Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1409.0535

Precision bounds in noisy quantum metrology

Jan Kołodyński

Published Sep 1, 2014• 22 citations
arXivHTMLPDF

Abstract

In an idealistic setting, quantum metrology protocols allow to sense physical parameters with mean squared error that scales as $1/N^2$ with the number of particles involved---substantially surpassing the $1/N$-scaling characteristic to classical statistics. A natural question arises, whether such an impressive enhancement persists when one takes into account the decoherence effects that are unavoidable in any real-life implementation. In this thesis, we resolve a major part of this issue by describing general techniques that allow to quantify the attainable precision in metrological schemes in the presence of uncorrelated noise. We show that the abstract geometrical structure of a quantum channel describing the noisy evolution of a single particle dictates then critical bounds on the ultimate quantum enhancement. Our results prove that an infinitesimal amount of noise is enough to restrict the precision to scale classically in the asymptotic $N$ limit, and thus constrain the maximal improvement to a constant factor. Although for low numbers of particles the decoherence may be ignored, for large $N$ the presence of noise heavily alters the form of both optimal states and measurements attaining the ultimate resolution. However, the established bounds are then typically achievable with use of techniques natural to current experiments. In this work, we thoroughly introduce the necessary concepts and mathematical tools lying behind metrological tasks, including both frequentist and Bayesian estimation theory frameworks. We provide examples of applications of the methods presented to typical qubit noise models, yet we also discuss in detail the phase estimation tasks in Mach-Zehnder interferometry both in the classical and quantum setting---with particular emphasis given to photonic losses while analysing the impact of decoherence.

DOI
10.48550/arxiv.1409.0535

Related papers

Efficient tools for quantum metrology with uncorrelated noise

Jan Kołodyński, Rafał Demkowicz-Dobrzański

Quantum metrology offers enhanced performance in experiments on topics such as gravitational wave-detection, magnetometry or atomic clock frequency calibration. The enhancement, however, requires a delicate tuning of relevant quantum features, such as entanglement or squeezing. For any practical application, the inevitable impact of decoherence needs to be taken into account in order to correctly quantify the ultimate attainable gain in precision. We compare the applicability and the effectiveness of various methods of calculating the ultimate precision bounds resulting from the presence of decoherence. This allows us to place a number of seemingly unrelated concepts into a common framework and arrive at an explicit hierarchy of quantum metrological methods in terms of the tightness of the bounds they provide. In particular, we show a way to extend the techniques originally proposed in Demkowicz-Dobrzański et al (2012 Nature Commun. 3 1063), so that they can be efficiently applied not only in the asymptotic but also in the finite number of particles regime. As a result, we obtain a simple and direct method, yielding bounds that interpolate between the quantum enhanced scaling characteristic for a small number of particles and the asymptotic regime, where quantum enhancement amounts to a constant factor improvement. Methods are applied to numerous models, including noisy phase and frequency estimation, as well as the estimation of the decoherence strength itself.

arXiv:1303.7271•New Journal of Physics•Jul 23, 2013•169 citations

The elusive Heisenberg limit in quantum-enhanced metrology

Rafał Demkowicz-Dobrzański, Jan Kołodyński, Mădălin Guţǎ

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.

arXiv:1201.3940•Nature Communications•Sep 18, 2012•773 citations

Protocols for optical quantum metrology

Zixin Huang

This thesis explores various protocols which can be applied to quantum metrology, such that precision of the measurement can beat the best classical approaches. We first demonstrate that ancilla-assisted schemes can improve phase estimation in the presence of noise, and we propose an experiment to demonstrate these effects. Given that the use of ancillae helps, it is no surprise that there are noisy channels for which the optimal strategy may depend on the noise level. We show that there is a non-trivial crossover between the ancilla-assisted strategy and the parallel-entangled strategy for Pauli noise channels and the amplitude damping channel. We see that under certain circumstances, the environment cannot access the parameter of interest. We then integrated techniques from quantum communication, to derive a general framework that renders quantum metrology protocols into cryptographic ones. Here the protocol is unconditionally secure if the trusted parties wish only to achieve the standard quantum limit. If Heisenberg-limited precision is to be achieved, they sacrifice some security, where adversaries can access information, but only at the risk of getting caught at it (cheat-sensitivity). In many scenarios, the quantum Fisher information is achieved for only one value of the phase. In a particular phase estimation scheme that measures the parity at the output of a Mach-Zehnder interferometer where the input is a two-mode squeezed vacuum, this problem is particularly imminent. We apply an adaptive technique to resolve this issue, showing that one can achieve precisions substantially below the shot noise limit. The key to a quadratically enhanced improvement in quantum metrology is entanglement. Finally, we demonstrate in a photonic system how high-dimensional entanglement can be certified, and provide an alternate interpretation of entanglement from an information-theoretic perspective.

•Apr 5, 2018•0 citations

True precision limits in quantum metrology

Marcin Jarzyna, Rafał Demkowicz-Dobrzański

We show that quantification of the performance of quantum-enhanced measurement schemes based on the concept of quantum Fisher information (QFI) yields results that are asymptotically equivalent to those from the rigorous Bayesian approach, provided generic uncorrelated noise is present in the setup. At the same time, we show that for the problem of decoherence-free phase estimation this equivalence breaks down, and the achievable estimation uncertainty calculated within the Bayesian approach is larger by a factor of π than that predicted from the QFI even in the large prior knowledge (small parameter fluctuation) regime, where the QFI is conventionally regarded as a reliable figure of merit. We conjecture that an analogous discrepancy is present in the arbitrary decoherence-free unitary parameter estimation scheme, and propose a general formula for the asymptotically achievable precision limit. We also discuss protocols utilizing states with an indefinite number of particles, and show that within the Bayesian approach it is legitimate to replace the number of particles with the mean number of particles in the formulas for the asymptotic precision, which as a consequence provides another argument for proposals based on the properties of the QFI of indefinite particle number states leading to sub-Heisenberg precisions not being practically feasible.

arXiv:1407.4805•New Journal of Physics•Jan 9, 2015•143 citations

Ultimate Precision Limits for Noisy Frequency Estimation

Andrea Smirne, Jan Kołodyński, Susana F. Huelga +1 more

Quantum metrology protocols allow us to surpass precision limits typical to classical statistics. However, in recent years, no-go theorems have been formulated, which state that typical forms of uncorrelated noise can constrain the quantum enhancement to a constant factor and, thus, bound the error to the standard asymptotic scaling. In particular, that is the case of time-homogeneous (Lindbladian) dephasing and, more generally, all semigroup dynamics that include phase covariant terms, which commute with the system Hamiltonian. We show that the standard scaling can be surpassed when the dynamics is no longer ruled by a semigroup and becomes time inhomogeneous. In this case, the ultimate precision is determined by the system short-time behavior, which when exhibiting the natural Zeno regime leads to a nonstandard asymptotic resolution. In particular, we demonstrate that the relevant noise feature dictating the precision is the violation of the semigroup property at short time scales, while non-Markovianity does not play any specific role.

arXiv:1511.02708•Physical Review Letters•Mar 24, 2016•177 citations

Fundamental noisy multiparameter quantum bounds

Shibdas Roy

Abstract Quantum multiparameter estimation involves estimating multiple parameters simultaneously and can be more precise than estimating them individually. Our interest here is to determine fundamental quantum limits to the achievable multiparameter estimation precision in the presence of noise. We first present a lower bound to the estimation error covariance for a noisy initial probe state evolving through a noiseless quantum channel. We then present a lower bound to the estimation error covariance in the most general form for a noisy initial probe state evolving through a noisy quantum channel. We show conditions and accordingly measurements to attain these estimation precision limits for noisy systems. We see that the Heisenberg precision scaling of 1/N can be achieved with a probe comprising N particles even in the presence of noise. In fact, some noise in the initial probe state or the quantum channel can serve as a feature rather than a bug, since the estimation precision scaling achievable in the presence of noise in the initial state or the channel in some situations is impossible in the absence of noise in the initial state or the channel. However, a lot of noise harms the quantum advantage achievable with N parallel resources, and allows for a best precision scaling of $${\bf{1}}/\sqrt{{\boldsymbol{N}}}$$ 1 / N . Moreover, the Heisenberg precision limit can be beaten with noise in the channel, and we present a super-Heisenberg precision limit with scaling of 1/N2 for optimal amount of noise in the channel, characterized by one-particle evolution operators. Furthermore, using γ-particle evolution operators for the noisy channel, where γ > 1, the best precision scaling attainable is 1/N2γ, which is otherwise known to be only possible using 2γ-particle evolution operators for a noiseless channel.

arXiv:1805.07902•Scientific Reports•Jan 31, 2019•9 citations
In an idealistic setting, quantum metrology protocols allow to sense physical parameters with mean squared error that scales as 1/N21/N^21/N2 with the number of particles involved---substantially surpassing the 1/N1/N1/N-scaling characteristic to classical statistics. A natural question arises, whether such an impressive enhancement persists when one takes into account the decoherence effects that are unavoidable in any real-life implementation. In this thesis, we resolve a major part of this issue by describing general techniques that allow to quantify the attainable precision in metrological schemes in the presence of uncorrelated noise. We show that the abstract geometrical structure of a quantum channel describing the noisy evolution of a single particle dictates then critical bounds on the ultimate quantum enhancement. Our results prove that an infinitesimal amount of noise is enough to restrict the precision to scale classically in the asymptotic NNN limit, and thus constrain the maximal improvement to a constant factor. Although for low numbers of particles the decoherence may be ignored, for large NNN the presence of noise heavily alters the form of both optimal states and measurements attaining the ultimate resolution. However, the established bounds are then typically achievable with use of techniques natural to current experiments. In this work, we thoroughly introduce the necessary concepts and mathematical tools lying behind metrological tasks, including both frequentist and Bayesian estimation theory frameworks. We provide examples of applications of the methods presented to typical qubit noise models, yet we also discuss in detail the phase estimation tasks in Mach-Zehnder interferometry both in the classical and quantum setting---with particular emphasis given to photonic losses while analysing the impact of decoherence.
Quantum metrology offers enhanced performance in experiments on topics such as gravitational wave-detection, magnetometry or atomic clock frequency calibration. The enhancement, however, requires a delicate tuning of relevant quantum features, such as entanglement or squeezing. For any practical application, the inevitable impact of decoherence needs to be taken into account in order to correctly quantify the ultimate attainable gain in precision. We compare the applicability and the effectiveness of various methods of calculating the ultimate precision bounds resulting from the presence of decoherence. This allows us to place a number of seemingly unrelated concepts into a common framework and arrive at an explicit hierarchy of quantum metrological methods in terms of the tightness of the bounds they provide. In particular, we show a way to extend the techniques originally proposed in Demkowicz-Dobrzański et al (2012 Nature Commun. 3 1063), so that they can be efficiently applied not only in the asymptotic but also in the finite number of particles regime. As a result, we obtain a simple and direct method, yielding bounds that interpolate between the quantum enhanced scaling characteristic for a small number of particles and the asymptotic regime, where quantum enhancement amounts to a constant factor improvement. Methods are applied to numerous models, including noisy phase and frequency estimation, as well as the estimation of the decoherence strength itself.
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.
This thesis explores various protocols which can be applied to quantum metrology, such that precision of the measurement can beat the best classical approaches. We first demonstrate that ancilla-assisted schemes can improve phase estimation in the presence of noise, and we propose an experiment to demonstrate these effects. Given that the use of ancillae helps, it is no surprise that there are noisy channels for which the optimal strategy may depend on the noise level. We show that there is a non-trivial crossover between the ancilla-assisted strategy and the parallel-entangled strategy for Pauli noise channels and the amplitude damping channel. We see that under certain circumstances, the environment cannot access the parameter of interest. We then integrated techniques from quantum communication, to derive a general framework that renders quantum metrology protocols into cryptographic ones. Here the protocol is unconditionally secure if the trusted parties wish only to achieve the standard quantum limit. If Heisenberg-limited precision is to be achieved, they sacrifice some security, where adversaries can access information, but only at the risk of getting caught at it (cheat-sensitivity). In many scenarios, the quantum Fisher information is achieved for only one value of the phase. In a particular phase estimation scheme that measures the parity at the output of a Mach-Zehnder interferometer where the input is a two-mode squeezed vacuum, this problem is particularly imminent. We apply an adaptive technique to resolve this issue, showing that one can achieve precisions substantially below the shot noise limit. The key to a quadratically enhanced improvement in quantum metrology is entanglement. Finally, we demonstrate in a photonic system how high-dimensional entanglement can be certified, and provide an alternate interpretation of entanglement from an information-theoretic perspective.
We show that quantification of the performance of quantum-enhanced measurement schemes based on the concept of quantum Fisher information (QFI) yields results that are asymptotically equivalent to those from the rigorous Bayesian approach, provided generic uncorrelated noise is present in the setup. At the same time, we show that for the problem of decoherence-free phase estimation this equivalence breaks down, and the achievable estimation uncertainty calculated within the Bayesian approach is larger by a factor of π than that predicted from the QFI even in the large prior knowledge (small parameter fluctuation) regime, where the QFI is conventionally regarded as a reliable figure of merit. We conjecture that an analogous discrepancy is present in the arbitrary decoherence-free unitary parameter estimation scheme, and propose a general formula for the asymptotically achievable precision limit. We also discuss protocols utilizing states with an indefinite number of particles, and show that within the Bayesian approach it is legitimate to replace the number of particles with the mean number of particles in the formulas for the asymptotic precision, which as a consequence provides another argument for proposals based on the properties of the QFI of indefinite particle number states leading to sub-Heisenberg precisions not being practically feasible.
Quantum metrology protocols allow us to surpass precision limits typical to classical statistics. However, in recent years, no-go theorems have been formulated, which state that typical forms of uncorrelated noise can constrain the quantum enhancement to a constant factor and, thus, bound the error to the standard asymptotic scaling. In particular, that is the case of time-homogeneous (Lindbladian) dephasing and, more generally, all semigroup dynamics that include phase covariant terms, which commute with the system Hamiltonian. We show that the standard scaling can be surpassed when the dynamics is no longer ruled by a semigroup and becomes time inhomogeneous. In this case, the ultimate precision is determined by the system short-time behavior, which when exhibiting the natural Zeno regime leads to a nonstandard asymptotic resolution. In particular, we demonstrate that the relevant noise feature dictating the precision is the violation of the semigroup property at short time scales, while non-Markovianity does not play any specific role.
Abstract Quantum multiparameter estimation involves estimating multiple parameters simultaneously and can be more precise than estimating them individually. Our interest here is to determine fundamental quantum limits to the achievable multiparameter estimation precision in the presence of noise. We first present a lower bound to the estimation error covariance for a noisy initial probe state evolving through a noiseless quantum channel. We then present a lower bound to the estimation error covariance in the most general form for a noisy initial probe state evolving through a noisy quantum channel. We show conditions and accordingly measurements to attain these estimation precision limits for noisy systems. We see that the Heisenberg precision scaling of 1/N can be achieved with a probe comprising N particles even in the presence of noise. In fact, some noise in the initial probe state or the quantum channel can serve as a feature rather than a bug, since the estimation precision scaling achievable in the presence of noise in the initial state or the channel in some situations is impossible in the absence of noise in the initial state or the channel. However, a lot of noise harms the quantum advantage achievable with N parallel resources, and allows for a best precision scaling of 1/N{\bf{1}}/\sqrt{{\boldsymbol{N}}}1/N​ 1 / N . Moreover, the Heisenberg precision limit can be beaten with noise in the channel, and we present a super-Heisenberg precision limit with scaling of 1/N2 for optimal amount of noise in the channel, characterized by one-particle evolution operators. Furthermore, using γ-particle evolution operators for the noisy channel, where γ > 1, the best precision scaling attainable is 1/N2γ, which is otherwise known to be only possible using 2γ-particle evolution operators for a noiseless channel.