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Cited by 773Quantum Information and CryptographyQuantum Mechanics and ApplicationsQuantum Computing Algorithms and Architecture

The elusive Heisenberg limit in quantum-enhanced metrology

Open access
Demkowicz-Dobrzański, Rafał, Kołodyński, Jan, Guţă, Mădălin·2012·Nature Communications
DOI 10.1038/ncomms2067PMC PMC3658100
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Abstract

Quantum precision enhancement is of fundamental importance for the development of advanced metrological optical experiments, such as gravitational wave detection and frequency calibration with atomic clocks. Precision in these experiments is strongly limited by the 1/√N shot noise factor with N being the number of probes (photons, atoms) employed in the experiment. Quantum theory provides tools to overcome the bound by using entangled probes. In an idealized scenario this gives rise to the Heisenberg scaling of precision 1/N. Here we show that when decoherence is taken into account, the maximal possible quantum enhancement in the asymptotic limit of infinite N amounts generically to a constant factor rather than quadratic improvement. We provide efficient and intuitive tools for deriving the bounds based on the geometry of quantum channels and semi-definite programming. We apply these tools to derive bounds for models of decoherence relevant for metrological applications including: depolarization, dephasing, spontaneous emission and photon loss. Quantum metrology employs the properties of quantum states to further enhance the accuracy of some of the most precise measurement schemes to date. Here, a method for estimating the upper bounds to achievable precision in quantum-enhanced metrology protocols in the presence of decoherence is presented.

AI snapshot

Written from the paper’s full text.

TL;DR

When decoherence acts independently on each probe, the asymptotic quantum enhancement in metrology generically collapses from Heisenberg scaling 1/N to a constant-factor improvement over the standard 1/√N scaling. The paper develops classical-simulation and channel-extension methods to bound this constant factor for depolarization, dephasing, spontaneous emission, and photon loss.

Key findings

  • In the asymptotic limit of infinite N, the maximal quantum enhancement under decoherence is generically a constant factor rather than the quadratic Heisenberg improvement.
  • The channel-extension (CE) method yields bounds at least as tight as the classical-simulation (CS) method and gives the tightest known asymptotic bounds for the relevant models; CS provides geometric intuition but can fail for some decoherence models.
  • New bounds are derived for depolarization and spontaneous emission, while dephasing and lossy interferometer bounds are re-derived with simplicity and match known results.
  • For Heisenberg scaling to survive, decoherence would need to decrease with N roughly as (1−η)≈1/N; the transition appears around N≈1/(1−η)=20 for η=0.95. For GEO600 with η=0.62, the reported experimentally observed reduction was a factor of 0.67.

Methods

  • The paper uses the quantum Cramér–Rao bound and quantum Fisher information, maximizing over input states to bound estimation uncertainty for unbiased estimators near a fixed parameter value.
  • CS bounds precision by writing the channel as a classical mixture and using the classical Fisher information; the tightest local CS uses two channels on the tangent line to the channel trajectory, with distances ε± from the boundary.
  • CE bounds precision by optimizing over equivalent Kraus representations via a Hermitian generator h, formulated as a semi-definite program and implemented with CVX for Matlab. Examples use two-level probes sensing a phase shift U(ϕ)=exp[(iσ3ϕ)/2] under depolarization, dephasing, spontaneous emission, and photon loss.

Limitations

  • The paper states that saturability of the bounds is not addressed: CS does not provide tight bounds in general, and the CE bounds are not proven saturable in the asymptotic limit.
  • The minimization over purifications may in principle yield a tight bound, but there is no effective algorithm to find the global minimum; one must show coincidence with a particular estimation strategy.
  • The methods focus on the asymptotic large-N regime; the bounds remain valid but weaker for finite N, and improving finite-N bounds is left for future work. The text also assumes decoherence acts independently on each probe and may be represented with a finite number of Kraus operators.

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

Integrity signals

Registry

Crossref · 10.1038/ncomms2067

Cites

60 of 60 references verified0 citation errors · 0 not checkable

Statistics

Nothing to recompute.

Claims the paper makes

0 of 25 cited sources checked · 0 supported · 0 flagged

  • Findingreasoning

    The CE method yields a standard-scaling bound for spontaneous emission, listed in Table 1.

    “Consequently, there is no need for minimization over h and from equation (19) we automatically obtain an SS bound listed in Table 1, which to our knowledge has not been reported in the literature before.”
    Quote in the paperSubstantial claim
  • Findingcitation

    The channel-extension bound in equation (16) is an upper bound on the quantum Fisher information that goes around the input state optimization.

    “This leads to an upper bound on , which goes around the input state optimization, yielding42: 16”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [42] Fujiwara, A. & Imai, H. A fibre bundle over manifolds of quantum channels and its application to quantum statistics. J. Phys. A: Math. Theor. 41, 255304 (2008). source

      Not checked in the free run.

  • Findingcitation

    For dephasing, the CS method yields a bound that is exactly the same as the one derived by Escher et al. or by the CE method.

    “Most importantly, the above bound, is exactly the same as the one derived by Escher et al.40 or by the CE method, where the minimum in equation (19) corresponds to .”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [40] Escher, B. M., de Matos Filho, R. L. & Davidovich, L. General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology. Nat. Phys. 7, 406–411 (2011). source

      Not checked in the free run.

  • Findingcitation

    Using the classical Cramér-Rao bound, the CS method yields a lower bound on the uncertainty of the original estimation problem.

    “Hence, using the classical CRB46, we obtain a lower bound on the uncertainty of the original problem equation (1) 7”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [46] Kay, S. M. Fundamentals of Statistical Signal Processing: Estimation Theory, Prentice Hall, 1993.

      Not checked in the free run.

  • Findingcitation

    For the GEO600 experiment with reported overall optical transmission η=0.62, the theoretically predicted maximal quantum enhancement is a factor reduction in estimation uncertainty compared with the classical limit.

    “For the reported overall optical transmission η=0.62, the theoretically predicted maximal quantum enhancement amounts to a factor reduction in estimation uncertainty compared with the classical limit.”
    Quote in the paperSubstantial claim0 of 1 sources checked
    • cites [17] LIGO Collaboration.. A gravitational wave observatory operating beyond the quantum shot-noise limit. Nat. Phys. 7, 962–965 (2011). source

      Not checked in the free run.

  • May combine two claims. 15
  • The quote only partly states this claim. 1
  • Classified as probably not a substantial claim. 6

  • May combine two claims.

    Findingcitation

    The CE method yields an optimal bound for the lossy interferometer, listed in Table 1, which is asymptotically equally tight to the best bounds known in the literature.

    “The optimal bound (see Table 1) corresponds to h with non-zero elements . This bound is asymptotically equally tight to the best bounds known in the literature383940, proving again that CE method despite its simplicity is able to provide powerful results in a straightforward manner.”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [38] Kołodyński, J. & Demkowicz-Dobrzański, R. Phase estimation without a priori phase knowledge in the presence of loss. Phys, Rev. A 82, 053804 (2010). source

      Not checked in the free run.

    • cites [39] Knysh, S., Smelyanskiy, V. N. & Durkin, G. A. Scaling laws for precision in quantum interferometry and the bifurcation landscape of the optimal state. Phys. Rev. A 83, 021804(R) (2011). source

      Not checked in the free run.

    • cites [40] Escher, B. M., de Matos Filho, R. L. & Davidovich, L. General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology. Nat. Phys. 7, 406–411 (2011). source

      Not checked in the free run.

  • May combine two claims.

    Findingnone given

    The lossy interferometer channel is non-extremal but φ-extremal.

    “Although the corresponding channel Λϕ is non-extremal, it is unfortunately ϕ-extremal and CS cannot be used.”
    Quote in the paperSubstantial claim

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

    Findingown data

    For depolarization, the CE method yields a tighter bound than the CS method, and the analytical result of the semi-definite optimization is shown in Table 1.

    “Applying the CE method, it is possible to further improve the bound and the analytical result of the semi-definite optimization is also shown in Table 1.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingown data

    Using the CS method, one infers standard scaling of precision and calculates the distances from the boundary of the space of quantum channels, resulting in the bound presented in Table 1.

    “Using CS method, we infer the SS of precision and calculate the 'distances' from the boundary of the SQC (see Methods) resulting in the bound presented in Table 1.”
    Quote in the paperSubstantial claim

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

    Findingcitation

    Depolarization involves four Kraus operators, making it a channel lying in the interior of the space of quantum channels.

    “Its description involves four Kraus operators 20 which makes it an example of a channel lying in the interior of the SQC.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Findingreasoning

    A quantitative standard-scaling bound can be obtained from equation (19), where the minimization runs over h satisfying equation (18), and can be formulated as a semi-definite program.

    “Here we go a step further and show that in this case we can obtain a quantitative SS bound 19 where the minimization runs over h satisfying equation (18), and can be formulated as a semi-definite program, as described in the Methods section.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    Dephasing is a non-extremal and φ-non-extremal channel, allowing the CS construction and yielding the bound given in Table 1.

    “It is, however, a non-extremal and more importantly ϕ-non-extremal channel allowing for the CS construction with (see Methods), yielding the bound given in Table 1.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Findingreasoning

    Dephasing involves only two Kraus operators, so it is not a full-rank channel and lies on the boundary of the space of quantum channels.

    “As it involves only two Kraus operators, it is not a full-rank channel and lies on the boundary of the SQC.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationcitation

    Channels that are φ-non-extremal obey standard precision scaling and include all full-rank channels.

    “Channels that have this additional property will be called 'ϕ-non-extremal'. They obey the standard precision scaling, and include all full-rank channels (that is, channels lying in the interior of the SQC)4142.”
    Quote in the paperSubstantial claim0 of 2 sources checked
    • cites [41] Matsumoto, K. On metric of quantum channel spaces. Preprint at arxiv:1006.0300 (2010).

      Not checked in the free run.

    • cites [42] Fujiwara, A. & Imai, H. A fibre bundle over manifolds of quantum channels and its application to quantum statistics. J. Phys. A: Math. Theor. 41, 255304 (2008). source

      Not checked in the free run.

  • May combine two claims.

    Premisecitation

    In the presence of decoherence, the optimal input states do not have an intuitive form and the maximization of the quantum Fisher information with rising N becomes hard even numerically.

    “In the presence of decoherence, the optimal input states do not have an intuitive form and the maximization of the QFI (with rising N) becomes hard even numerically202930.”
    Quote in the paperSubstantial claim0 of 3 sources checked
    • cites [20] Huelga, S. F. et al. Improvement of frequency standards with quantum entanglement. Phys. Rev. Lett. 79, 3865–3868 (1997). source

      Not checked in the free run.

    • cites [29] Dorner, U. et al. Optimal quantum phase estimation. Phys. Rev. Lett. 102, 040403 (2009). source

      Not checked in the free run.

    • cites [30] Demkowicz-Dobrzański, R. et al. Quantum phase estimation with lossy interferometers. Phys. Rev. A 80, 013825 (2009). source

      Not checked in the free run.

  • May combine two claims.

    Interpretationreasoning

    For extremal channels, ɛ±=0 and the bound is not useful.

    “For extremal channels, ɛ±=0 and the bound is not useful.”
    Quote in the paperSubstantial claim

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

    Interpretationnone given

    The CS method provides an intuitive geometric insight into the absence of asymptotic Heisenberg scaling in the presence of decoherence.

    “Yet the CS method, which may fail to provide equally tight bounds, provides an intuitive geometric insight into the absence of asymptotic HS in the presence of decoherence.”
    Quote in the paperNot a substantial claim

  • May combine two claims.

    Interpretationnone given

    The CE method is more powerful and, for the most relevant metrological models, yields in the asymptotic limit of large number of probes the tightest bounds known in the literature.

    “The CE method is more powerful and for the most relevant metrological models yields in the asymptotic limit of large number of probes the tightest bounds known in the literature.”
    Quote in the paperNot a substantial claim

  • Classified as probably not a substantial claim.

    Interpretationreasoning

    The only way to convince oneself that the global minimum has been achieved is to show that the theoretical lower bound on estimation uncertainty coincides with the performance of a particular estimation strategy.

    “and the only way to convince oneself that one has achieved the global minimum is again to show that the theoretical lower bound on estimation uncertainty coincides with the performance of a particular estimation strategy.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Premisenone given

    The derived bounds are lower bounds on estimation uncertainties, and an upper bound can always be found by choosing a particular estimation method.

    “We should stress however, that these are lower bounds on estimation uncertainties and an upper bound can always be found by choosing a particular estimation method.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    The GEO600 experiment operates close to the fundamental quantum limit, and any significant improvement is possible only if the optical loss is further reduced.

    “The reported experimentally observed reduction was a factor of 0.67, which is an indication that the experiment operates close to the fundamental quantum limit and any significant improvement is possible only if the optical loss is further reduced.”
    Quote in the paperSubstantial claim

  • May combine two claims.

    Interpretationreasoning

    In the decoherence-free case where the channels are unitary, which are extremal points of the space of quantum channels, the only admissible pφ is the irregular Dirac delta distribution, so Heisenberg scaling is possible.

    “This happens, for example, in the decoherence-free case where Λϕ are unitary channels, which are extremal points of the SQC and the only admissible pϕ in equation (4) is the irregular Dirac delta distribution being zero on all channels except Λϕ .”
    Quote in the paperSubstantial claim

  • Probably not a substantial claim: may not pass S3 (load-bearing), S4 (based).

    Premisenone given

    The most relevant models in quantum-enhanced metrology belong to the class of channels formed by a unitary rotation followed by φ-independent decoherence.

    “The most relevant models in quantum-enhanced metrology belong to this class, but the methods presented may be applied to more general models as well.”
    Quote in the paperNot a substantial claim
    • Rests on an assumption: The most relevant models in quantum-enhanced metrology belong to the class of channels formed by a unitary rotation followed by ϕ-independent decoherence.

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

    Premisenone given

    All examples of channels presented are of the form Λφ[ρ]=Λ[Uφ ρUφ†], a concatenation of a unitary rotation encoding the estimated parameter and a φ-independent decoherence process.

    “All examples of channels presented below are of the form Λϕ[ρ]=Λ[Uϕ ρUϕ†], that is, a concatenation of a unitary rotation encoding the estimated parameter ϕ and an ϕ-independent decoherence process.”
    Quote in the paperNot a substantial claim
    • The authors state this as an assumption: The decoherence process in the examples is ϕ-independent.

  • Probably not a substantial claim: may not pass S2 (checkable), S3 (load-bearing), S4 (based).

    Interpretationnone given

    The methods presented may be applied to more general models as well.

    “The most relevant models in quantum-enhanced metrology belong to this class, but the methods presented may be applied to more general models as well.”
    Quote in the paperNot a substantial claim
20 of 20

  • The authors state this as an assumption.

    The decoherence process acts independently on each of the probes.

    “Note that since the decoherence process is assumed to act independently on each of the probes, the global channel is described by the tensor product .”

  • The paper relies on this without support.

    In most metrology applications, N is larger by several orders of magnitude than the transition point.

    “As in most metrology applications N is larger by several order of magnitude, we expect that the SS scaling provides a reliable bound for the optimal estimation precision.”

  • The authors state this as an assumption.

    The decoherence process in the examples is ϕ-independent.

    “All examples of channels presented below are of the form Λϕ[ρ]=Λ[Uϕ ρUϕ†], that is, a concatenation of a unitary rotation encoding the estimated parameter ϕ and an ϕ-independent decoherence process.”

  • The paper relies on this without support.

    The most relevant models in quantum-enhanced metrology belong to the class of channels formed by a unitary rotation followed by ϕ-independent decoherence.

    “The most relevant models in quantum-enhanced metrology belong to this class, but the methods presented may be applied to more general models as well.”
4 of 4

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?

  • The paper evaluated the prediction for optimal interrogation times of product and cat states for a model of non-Markovian dephasing as proposed in [39] and obtained the ratio t_GHZ = t_p / N, just as in the Markov case.

    “We have evaluated the prediction for the optimal interrogation times of product and cat states for a model of non-Markovian dephasing of this type, as proposed in [39], and obtained the ratio t GHZ = tp / N , just as in the Markov case.”

    From Quantum Speed Limits in Open System Dynamics (2013) · cites this paper as [39]

Cites

Flagged first, then by citations

Research works printed but not linked: 1

  1. Nielsen, M. A. & Chuang, I. L. Quantum Computing and Quantum Information (Cambridge University Press, 2000).

Cited by

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Related

Authors
Demkowicz-Dobrzański, R., Kołodyński, J., & Guţă, M.
Affiliations
University of Warsaw (PL)
Corresponding author
Demkowicz-Dobrzański, Rafał
ORCID
Venue
Nature Communications
Publisher
Springer Science and Business Media LLC
Published
2012-09-18
Volume
3
Issue
1
Pages
1063-1063
ISSN
2041-1723 (ISSN-L)
DOI
10.1038/ncomms2067
PMC
PMC3658100
OpenAlex
W1993488859
Semantic Scholar
d1a85cf8ff0d2b33284248bcf0e3b3dcd7e5afb7
Cited by
773
References
60
Citations by year
2026: 48 · 2025: 63 · 2024: 68 · 2023: 78 · 2022: 36 · 2021: 68 · 2020: 69 · 2019: 52 · 2018: 53 · 2017: 68 · 2016: 53 · 2015: 45 · 2014: 45 · 2013: 25 · 2012: 2
FWCI
36.76
Type
journal article
Language
en
Open access
Licence
Locations
  • Nature Communications (publishedVersion, cc-by-nc-sa)
  • Europe PMC (PubMed Central) (publishedVersion, cc-by-nc-sa)
  • PubMed Central (publishedVersion, cc-by-nc-sa)
  • PubMed (publishedVersion)
  • arXiv (Cornell University) (submittedVersion)
  • RePEc: Research Papers in Economics (submittedVersion)
  • arXiv (Cornell University)
Funders
  • European Commission
  • Fundacja na rzecz Nauki Polskiej
  • Engineering and Physical Sciences Research Council (EP/E052290/1)
  • CHIST-ERA
Topics
  • 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
  • Physical Sciences › Computer Science › Artificial Intelligence › Quantum Computing Algorithms and Architecture
Keywords
Heisenberg limit, quantum-enhanced sensing, decoherence, semidefinite programming, quantum Fisher information, precision bounds, depolarization, dephasing, spontaneous emission, photon loss, gravitational wave detection

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