Ziv-Zakai error bounds for quantum parameter estimation.

I propose quantum versions of the Ziv-Zakai bounds as alternatives to the widely used quantum Cramér-Rao bounds for quantum parameter estimation. From a simple form of the proposed bounds, I derive both a Heisenberg error limit that scales with the average energy and a limit similar to the quantum Cramér-Rao bound that scales with the energy variance. These results are further illustrated by applying the bound to a few examples of optical phase estimation, which show that a quantum Ziv-Zakai bound can be much higher and thus tighter than a quantum Cramér-Rao bound for states with highly non-gaussian photon-number statistics in certain regimes and also stay close to the latter where the latter is expected to be tight.

• Publication year

  2011

• Venue

  Physical Review Letters

• Publication date

  2011-11-15

• Fields of study

  Medicine, Physics

• Identifiers
  DOI 10.1103/PhysRevLett.108.230401arXiv 1111.3568PMID 23003924
• External record

  Open on Semantic Scholar

• Source metadata

  Semantic Scholar, PubMed

• No claims are published for this paper.

• No concepts are published for this paper.

• Optical interferometry in the presence of large phase diffusion

  2012cited by this paper
• Gravitational wave astronomy of single sources with a pulsar timing array

  2011cited by this paper
• Erratum: General Optimality of the Heisenberg Limit for Quantum Metrology [Phys. Rev. Lett. 105, 180402 (2010)]

  2011cited by this paper
• Continuous quantum hypothesis testing.

  2011cited by this paper
• Universality of the Heisenberg limit for estimates of random phase shifts

  2011cited by this paper
• Advances in quantum metrology

  2011cited by this paper
• Optical phase estimation in the presence of phase diffusion.

  2010cited by this paper
• Fundamental quantum limit to waveform estimation

  2010cited by this paper
• General optimality of the Heisenberg limit for quantum metrology.

  2010cited by this paper
• Entanglement and sensitivity in precision measurements with states of a fluctuating number of particles.

  2010cited by this paper
• Quantum metrology with two-mode squeezed vacuum: parity detection beats the Heisenberg limit.

  2009cited by this paper
• Quantum theory of optical temporal phase and instantaneous frequency. II. Continuous-time

  2008cited by this paper
• Bayesian Bounds for Parameter Estimation and Nonlinear Filtering/Tracking

  2007cited by this paper
• Generalized limits for single-parameter quantum estimation.

  2006cited by this paper
• Quantum-Enhanced Measurements: Beating the Standard Quantum Limit

  2004cited by this paper
• Quantum limits to dynamical evolution

  2002cited by this paper
• Phase measurements at the theoretical limit

  2000cited by this paper
• Bayesian bounds on parameter extraction accuracy for compact coalescing binary gravitational wave signals

  1997cited by this paper
• Cryptographic Distinguishability Measures for Quantum-Mechanical States

  1997cited by this paper
• FUNDAMENTAL QUANTUM LIMIT IN PRECISION PHASE MEASUREMENT

  1997cited by this paper
• Adaptive single-shot phase measurements: The full quantum theory

  1997cited by this paper
• Complementarity and Fundamental Limit in Precision Phase Measurement.

  1996cited by this paper
• Optimal frequency measurements with maximally correlated states.

  1996cited by this paper
• Generalized uncertainty relations: Theory, examples, and Lorentz invariance

  1995cited by this paper
• Optimal quantum measurements for phase estimation.

  1995cited by this paper
• Statistical distance and the geometry of quantum states.

  1994cited by this paper
• Phase Resolution and Coherent Phase States

  1993cited by this paper
• Phase noise in a squeezed state

  1993cited by this paper
• Maximum-likelihood analysis of multiple quantum phase measurements.

  1992cited by this paper
• Quantum phase measurement: A system-theory perspective.

  1991cited by this paper
• Ultimate quantum limits on phase measurement.

  1989cited by this paper
• Relations between Belini-Tartara, Chazan-Zakai-Ziv, and Wax-Ziv lower bounds

  1988cited by this paper
• SU(2) and SU(1,1) interferometers.

  1986cited by this paper
• Probabilistic and Statistical Aspects of Quantum Theory

  1982cited by this paper
• Improved Lower Bounds on Signal Parameter Estimation

  1975cited by this paper
• Bounds on Error in Signal Parameter Estimation

  1974cited by this paper
• A unitarity bound on the evolution of nonstationary states

  1973cited by this paper
• Multiple-parameter quantum estimation and measurement of nonselfadjoint observables

  1973cited by this paper
• Performance limitations and error calculations for parameter estimation

  1970cited by this paper
• Some lower bounds on signal parameter estimation

  1969cited by this paper
• Quantum detection and estimation theory

  1969cited by this paper
• Detection, Estimation, and Modulation Theory, Part I

  1968cited by this paper

• Doppler Ambiguity-Resolving Waveform Design Based on Ziv–Zakai Bound Optimization

  2026cites this paper
• Generalized Zernike Phase-Contrast Imaging

  2026cites this paper
• Excessive precision compromises accuracy even with unlimited resources due to the trade-off in quantum metrology

  2025cites this paper
• Noise Constraints on Sensitivity Scaling in Quantum Nonlinear Metrology.

  2025cites this paper
• Tighter Heisenberg limits for phase estimation in noisy environments based on the quantum Ziv-Zakai bound.

  2025cites this paper
• Measurement incompatibility in Bayesian multiparameter quantum estimation

  2025cites this paper
• Quantum Interferometric Power Versus Local Quantum Uncertainty in Bell Coherent States Under Amplitude Damping

  2025cites this paper
• Non-Hermitian sensing from the perspective of postselected measurements

  2025cites this paper
• Nonlinear Heisenberg Limit via Uncertainty Principle in Quantum Metrology

  2025cites this paper
• On the role of symmetry and geometry in global quantum sensing

  2025cites this paper
• The Cramér-Rao approach and global quantum estimation of bosonic states

  2024cites this paper
• Estimating purity and mixing parameter of the initial state for Tavis–Cumming and dephasing models in multiparameter metrological schemes

  2024cites this paper
• Noiseless linear amplification-based quantum Ziv-Zakai bound for phase estimation and its Heisenberg error limits in noisy scenarios

  2024influential citation
• Benchmarking Bayesian quantum estimation

  2024cites this paper
• First-principles construction of symmetry-informed quantum metrologies

  2024cites this paper
• A Comprehensive Study on Ziv-Zakai Lower Bounds on the MMSE

  2024cites this paper
• Review: Quantum metrology and sensing with many-body systems

  2024cites this paper
• Bayesian quantum phase estimation with fixed photon states

  2023cites this paper
• Quantum Limits of Transmission Electron Microscopy.

  2023cites this paper
• Bayesian Nagaoka-Hayashi Bound for Multiparameter Quantum-State Estimation Problem

  2023cites this paper
• Hierarchies of Frequentist Bounds for Quantum Metrology: From Cramér-Rao to Barankin.

  2023cites this paper
• Functional Properties of the Ziv-Zakai bound with Arbitrary Inputs

  2023cites this paper
• Global quantum thermometry based on the optimal biased bound

  2023cites this paper
• Ziv-Zakai-type error bounds for general statistical models

  2023cites this paper
• Optimal finite-dimensional probe states for quantum phase estimation

  2023cites this paper
• Evaluating the quantum optimal biased bound in a unitary evolution process

  2023cites this paper
• QuanEstimation: An open-source toolkit for quantum parameter estimation

  2022cites this paper
• Enhancement in the mean square range delay accuracy by means of multiple entangled photon states quantum illumination

  2022cites this paper
• Evaluating the quantum Ziv-Zakai bound for phase estimation in noisy environments.

  2022influential citation
• Entanglement-assisted multi-aperture pulse-compression radar for angle resolving detection

  2022cites this paper
• Quantum Metrology with Limited Measurement Resources

  2021cites this paper
• Extreme expected values and their applications in quantum metrology

  2021cites this paper
• Decoherence effects on quantum Fisher information for moving two four-level atoms in the presence of Stark effect and Kerr-like medium

  2021cites this paper
• On the Analytical Derivation of Quantum Fisher Information and Skew Information for two Qubit X States

  2021cites this paper
• Ultimate Accuracy Limit of Quantum Pulse-Compression Ranging.

  2021cites this paper
• Qubit Quantum Metrology with Limited Measurement Resources

  2021cites this paper
• Average number is an insufficient metric for interferometry

  2021cites this paper
• Bayesian Phase Estimation via Active Learning

  2021cites this paper
• Non-asymptotic quantum metrology : extracting maximum information from limited data

  2020influential citation
• Ju n 20 19 Quantum Semiparametric Estimation

  2019cites this paper
• Non-asymptotic quantum metrology

  2019influential citation
• The Holevo Cram\'er-Rao bound is at most thrice the Helstrom version

  2019cites this paper
• Generalized-mean Cramér-Rao bounds for multiparameter quantum metrology

  2019cites this paper
• Dynamic of quantum Fisher information and quantum interferometric power in multipartite coherent states

  2019cites this paper
• Quantum Fisher information matrix and multiparameter estimation

  2019cites this paper
• Sensitivity bounds for interferometry with Ising Hamiltonians

  2019cites this paper
• Stochastic Resonance with Unital Quantum Noise

  2019cites this paper
• Bayesian multiparameter quantum metrology with limited data

  2019cites this paper
• Using states with a large photon number variance to increase quantum Fisher information in single-mode phase estimation

  2019influential citation
• Quantum Semiparametric Estimation

  2019cites this paper
• Quantum Fisher information of atomic system interacting with a single cavity mode in the presence of Kerr medium

  2019cites this paper
• Quantum metrology in the presence of limited data

  2018cites this paper
• Quantum enhanced metrology: quantum mechanical correlationsand uncertainty relations

  2018cites this paper
• Resolving unattainable Cramer–Rao bounds for quantum sensors

  2018cites this paper
• Frequentist and Bayesian Quantum Phase Estimation

  2018cites this paper
• Quantum Fisher Information of Decohered W and GHZ Superposition States with Arbitrary Relative Phase

  2017cites this paper
• Detecting Violation of Bell Inequalities using LOCC Maximized Quantum Fisher Information and Entanglement Measures

  2017cites this paper
• Quantum Fisher Information: Theory and Applications

  2017cites this paper
• Adaptive estimation of a time-varying phase with coherent states: Smoothing can give an unbounded improvement over filtering

  2017cites this paper
• The influence of non-Gaussian noise on the accuracy of parameter estimation

  2017cites this paper
• Quantum Fisher Information of Decohered W and GHZ Superposition States with Arbitrary Relative Phase

  2017cites this paper
• UvA-DARE ( Digital Academic Repository ) Adaptive Quantum Metrology under General Markovian Noise

  2017cites this paper
• Digital Quantum Estimation.

  2017cites this paper
• Quantum interferometric power and the role of nonclassical correlations in quantum metrology for a special class of two-qubit states

  2017cites this paper
• Non-asymptotic analysis of quantum metrology protocols beyond the Cramér–Rao bound

  2017cites this paper
• Quantum Fisher Information of W and GHZ State Superposition under Decoherence

  2017cites this paper
• A Proposal for Quantum Fisher Information Optimization and its Relation with Entanglement Measures

  2017cites this paper
• Quantum metrology with full and fast quantum control

  2016influential citation
• Sequential Feedback Scheme Outperforms the Parallel Scheme for Hamiltonian Parameter Estimation.

  2016cites this paper
• Spectrum analysis with quantum dynamical systems

  2016cites this paper
• Optimal adaptive control for quantum metrology with time-dependent Hamiltonians

  2016cites this paper
• Breaking the weak Heisenberg limit

  2016cites this paper
• Valid lower bound for all estimators in quantum parameter estimation

  2016cites this paper
• Sharp transitions in low-number quantum dots Bayesian magnetometry

  2016cites this paper
• Conservative classical and quantum resolution limits for incoherent imaging

  2016cites this paper
• Bayesian big bang: Cosmic acceleration

  2016cites this paper
• Discontinuity and Protection of Quantum Fisher Information for a Two-Qubit System

  2015cites this paper
• Sub-Poissonian and anti-bunching criteria via majorization of statistics

  2015cites this paper
• Improved mirror position estimation using resonant quantum smoothing

  2015cites this paper
• Quantum Weiss-Weinstein bounds for quantum metrology

  2015cites this paper
• Quantum limits in optical interferometry

  2014cites this paper
• Quantum Metrological Bounds for Vector Parameter in Presence of Noise

  2014cites this paper
• Derivation of quantum Chernoff metric with perturbation expansion method

  2014cites this paper
• Optimal quantum-enhanced interferometry

  2014cites this paper
• Quantum multiparameter metrology with generalized entangled coherent state

  2014cites this paper
• Estimation of atomic interaction parameters by photon counting

  2014cites this paper
• Quantum limits to optical point-source localization

  2014cites this paper
• Precision bounds in noisy quantum metrology

  2014cites this paper
• Quantum Bell-Ziv-Zakai Bounds and Heisenberg Limits for Waveform Estimation

  2014cites this paper
• True precision limits in quantum metrology

  2014cites this paper
• Analysis of Entanglement Measures and LOCC Maximized Quantum Fisher Information of General Two Qubit Systems

  2014cites this paper
• A Heisenberg limit for quantum region estimation

  2014cites this paper
• Lower Bounds for Quantum Parameter Estimation

  2013cites this paper
• Quantum Fisher Information for Density Matrices with Arbitrary Ranks

  2013cites this paper
• Quantum limits on postselected, probabilistic quantum metrology

  2013cites this paper
• Towards quantum-enhanced precision measurements: Promise and challenges

  2013cites this paper
• Quantum metrology with open dynamical systems

  2013cites this paper
• Quantum transition-edge detectors

  2013cites this paper
• Macroscopic quantum mechanics: theory and experimental concepts of optomechanics

  2013cites this paper
• Testing quantum mechanics: a statistical approach

  2013cites this paper