Abstract In this paper we design the following two-step scheme to estimate the model parameter ω 0 of the quantum system: first we utilize the Fisher information with respect to an intermediate variable v = cos ( ω 0 t ) to determine an optimal initial state and to seek optimal parameters of the POVM measurement operators; second we explore how to estimate ω 0 from v by choosing t when a priori information knowledge of ω 0 is available. Our optimal initial state can achieve the maximum quantum Fisher information. The formulation of the optimal time t is obtained and the complete algorithm for parameter estimation is presented. We further explore how the lower bound of the estimation deviation depends on the a priori information of the model .
Abstract In this paper we design the following two-step scheme to estimate the model parameter ω 0 of the quantum system: first we utilize the Fisher information with respect to an intermediate variable v = cos ( ω 0 t ) to determine an optimal initial state and to seek optimal parameters of the POVM measurement operators; second we explore how to estimate ω 0 from v by choosing t when a priori information knowledge of ω 0 is available. Our optimal initial state can achieve the maximum quantum Fisher information. The formulation of the optimal time t is obtained and the complete algorithm for parameter estimation is presented. We further explore how the lower bound of the estimation deviation depends on the a priori information of the model .
Quantum parameter estimation holds significant promise for achieving high precision through the utilization of the most informative measurements. While various lower bounds have been developed to assess the best accuracy for estimates, they are not tight, nor provide a construction of the optimal measurement in general. Thus, determining the explicit forms of optimal measurements has been challenging due to the non-trivial optimization. In this study, we introduce an algorithm, termed QestOptPOVM, designed to directly identify optimal positive operator-valued measure (POVM) using the steepest descent method. Through rigorous testing on several examples for multiple copies of qubit states (up to six copies), we demonstrate the efficiency and accuracy of our proposed algorithm. Moreover, a comparative analysis between numerical results and established lower bounds serves to validate the tightness of the Nagaoka-Hayashi bound in finite-sample quantum metrology for our examples. Concurrently, our algorithm functions as a tool for elucidating the explicit forms of optimal POVMs, thereby enhancing our understanding of quantum parameter estimation methodologies.
A unified framework is proposed in this paper for parameter estimation using convex optimization and experiment design applying convex maximization for Pauli channels, that can be extended to generalized Pauli channels, too. In the case of known channel directions, an affine parametrization of the Choi matrix turns the LS parameter estimation into a convex optimization problem also for the generalized Pauli channels. A simple iterative algorithm for estimating the channel directions is also given for qubit Pauli channels. The experiment design was performed by maximizing the trace of the Fisher information matrix of the output quantum system to find optimal input state and measurement POVM for the channel estimation. For the known channel direction case it was found that the optimal input state should be pure and the optimal measurement POVM is extremal. It was also shown that both the input state and the POVM elements in the optimal configuration should be parallel to the channel directions in the qubit Pauli channel case. The proposed methods and algorithms are illustrated by simple numerical examples.
In this technical note, we study the problem of parameter estimation for quantum channels, which are (completely positive) maps acting on quantum states. We focus on an important subclass of channels called Pauli channels which are characterized by a certain set of vectors (directions) and a number of scalar parameters. For the case where the directions are known, a special parametrization turns the parameter estimation problem into a convex optimization problem. For the case of unknown directions we give a simple algorithm to estimate the directions for qubit Pauli channels. These results assume that the identification experiment configuration is given, namely, a set of quantum states are given on which the channel acts, and a set of positive measurement (operator)s are fixed from which information is gathered. In the second part of the technical note we consider the problem of determining the experiment design, namely, determining states and measurements for optimal parameter identification of the channels. We formalize this problem as a maximization problem for Fisher information and, assuming known channel directions, prove that this problem is convex. We also prove that the optimal states to be used in experiments are pure and the optimal measurements are extremal. For qubit Pauli channels we prove that both the optimal pure input states and projective measurements should be directed towards the channel directions. We illustrate the results of the technical note with numerical examples and simulations.
Informationally complete measurements form the foundation of universal quantum state reconstruction, while quantum parameter estimation is based on the local structure of the manifold of quantum states. Here we establish a general link between these two aspects, in the context of a single informationally complete measurement, by employing a suitably adapted operator frame theory. In particular, we bound the ratio between the classical and quantum Fisher information in terms of the spectral decomposition of the associated frame operator, and connect these bounds to the optimal and least optimal directions for parameter encoding. The geometric and operational characterization of information extraction thus obtained reveals the fundamental tradeoff imposed by informational completeness on local quantum parameter estimation.
The quantum Fisher information (QFI) quantifies the ultimate precision of estimating a parameter from a quantum state and can be regarded as a reliability measure of a quantum system as a quantum sensor. However, estimation of the QFI for a mixed state is in general a computationally demanding task. In this paper we present a variational quantum algorithm called variational quantum Fisher information estimation (VQFIE) to address this task. By estimating lower and upper bounds on the QFI, based on bounding the fidelity, VQFIE outputs a range in which the actual QFI lies. This result can then be used to variationally prepare the state that maximizes the QFI, for the application of quantum sensing. In contrast to previous approaches, VQFIE does not require knowledge of the explicit form of the sensor dynamics. We simulate the algorithm for a magnetometry setup and demonstrate the tightening of our bounds as the state purity increases. For this example, we compare our bounds with literature bounds and show that our bounds are tighter.
Squeezed states of light are a crucial resource for improving the precision of parameter estimation [1], and this has made an impact in areas as wide ranging as biological imaging [2] and gravitational wave detection [3]. However, the optimal bounds of the Fisher information (FI) in continuous variable (CV) quantum metrology have, so far, only been abstractly defined and do not account for many important experimental parameters, such as the quantum efficiency and bandwidth of the detector [4]. We show that classical FI calculations can be used to compute these optimal bounds with more applicability to experimental measurement. This is demonstrated in the case of single parameter estimation of a modulated optical loss, using a bright amplitude squeezed state as a probe.