Metrological symmetries in singular quantum multi-parameter estimation

Abstract

The theoretical foundation of quantum sensing is rooted in the Cramér-Rao formalism, which establishes quantitative precision bounds for a given quantum probe. In many practical scenarios, where more than one parameter is unknown, the multi-parameter Cramér-Rao bound (CRB) applies. Since this is a matrix inequality involving the inverse of the quantum Fisher information matrix (QFIM), the formalism breaks down when the QFIM is singular. In this paper, we examine the physical origins of such singularities, showing that they result from an over-parametrization on the metrological level. This is itself caused by emergent metrological symmetries, whereby the same set of measurement outcomes are obtained for different combinations of system parameters. Although the number of effective parameters is equal to the number of non-zero QFIM eigenvalues, the Cramér-Rao formalism typically does not provide information about the effective parameter encoding. Instead, we demonstrate through a series of concrete examples that Bayesian estimation can provide deep insights. In particular, the metrological symmetries appear in the Bayesian posterior distribution as lines of persistent likelihood running through the space of unknown parameters. These lines are contour lines of the effective parameters which, through suitable parameter transformations, can be estimated and follow their own effective CRBs.

24 pages with 8 figures

References in corpus (83)

• Quantum sensing
• Advances in Quantum Metrology
• Quantum-enhanced measurements: beating the standard quantum limit
• Quantum metrology
• Sensitivity Optimization for NV-Diamond Magnetometry
• Nonlinear atom interferometer surpasses classical precision limit
• Quantum Fisher information matrix and multiparameter estimation
• Precision Measurement of the Newtonian Gravitational Constant Using Cold Atoms
• A robust, scanning quantum system for nanoscale sensing and imaging
• Single-Ion Atomic Clock with $3\times 10^{-18}$ Systematic Uncertainty
• Quantum critical scaling of the geometric tensors
• Quantum enhanced measurements without entanglement
• Interferometry with Bose-Einstein Condensates in Microgravity
• Quantum criticality as a resource for quantum estimation
• An Entanglement-Enhanced Microscope
• Individual quantum probes for optimal thermometry
• Multi-parameter estimation in networked quantum sensors
• Compatibility in Multiparameter Quantum Metrology
• Sub-nanoscale Temperature, Magnetic Field and Pressure sensing with Spin Centers in 2D hexagonal Boron Nitride
• Critical Quantum metrology with a finite-component quantum phase transition
• A perspective on multiparameter quantum metrology: from theoretical tools to applications in quantum imaging
• Quantum-enhanced sensing of displacements and electric fields with large trapped-ion crystals
• Multi-parameter estimation beyond Quantum Fisher Information
• Thermometry in the quantum regime: Recent theoretical progress
• Dynamic framework for criticality-enhanced quantum sensing
• Enhanced metrology at the critical point of a many-body Rydberg atomic system
• Optimized quantum sensing with a single electron spin using real-time adaptive measurements
• On quantumness in multi-parameter quantum estimation
• Phase detection at the quantum limit with multi-photon Mach-Zehnder interferometry
• Tunable Quantum Criticality and Super-ballistic Transport in a `Charge' Kondo Circuit
• At the limits of criticality-based quantum metrology: apparent super-Heisenberg scaling revisited
• An Optical Atomic Clock Based on a Highly Charged Ion
• Critical parametric quantum sensing
• Bayesian parameter inference from continuously monitored quantum systems
• The quantum Ising chain for beginners
• Global sensing and its impact for quantum many-body probes with criticality
• Quantum Semiparametric Estimation
• Review: Quantum Metrology and Sensing with Many-Body Systems
• Heisenberg limited single-mode quantum metrology
• Composite-pulse magnetometry with a solid-state quantum sensor
• Bayesian parameter estimation by continuous homodyne detection
• Bayesian multi-parameter quantum metrology with limited data
• Quantum state estimation with nuisance parameters
• Two-channel Kondo physics in two-impurity Kondo models
• Probe incompatibility in multiparameter noisy quantum metrology
• Efficient computation of the Nagaoka--Hayashi bound for multi-parameter estimation with separable measurements
• Quantum simulation of an exotic quantum critical point in a two-site charge Kondo circuit
• Quantum sensing networks for the estimation of linear functions
• Optimal Probes for Global Quantum Thermometry
• Fundamental limits in Bayesian thermometry and attainability via adaptive strategies
• Optimal probes and error-correction schemes in multi-parameter quantum metrology
• On the discontinuity of the quantum Fisher information for quantum statistical models with parameter dependent rank
• Critical quantum metrology assisted by real-time feedback control
• Thermometry of Strongly Correlated Fermionic Quantum Systems using Impurity Probes
• Sequential measurements for quantum-enhanced magnetometry in spin chain probes
• Multiparameter critical quantum metrology with impurity probes
• Optimal cold atom thermometry using adaptive Bayesian strategies
• Two-channel Kondo physics in odd impurity chains
• Quantum Science and the Search for Axion Dark Matter
• Optimality and Noise-Resilience of Critical Quantum Sensing
• Quantum Sensing in Tweezer Arrays: Optical Magnetometry on an Individual-Atom Sensor Grid
• Multiple-phase quantum interferometry -- real and apparent gains of measuring all the phases simultaneously
• Uninformed Bayesian Quantum Thermometry
• Protocols for estimating multiple functions with quantum sensor networks: geometry and performance
• Quantum Fisher information matrix in Heisenberg XY model
• Fundamental limits of metrology at thermal equilibrium
• Squeezing and overcoming the Heisenberg scaling with spin-orbit coupled quantum gases
• Current Trends in Global Quantum Metrology
• Multiparameter quantum metrology in the Heisenberg Limit regime: many repetition scenario vs. full optimization
• Designing optimal protocols in Bayesian quantum parameter estimation with higher-order operations
• Uncertain Quantum Critical Metrology: From Single to Multi Parameter Sensing
• Thermometry by correlated dephasing of impurities in a 1D Fermi gas
• Benchmarking Bayesian quantum estimation
• First-principles construction of symmetry-informed quantum metrologies
• Overcoming Quantum Metrology Singularity through Sequential Measurements
• Many-body quantum interference route to the two-channel Kondo effect: Inverse design for molecular junctions and quantum dot devices
• Fisher information rates in sequentially measured quantum systems
• Experimental investigation of Bayesian bounds in multiparameter estimation
• Enhancing precision thermometry with nonlinear qubits
• Quantum Sensing from Gravity as Universal Dephasing Channel for Qubits
• Nanoscale single-electron box with a floating lead for quantum sensing: modelling and device characterization
• On the role of symmetry and geometry in global quantum sensing
• Existence of unbiased resilient estimators in discrete quantum systems

Cited by in corpus (5)

• Critical Quantum Sensing: a tutorial on parameter estimation near quantum phase transitions
• Correlated decoherence and thermometry with mobile impurities in a 1D Fermi gas
• Minimal Trade-off and Optimal Measurement for Multiparameter Quantum Estimation
• Sensitivity Bounds of Multiparameter Metrology at Thermal Equilibrium
• Metrological approach to the emergence of classical objectivity