpretationown data

    This is in stark contrast to unitary evolutions and constitutes a first demonstration of the usefulness of the generalized approach to speed limits in quantum dynamics.

    “This is in stark contrast to the case of unitary evolutions, discussed in the previous section, and constitutes a first demonstration of the usefulness of our generalized approach to speed limits in quantum dynamics.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisecitation

    Deffner and Lutz extended the ML bound to open quantum systems using a geometric approach with the Bures angle, introduced a new bound tighter than both ML and MT, and showed that non-Markovianity can speed up quantum evolution.

    “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.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [33] source

      Not checked in the free run.

  • May combine two claims.

    Interpretationcitation

    The present study allows one to certify that in a particular implementation, quantum states have been driven at the ultimate speed limit and their evolution cannot be sped up further whenever saturation of one of the bounds is demonstrated.

    “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.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [42] source

      Not checked in the free run.

  • May combine two claims.

    Interpretationown data

    The generalized analysis reveals a physical mechanism, distinct from dephasing, in which it leads to significantly tighter bounds than those established in previous literature.

    “This reveals another important physical mechanism, distinct from dephasing, in which our generalized analysis leads to significantly tighter bounds than those established in previous literature, which, in this case, is clearly”
    Quote in the paperNot a substantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Premisecitation

    The analytical formulas of the fidelity F(ρ0,ρτ) and the affinity A(ρ0,ρτ) for any pair of one-qubit states are given by Eqs. (A12) and (A13).

    “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]:”
    Quote in the paperNot a substantial claim0 of 3 sources checked
    • cites [29] source

      Not checked in the free run.

    • cites [73] source

      Not checked in the free run.

    • cites [74] source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation

    Dephasing and amplitude damping are naturally occurring sources of decoherence in NMR implementation, and the results can be accessed by spin ensemble measurements.

    “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].”
    Quote in the paperNot a substantial claim0 of 2 sources checked
    • cites [71]

      Not checked in the free run.

    • cites [72] source

      Not checked in the free run.

  • Probably not a substantial claim: may not pass S3 (load-bearing).

    Interpretationnone given

    Equation (A17) seems hard to generalize to other contractive Riemannian metrics because the analytical expressions of the corresponding geodesic lengths are still unknown.

    “On the other hand, Eq. (A17) seems hard to generalize to other contractive Riemannian metrics since the analytical expressions of the corresponding geodesic lengths are still unknown.”
    Quote in the paperNot a substantial claim
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

Each is checked against this paper’s own text: does it say what the citing paper says it does?

  • Tracing over the environment generally turns unitary dynamics between purifications into nonunitary dynamics between the mixed states, so the Bures metric need not select geodesics generated by unitary operations even when the states share a spectrum.

    “However, tracing over H B , in general, turns unitary dynamics between jψ 1 i and jψ 2 i into nonunitary dynamics between ρ 1 and ρ 2 [50].Consequently, the Bures metric does not necessarily select geodesics generated by unitary operations, even if ρ 1 and ρ 2 have the same spectrum.”

    From Tightening Quantum Speed Limits for Almost All States (2018) · cites this paper as [50]

  • In the eigenbasis of the density matrix ρ, g_μν takes the form g_μν = 1/4 Σ_i <λ_i|dρ|λ_i>^2/λ_i + 1/2 Σ_{i<j} |<λ_i|dρ|λ_j>|^2/λ_j h(λ_i/λ_j), where h is the Morozova-Čencov function.

    “In the eigenbasis of the density matrix ρ, g µν takes the form as [91][92][93] g µν = 1 4 i λ i |dρ|λ i 2 λ i + 1 2 i<j | λ i |dρ|λ j | 2 λ j h(λ i /λ j ) ,(78”

    From Quantum Fisher information matrix and multiparameter estimation (2020) · cites this paper as [93]

  • Pires et al established an infinite family of quantum speed limits based on a contractive Riemannian metric.

    “Recently, Pires et al [93] established an infinite family of quantum speed limits based on a contractive Riemannian metric discussed in section 2.4.2.”

    From Quantum Fisher information matrix and multiparameter estimation (2020) · cites this paper as [93]

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

Cites

Flagged first, then by citations

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

Cited by

Sorted by citations

Related

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 · 2015: 1
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 › Computer Science › Artificial Intelligence › 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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