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:1311.6600 · Journal of Physics A Mathematical and Theoretical

Optimal condition for measurement observable via error-propagation

Wei Zhong, Xiao-Ming Lu, Xiao-Xing Jing, Xiaoguang Wang

Published Sep 10, 2014• 22 citations
arXivHTMLPDF

Abstract

Propagation of error is a widely used estimation tool in experiments, where the estimation precision of the parameter depends on the fluctuation of the physical observable. Thus which observable is chosen will greatly affect the estimation sensitivity. Here we study the optimal observable for the ultimate sensitivity bounded by the quantum Cramér-Rao theorem in parameter estimation. By invoking the Schrödinger-Robertson uncertainty relation, we derive the necessary and sufficient condition for the optimal observables saturating the ultimate sensitivity for single parameter estimate. By applying this condition to Greenberg-Horne-Zeilinger states, we obtain the general expression of the optimal observable for separable measurements to achieve the Heisenberg-limit precision and show that it is closely related to the parity measurement. However, Jose {\em et al} [Phys. Rev. A {\bf 87}, 022330 (2013)] have claimed that the Heisenberg limit may not be obtained via separable measurements. We show this claim is incorrect.

DOI
10.1088/1751-8113/47/38/385304

Related papers

Optimal probes and error-correction schemes in multi-parameter quantum metrology

Wojciech Górecki, Sisi Zhou, Liang Jiang +1 more

We derive a necessary and sufficient condition for the possibility of achieving the Heisenberg scaling in general adaptive multi-parameter estimation schemes in presence of Markovian noise. In situations where the Heisenberg scaling is achievable, we provide a semidefinite program to identify the optimal quantum error correcting (QEC) protocol that yields the best estimation precision. We overcome the technical challenges associated with potential incompatibility of the measurement optimally extracting information on different parameters by utilizing the Holevo Cramér-Rao (HCR) bound for pure states. We provide examples of significant advantages offered by our joint-QEC protocols, that sense all the parameters utilizing a single error-corrected subspace, over separate-QEC protocols where each parameter is effectively sensed in a separate subspace.

arXiv:1901.00896•Quantum•Jul 2, 2020•51 citations

Enhanced parameter estimation by measurement of non-Hermitian operators

Jianning Li, Haodi Liu, Zhihai Wang +1 more

Abstract Quantum metrology aims at delivering new quantum-mechanical improvement to technologies of parameter estimations with precision bounded by the quantum Cramér-Rao bound. The currently used quantum Cramér-Rao bound was established with measurements of observables restricted to be Hermitian. This constrains the bound and limits the precision of parameter estimation. In this paper, we lift the constraint and derive a previously unknown quantum Cramér-Rao bound. We find that the new bound can reach arbitrary small value with mixed states and it breaks the Heisenberg limit in some cases. We construct a setup to measure non-Hermitian operators and discuss the saturation of the present bound. Two examples—the phase estimation with Greenberger-Horne-Zeilinger states of trapped ions and the adiabatic quantum parameter estimation with the nuclear magnetic resonance—are employed to demonstrate the theory. The present study might open a new research direction—non-Hermitian quantum metrology.

AAPPS bulletin•Oct 4, 2023•15 citations

The Most Informative Cramér--Rao Bound for Quantum Two-Parameter Estimation with Pure State Probes

Simon K. Yung, C. M. Yung, Lorcán O. Conlon +1 more

Optimal measurements for quantum multiparameter estimation are complicated by the uncertainty principle. Generally, there is a trade-off between the precision with which different parameters can be simultaneously estimated. The task of determining the minimum achievable estimation error is a central task of multiparameter quantum metrology. For estimating parameters encoded in pure quantum states, the ultimate limit is known, but is given by the solution of a non-trivial minimisation problem. We present a new expression for the achievable bound for two-parameter estimation with pure states that is considerably simpler. We also determine the optimal measurements, completing the problem of two-parameter estimation with pure state probes. To demonstrate the utility of our result, we determine the precision limit for estimating displacements using grid states.

quant-pharXiv:2511.14950•Nov 20, 2025•0 citations

Quantum Measurement Bounds beyond the Uncertainty Relations

Vittorio Giovannetti, Seth Lloyd, Lorenzo Maccone

In quantum mechanics, the Heisenberg uncertainty relations and the Cramér-Rao inequalities typically limit the precision in the estimation of a parameter through the standard deviation of a conjugate observable. Here we extend these relations by giving a bound to the precision of a parameter in terms of the expectation value of the conjugate observable. This has both foundational and practical consequences: in quantum optics it resolves a controversy over which is the ultimate precision in interferometry.

arXiv:1109.5661•Physical Review Letters•Jun 26, 2012•70 citations

Finite-Shot Sensitivity for Moment Estimation in Quantum Metrology

Shaowei Du, Shuheng Liu, Weidong Li +3 more

The quantum Cramér-Rao bound can be saturated only asymptotically and does not specify how many measurements are needed for a concrete estimator to approach it. We develop a finite-measurement theory for method-of-moments estimation, where the parameter is inferred from the sample mean of a calibrating observable rather than from the full likelihood. For general quantum statistical models, the expansion is written in terms of the calibration curve and the central moments of the measured observable. Nonlinear calibration curves make the usual moment estimator biased at finite measurement number; we construct a bias-corrected estimator with bias $O(ν^{-3})$. This gives sensitivity corrections beyond the leading error-propagation term of the chosen moment protocol. We identify a general density-matrix condition under which the full $1/ν^2$ correction vanishes. In unitary examples, the leading residual correction appears at order $1/ν^3$, is governed by calibration curvature, and can be reduced or cancelled by higher-rank components of the same measured observable. The resulting thresholds quantify how many measurements are needed before the asymptotic sensitivity of a moment-estimation protocol is operationally visible.

quant-pharXiv:2606.25920•Jun 25, 2026•0 citations

Optimal measurements for quantum multiparameter estimation with general states

Jing Yang, Shengshi Pang, Yiyu Zhou +1 more

We generalize the approach by Braunstein and Caves, [Phys. Rev. Lett. 72, 3439 (1994)]. to quantum multiparameter estimation with general states. We derive a matrix bound of the classical Fisher information matrix due to each measurement operator. The saturation of all these bounds results in the saturation of the matrix Helstrom Cram\'er-Rao bound. Remarkably, the saturation of the matrix bound is equivalent to the saturation of the scalar bound with respect to any given positive definite weight matrix. Necessary and sufficient conditions are obtained for the optimal measurements that give rise to the Helstrom Cram\'er-Rao bound associated with a general quantum state. To saturate the Helstrom bound with separable measurements or collective measurement entangling only a small number of identical states, we find it is necessary for the symmetric logarithmic derivatives to commute on the support of the state. As an important application of our results, we construct several local optimal measurements for the problem of estimating the three-dimensional separation of two incoherent optical point sources.

arXiv:1806.07337•Physical Review A•Sep 3, 2019•86 citations
Propagation of error is a widely used estimation tool in experiments, where the estimation precision of the parameter depends on the fluctuation of the physical observable. Thus which observable is chosen will greatly affect the estimation sensitivity. Here we study the optimal observable for the ultimate sensitivity bounded by the quantum Cramér-Rao theorem in parameter estimation. By invoking the Schrödinger-Robertson uncertainty relation, we derive the necessary and sufficient condition for the optimal observables saturating the ultimate sensitivity for single parameter estimate. By applying this condition to Greenberg-Horne-Zeilinger states, we obtain the general expression of the optimal observable for separable measurements to achieve the Heisenberg-limit precision and show that it is closely related to the parity measurement. However, Jose {\em et al} [Phys. Rev. A {\bf 87}, 022330 (2013)] have claimed that the Heisenberg limit may not be obtained via separable measurements. We show this claim is incorrect.
We derive a necessary and sufficient condition for the possibility of achieving the Heisenberg scaling in general adaptive multi-parameter estimation schemes in presence of Markovian noise. In situations where the Heisenberg scaling is achievable, we provide a semidefinite program to identify the optimal quantum error correcting (QEC) protocol that yields the best estimation precision. We overcome the technical challenges associated with potential incompatibility of the measurement optimally extracting information on different parameters by utilizing the Holevo Cramér-Rao (HCR) bound for pure states. We provide examples of significant advantages offered by our joint-QEC protocols, that sense all the parameters utilizing a single error-corrected subspace, over separate-QEC protocols where each parameter is effectively sensed in a separate subspace.
Abstract Quantum metrology aims at delivering new quantum-mechanical improvement to technologies of parameter estimations with precision bounded by the quantum Cramér-Rao bound. The currently used quantum Cramér-Rao bound was established with measurements of observables restricted to be Hermitian. This constrains the bound and limits the precision of parameter estimation. In this paper, we lift the constraint and derive a previously unknown quantum Cramér-Rao bound. We find that the new bound can reach arbitrary small value with mixed states and it breaks the Heisenberg limit in some cases. We construct a setup to measure non-Hermitian operators and discuss the saturation of the present bound. Two examples—the phase estimation with Greenberger-Horne-Zeilinger states of trapped ions and the adiabatic quantum parameter estimation with the nuclear magnetic resonance—are employed to demonstrate the theory. The present study might open a new research direction—non-Hermitian quantum metrology.
Optimal measurements for quantum multiparameter estimation are complicated by the uncertainty principle. Generally, there is a trade-off between the precision with which different parameters can be simultaneously estimated. The task of determining the minimum achievable estimation error is a central task of multiparameter quantum metrology. For estimating parameters encoded in pure quantum states, the ultimate limit is known, but is given by the solution of a non-trivial minimisation problem. We present a new expression for the achievable bound for two-parameter estimation with pure states that is considerably simpler. We also determine the optimal measurements, completing the problem of two-parameter estimation with pure state probes. To demonstrate the utility of our result, we determine the precision limit for estimating displacements using grid states.
In quantum mechanics, the Heisenberg uncertainty relations and the Cramér-Rao inequalities typically limit the precision in the estimation of a parameter through the standard deviation of a conjugate observable. Here we extend these relations by giving a bound to the precision of a parameter in terms of the expectation value of the conjugate observable. This has both foundational and practical consequences: in quantum optics it resolves a controversy over which is the ultimate precision in interferometry.
The quantum Cramér-Rao bound can be saturated only asymptotically and does not specify how many measurements are needed for a concrete estimator to approach it. We develop a finite-measurement theory for method-of-moments estimation, where the parameter is inferred from the sample mean of a calibrating observable rather than from the full likelihood. For general quantum statistical models, the expansion is written in terms of the calibration curve and the central moments of the measured observable. Nonlinear calibration curves make the usual moment estimator biased at finite measurement number; we construct a bias-corrected estimator with bias O(ν−3)O(ν^{-3})O(ν−3). This gives sensitivity corrections beyond the leading error-propagation term of the chosen moment protocol. We identify a general density-matrix condition under which the full 1/ν21/ν^21/ν2 correction vanishes. In unitary examples, the leading residual correction appears at order 1/ν31/ν^31/ν3, is governed by calibration curvature, and can be reduced or cancelled by higher-rank components of the same measured observable. The resulting thresholds quantify how many measurements are needed before the asymptotic sensitivity of a moment-estimation protocol is operationally visible.
We generalize the approach by Braunstein and Caves, [Phys. Rev. Lett. 72, 3439 (1994)]. to quantum multiparameter estimation with general states. We derive a matrix bound of the classical Fisher information matrix due to each measurement operator. The saturation of all these bounds results in the saturation of the matrix Helstrom Cram\'er-Rao bound. Remarkably, the saturation of the matrix bound is equivalent to the saturation of the scalar bound with respect to any given positive definite weight matrix. Necessary and sufficient conditions are obtained for the optimal measurements that give rise to the Helstrom Cram\'er-Rao bound associated with a general quantum state. To saturate the Helstrom bound with separable measurements or collective measurement entangling only a small number of identical states, we find it is necessary for the symmetric logarithmic derivatives to commute on the support of the state. As an important application of our results, we construct several local optimal measurements for the problem of estimating the three-dimensional separation of two incoherent optical point sources.