Role of the extended Hilbert space in the attainability of the Quantum Cramér-Rao bound for multiparameter estimation

• URL: http://arxiv.org/abs/2404.01520v1
• Date: Mon, 1 Apr 2024 23:01:03 GMT
• Title: Role of the extended Hilbert space in the attainability of the Quantum Cramér-Rao bound for multiparameter estimation
• Authors: Lorcan O. Conlon, Jun Suzuki, Ping Koy Lam, Syed M. Assad,
• Abstract summary: We show that the Nagaoka-Hayashi Cram'er-Rao bound is not always attainable. We provide necessary and sufficient conditions for the attainability of the SLDCRB in the case when the kernel space is one-dimensional.
• Score: 14.334779130141452
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The symmetric logarithmic derivative Cram\'er-Rao bound (SLDCRB) provides a fundamental limit to the minimum variance with which a set of unknown parameters can be estimated in an unbiased manner. It is known that the SLDCRB can be saturated provided the optimal measurements for the individual parameters commute with one another. However, when this is not the case the SLDCRB cannot be attained in general. In the experimentally relevant setting, where quantum states are measured individually, necessary and sufficient conditions for when the SLDCRB can be saturated are not known. In this setting the SLDCRB is attainable provided the SLD operators can be chosen to commute on an extended Hilbert space. However, beyond this relatively little is known about when the SLD operators can be chosen in this manner. In this paper we present explicit examples which demonstrate novel aspects of this condition. Our examples demonstrate that the SLD operators commuting on any two of the following three spaces: support space, support-kernel space and kernel space, is neither a necessary nor sufficient condition for commutativity on the extended space. We present a simple analytic example showing that the Nagaoka-Hayashi Cram\'er-Rao bound is not always attainable. Finally, we provide necessary and sufficient conditions for the attainability of the SLDCRB in the case when the kernel space is one-dimensional. These results provide new information on the necessary and sufficient conditions for the attainability of the SLDCRB.

Related papers

• Saturation of the Multiparameter Quantum Cramér-Rao Bound at the Single-Copy Level with Projective Measurements [0.0]
  It was not known when the quantum Cram'er-Rao bound can be saturated (achieved) when only a single copy of the quantum state is available. In this paper, key structural properties of optimal measurements that saturate the quantum Cram'er-Rao bound are illuminated.
  arXiv  Detail & Related papers  (2024-05-02T17:04:13Z)
• Saturability of the Quantum Cramér-Rao Bound in Multiparameter Quantum Estimation at the Single-Copy Level [0.0]
  The quantum Cram'er-Rao bound (QCRB) is the ultimate lower bound for precision in quantum parameter estimation. This paper establishes necessary and sufficient conditions for saturability of the QCRB in the single-copy setting.
  arXiv  Detail & Related papers  (2024-02-18T12:30:04Z)
• Optimal function estimation with photonic quantum sensor networks [0.0]
  We solve the problem of optimally measuring an analytic function of unknown local parameters each linearly coupled to a qubit sensor. In particular, we derive lower bounds on the achievable mean square error in estimating a linear function of either local phase shifts or quadrature displacements. For quadrature displacements, we extend bounds to the case of arbitrary linear functions.
  arXiv  Detail & Related papers  (2024-01-29T19:00:00Z)
• Optimal phase estimation in finite-dimensional Fock space [3.411077163447709]
  In the finite-dimensional Fock space the NOON state ceases to be optimal when the particle number is fixed yet not equal to the space dimension minus one. We present three theorems to answer this question and provide a complete optimal scheme to realize the ultimate precision limit in practice.
  arXiv  Detail & Related papers  (2023-12-04T15:24:14Z)
• Continuous percolation in a Hilbert space for a large system of qubits [58.720142291102135]
  The percolation transition is defined through the appearance of the infinite cluster. We show that the exponentially increasing dimensionality of the Hilbert space makes its covering by finite-size hyperspheres inefficient. Our approach to the percolation transition in compact metric spaces may prove useful for its rigorous treatment in other contexts.
  arXiv  Detail & Related papers  (2022-10-15T13:53:21Z)
• Optimal Scaling for Locally Balanced Proposals in Discrete Spaces [65.14092237705476]
  We show that efficiency of Metropolis-Hastings (M-H) algorithms in discrete spaces can be characterized by an acceptance rate that is independent of the target distribution. Knowledge of the optimal acceptance rate allows one to automatically tune the neighborhood size of a proposal distribution in a discrete space, directly analogous to step-size control in continuous spaces.
  arXiv  Detail & Related papers  (2022-09-16T22:09:53Z)
• The gap persistence theorem for quantum multiparameter estimation [14.334779130141452]
  We show that it is impossible to saturate the Holevo Cram'er-Rao bound (HCRB) for several physically motivated problems. We further prove that if the SLDCRB cannot be reached with a single copy of the probe state, it cannot be reached with collective measurements on any finite number of copies of the probe state.
  arXiv  Detail & Related papers  (2022-08-15T18:01:22Z)
• Beyond the Edge of Stability via Two-step Gradient Updates [49.03389279816152]
  Gradient Descent (GD) is a powerful workhorse of modern machine learning. GD's ability to find local minimisers is only guaranteed for losses with Lipschitz gradients. This work focuses on simple, yet representative, learning problems via analysis of two-step gradient updates.
  arXiv  Detail & Related papers  (2022-06-08T21:32:50Z)
• Sample-Efficient Reinforcement Learning Is Feasible for Linearly Realizable MDPs with Limited Revisiting [60.98700344526674]
  Low-complexity models such as linear function representation play a pivotal role in enabling sample-efficient reinforcement learning. In this paper, we investigate a new sampling protocol, which draws samples in an online/exploratory fashion but allows one to backtrack and revisit previous states in a controlled and infrequent manner. We develop an algorithm tailored to this setting, achieving a sample complexity that scales practicallyly with the feature dimension, the horizon, and the inverse sub-optimality gap, but not the size of the state/action space.
  arXiv  Detail & Related papers  (2021-05-17T17:22:07Z)
• On Lower Bounds for Standard and Robust Gaussian Process Bandit Optimization [55.937424268654645]
  We consider algorithm-independent lower bounds for the problem of black-box optimization of functions having a bounded norm. We provide a novel proof technique for deriving lower bounds on the regret, with benefits including simplicity, versatility, and an improved dependence on the error probability.
  arXiv  Detail & Related papers  (2020-08-20T03:48:14Z)
• Dimension-free convergence rates for gradient Langevin dynamics in RKHS [47.198067414691174]
  Gradient Langevin dynamics (GLD) and SGLD have attracted considerable attention lately. We provide a convergence analysis GLD and SGLD when the space is an infinite dimensional space.
  arXiv  Detail & Related papers  (2020-02-29T17:14:13Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.