On the properties of the asymptotic incompatibility measure in multiparameter quantum estimation

Abstract

We address the use of asymptotic incompatibility (AI) to assess the quantumness of a multiparameter quantum statistical model. AI is a recently introduced measure which quantifies the difference between the Holevo and the SLD scalar bounds, and can be evaluated using only the symmetric logarithmic derivative (SLD) operators of the model. At first, we evaluate analytically the AI of the most general quantum statistical models involving two-level (qubit) and single-mode Gaussian continuous-variable quantum systems, and prove that AI is a simple monotonous function of the state purity. Then, we numerically investigate the same problem for qudits ($d$-dimensional quantum systems, with $2<d\le 4$), showing that, while in general AI is not in general a function of purity, we have enough numerical evidence to conclude that the maximum amount of AI is attainable only for quantum statistical models characterized by a purity larger than ${\mu }_{\mathsf{min}}=1/(d-1)$. In addition, by parametrizing qudit states as thermal (Gibbs) states, numerical results suggest that, once the spectrum of the Hamiltonian is fixed, the AI measure is in one-to-one correspondence with the fictitious temperature parameter $\beta$ characterizing the family of density operators. Finally, by studying in detail the definition and properties of the AI measure we find that: i) given a quantum statistical model, one can readily identify the maximum number of asymptotically compatibile parameters; ii) the AI of a quantum statistical model bounds from above the AI of any sub-model that can be defined by fixing one or more of the original unknown parameters (or functions thereof), leading to possibly useful bounds on the AI of models involving noisy quantum dynamics.

arXiv comment21 pages, 2 figures

References in corpus (51)

• Quantum sensing
• Advances in Quantum Metrology
• Advances in Photonic Quantum Sensing
• Quantum Fisher information matrix and multiparameter estimation
• Quantum limits in optical interferometry
• Quantum Enhanced Multiple Phase Estimation
• Compatibility in Multiparameter Quantum Metrology
• Multi-parameter Quantum Metrology
• Quantum fidelity for arbitrary Gaussian states
• Quantum enhanced estimation of a multi-dimensional field
• Quantum parameter estimation using general single-mode Gaussian states
• Quantum parameter estimation using general single-mode Gaussian states
• A Geometric Perspective on Quantum Parameter Estimation
• A perspective on multiparameter quantum metrology: from theoretical tools to applications in quantum imaging
• A new approach to the Cramer-Rao type bound of the pure state model
• Optimal measurements for simultaneous quantum estimation of multiple phases
• Gaussian states in continuous variable quantum information
• Quantum Fisher information for states in exponential form
• On quantumness in multi-parameter quantum estimation
• Optimal estimation of joint parameters in phase space
• A tradeoff in simultaneous quantum-limited phase and loss estimation in interferometry
• Multiparameter Gaussian Quantum Metrology
• Discontinuities of the quantum Fisher information and the Bures metric
• Evaluating the Holevo Cramér-Rao bound for multi-parameter quantum metrology
• Attaining the ultimate precision limit in quantum state estimation
• Estimation of Gaussian quantum states
• Gaussian systems for quantum enhanced multiple phase estimation
• Local asymptotic normality for qubit states
• Local versus Global Strategies in Multi-parameter Estimation
• Towards superresolution surface metrology: Quantum estimation of angular and axial separations
• Explicit formula for the Holevo bound for two-parameter qubit estimation problem
• Tight bounds on the simultaneous estimation of incompatible parameters
• Estimation of unitary quantum operations
• Quantum limited superresolution of an incoherent source pair in three dimensions
• Optimal estimation of entanglement
• Multiparameter quantum estimation of noisy phase shifts
• Entangling measurements for multiparameter estimation with two qubits
• Efficient computation of the Nagaoka--Hayashi bound for multi-parameter estimation with separable measurements
• On superresolution imaging as a multiparameter estimation problem
• Ultimate precision of joint quadrature parameter estimation with a Gaussian probe
• Quantum estimation of a two-phase spin rotation
• On the discontinuity of the quantum Fisher information for quantum statistical models with parameter dependent rank
• Classification and characterization of quantum parametric models in quantum estimation theory
• General expressions for the quantum Fisher information matrix with applications to discrete quantum imaging
• Incompatibility in Quantum Parameter Estimation
• Weak measurements and the joint estimation of phase and phase diffusion
• On the quantumness of multiparameter estimation problems for qubit systems
• A tight Cramér-Rao bound for joint parameter estimation with a pure two-mode squeezed probe
• Accessible precisions for estimating two conjugate parameters using Gaussian probes
• Maximum logarithmic derivative bound on quantum state estimation as a dual of the Holevo bound
• An upper bound on the number of compatible parameters in simultaneous quantum estimation

Cited by in corpus (3)

• Quantum scale estimation
• Incompatibility measures in multi-parameter quantum estimation under hierarchical quantum measurements
• Performance-tradeoff relation for locating two incoherent optical point sources