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

Heisenberg Limit of Phase Measurements with a Fluctuating Number of Photons

Y. R. Zhang, Guangri Jin, Junpeng Cao, Wentao Liu, Hao Fan, Seth Lloyd

Published May 16, 2011• 0 citations
OpenAlex

Abstract

The minimum error of unbiased parameter estimation is quantified by the quantum Fisher information in accordance to the Cramer-Rao bound. We indicate that only superposed NOON states by simultaneous measurements can achieve the maximum quantum Fisher information with form $ $ for a given photon number distribution by a POVM in linear two-path interferometer phase measurement. We present a series of specified superposed states with infinite quantum Fisher information but with finite average photon numbers. The advantage of this unbounded quantum Fisher information will be beneficial to many applications in quantum technology.

Related papers

Unbounded quantum Fisher information in two-path interferometry with finite photon number

Yu-Ran Zhang, Guangri Jin, Junpeng Cao +2 more

The minimum error of unbiased parameter estimation is quantified by the quantum Fisher information in accordance to the Cramér–Rao bound. We indicate that only superposed NOON states by simultaneous measurements can achieve the maximum quantum Fisher information with form for a given photon number distribution by a POVM in linear two-path interferometer phase measurement. We present a series of specified superposed states with infinite quantum Fisher information but with finite average photon numbers. The advantage of this unbounded quantum Fisher information will be beneficial to many applications in quantum technology.

arXiv:1105.2990•Journal of Physics A Mathematical and Theoretical•Dec 18, 2012•33 citations

Quantum Cramer-Rao bound and parity measurement

Aravind Chiruvelli, Hwang Lee

We show, by using parity detection, that the phase estimate obtained from the linear error propagation formula, indeed saturates the lower bound set by maximizing quantum Fisher information for several of the well-known states, proposed for the Heisenberg limited interferometry. Our results show that one can achieve the quantum Cram´er-Rao bound, by simply counting the number of photons which is not very far from actual realization.

Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE•Feb 8, 2010•2 citations

Quantum correlations in optical metrology: Heisenberg-limited phase estimation without mode entanglement

Jaspreet Sahota, Nicolás Quesada

The quantum Fisher information and quantum correlation parameters are employed to study the application of nonclassical light to the problem of parameter estimation. It is shown that the optimal measurement sensitivity of a quantum state is determined by its intermode correlations (which depends on path entanglement) and intramode correlations (which depends on the photon statistics). In light of these results, we consider the performance of quantum-enhanced optical interferometers. Furthermore, we propose a Heisenberg-limited metrology protocol involving standard elements from passive and active linear optics, for which the quantum Cram\'er-Rao bound is saturated with an intensity measurement. Interestingly, the quantum advantage for this scheme is derived solely from the nonclassical photon statistics of the probe state and does not depend on entanglement. We study the performance of this scheme in the presence of realistic losses and consequently predict a substantial enhancement over the shot-noise limit with current technological capabilities.

arXiv:1404.7110•Physical Review A•Jan 6, 2015•69 citations

Heisenberg Limit beyond Quantum Fisher Information

Wojciech Górecki

The Heisenberg limit provides a fundamental bound on the achievable estimation precision with a limited number of $N$ resources used (e.g., atoms, photons, etc.). Using entangled quantum states makes it possible to scale the precision with $N$ better than when resources would be used independently. Consequently, the optimal use of all resources involves accumulating them in a single execution of the experiment. Unfortunately, that implies that the most common theoretical tool used to analyze metrological protocols - quantum Fisher information (QFI) - does not allow for a reliable description of this problem, as it becomes operationally meaningful only with multiple repetitions of the experiment. In this thesis, using the formalism of Bayesian estimation and the minimax estimator, I derive asymptotically saturable bounds on the precision of the estimation for the case of noiseless unitary evolution. For the case where the number of resources $N$ is strictly constrained, I show that the final measurement uncertainty is $π$ times larger than would be implied by a naive use of QFI. I also analyze the case where a constraint is imposed only on the average amount of resources, the exact value of which may fluctuate (in which case QFI does not provide any universal bound for precision). In both cases, I study the asymptotic saturability and the rate of convergence of these bounds. In the following part, I analyze the problem of the Heisenberg limit when multiple parameters are measured simultaneously on the same physical system. In particular, I investigate the existence of a gain from measuring all parameters simultaneously compared to distributing the same amount of resources to measure them independently. I focus on two examples - the measurement of multiple phase shifts in a multi-arm interferometer and the measurement of three magnetic field components.

arXiv:2304.14370•Apr 27, 2023•1 citation

Phase estimation with photon number constraint

Masahito HAYASHI

Many researches proposed the use of the noon state as the input state for phase estimation, which is one topic of quantum metrology.This is because the input noon state provides the maximum Fisher information at the specific point.However, the Fisher information does not necessarily give the attainable bound for estimation error.In this paper, we adopt the local asymptotic mini-max criterion as well as the mini-max criterion, and show that the maximum Fisher information does not give the attainable bound for estimation error under these criteria in the phase estimation.We also propose the optimal input state under the constraints for photon number of the input state instead of the noon state.

arXiv:1011.2546•Progress in Informatics•Mar 1, 2011•19 citations

Quantum Fisher information measurement and verification of the quantum Cramér–Rao bound in a solid-state qubit

Min Yu, Yu Liu, Pengcheng Yang +8 more

Abstract The quantum Cramér–Rao bound sets a fundamental limit on the accuracy of unbiased parameter estimation in quantum systems, relating the uncertainty in determining a parameter to the inverse of the quantum Fisher information. We experimentally demonstrate near saturation of the quantum Cramér–Rao bound in the phase estimation of a solid-state spin system, provided by a nitrogen-vacancy center in diamond. This is achieved by comparing the experimental uncertainty in phase estimation with an independent measurement of the related quantum Fisher information. The latter is independently extracted from coherent dynamical responses of the system under weak parametric modulations, without performing any quantum-state tomography. While optimal parameter estimation has already been observed for quantum devices involving a limited number of degrees of freedom, our method offers a versatile and powerful experimental tool to explore the Cramér–Rao bound and the quantum Fisher information in systems of higher complexity, as relevant for quantum technologies.

arXiv:2003.08373•npj Quantum Information•May 12, 2022•71 citations
The minimum error of unbiased parameter estimation is quantified by the quantum Fisher information in accordance to the Cramer-Rao bound. We indicate that only superposed NOON states by simultaneous measurements can achieve the maximum quantum Fisher information with form for a given photon number distribution by a POVM in linear two-path interferometer phase measurement. We present a series of specified superposed states with infinite quantum Fisher information but with finite average photon numbers. The advantage of this unbounded quantum Fisher information will be beneficial to many applications in quantum technology.
The minimum error of unbiased parameter estimation is quantified by the quantum Fisher information in accordance to the Cramér–Rao bound. We indicate that only superposed NOON states by simultaneous measurements can achieve the maximum quantum Fisher information with form for a given photon number distribution by a POVM in linear two-path interferometer phase measurement. We present a series of specified superposed states with infinite quantum Fisher information but with finite average photon numbers. The advantage of this unbounded quantum Fisher information will be beneficial to many applications in quantum technology.
The quantum Fisher information and quantum correlation parameters are employed to study the application of nonclassical light to the problem of parameter estimation. It is shown that the optimal measurement sensitivity of a quantum state is determined by its intermode correlations (which depends on path entanglement) and intramode correlations (which depends on the photon statistics). In light of these results, we consider the performance of quantum-enhanced optical interferometers. Furthermore, we propose a Heisenberg-limited metrology protocol involving standard elements from passive and active linear optics, for which the quantum Cram\'er-Rao bound is saturated with an intensity measurement. Interestingly, the quantum advantage for this scheme is derived solely from the nonclassical photon statistics of the probe state and does not depend on entanglement. We study the performance of this scheme in the presence of realistic losses and consequently predict a substantial enhancement over the shot-noise limit with current technological capabilities.
The Heisenberg limit provides a fundamental bound on the achievable estimation precision with a limited number of NNN resources used (e.g., atoms, photons, etc.). Using entangled quantum states makes it possible to scale the precision with NNN better than when resources would be used independently. Consequently, the optimal use of all resources involves accumulating them in a single execution of the experiment. Unfortunately, that implies that the most common theoretical tool used to analyze metrological protocols - quantum Fisher information (QFI) - does not allow for a reliable description of this problem, as it becomes operationally meaningful only with multiple repetitions of the experiment. In this thesis, using the formalism of Bayesian estimation and the minimax estimator, I derive asymptotically saturable bounds on the precision of the estimation for the case of noiseless unitary evolution. For the case where the number of resources NNN is strictly constrained, I show that the final measurement uncertainty is πππ times larger than would be implied by a naive use of QFI. I also analyze the case where a constraint is imposed only on the average amount of resources, the exact value of which may fluctuate (in which case QFI does not provide any universal bound for precision). In both cases, I study the asymptotic saturability and the rate of convergence of these bounds. In the following part, I analyze the problem of the Heisenberg limit when multiple parameters are measured simultaneously on the same physical system. In particular, I investigate the existence of a gain from measuring all parameters simultaneously compared to distributing the same amount of resources to measure them independently. I focus on two examples - the measurement of multiple phase shifts in a multi-arm interferometer and the measurement of three magnetic field components.
Many researches proposed the use of the noon state as the input state for phase estimation, which is one topic of quantum metrology.This is because the input noon state provides the maximum Fisher information at the specific point.However, the Fisher information does not necessarily give the attainable bound for estimation error.In this paper, we adopt the local asymptotic mini-max criterion as well as the mini-max criterion, and show that the maximum Fisher information does not give the attainable bound for estimation error under these criteria in the phase estimation.We also propose the optimal input state under the constraints for photon number of the input state instead of the noon state.
Abstract The quantum Cramér–Rao bound sets a fundamental limit on the accuracy of unbiased parameter estimation in quantum systems, relating the uncertainty in determining a parameter to the inverse of the quantum Fisher information. We experimentally demonstrate near saturation of the quantum Cramér–Rao bound in the phase estimation of a solid-state spin system, provided by a nitrogen-vacancy center in diamond. This is achieved by comparing the experimental uncertainty in phase estimation with an independent measurement of the related quantum Fisher information. The latter is independently extracted from coherent dynamical responses of the system under weak parametric modulations, without performing any quantum-state tomography. While optimal parameter estimation has already been observed for quantum devices involving a limited number of degrees of freedom, our method offers a versatile and powerful experimental tool to explore the Cramér–Rao bound and the quantum Fisher information in systems of higher complexity, as relevant for quantum technologies.