Maximizing the precision in estimating parameters in a quantum system subject to instrumentation constraints is cast as a convex optimization problem. We account for prior knowledge about the parameter range by developing a worst-case and average case objective for optimizing the precision. Focusing on the single parameter case, we show that the optimization problems are {\em linear programs}. For the average case the solution to the linear program can be expressed analytically and involves a simple search: finding the largest element in a list. An example is presented which compares what is possible under constraints against the ideal with no constraints, the Quantum Fisher Information.
Maximizing the precision in estimating parameters in a quantum system subject to instrumentation constraints is cast as a convex optimization problem. We account for prior knowledge about the parameter range by developing a worst-case and average case objective for optimizing the precision. Focusing on the single parameter case, we show that the optimization problems are {\em linear programs}. For the average case the solution to the linear program can be expressed analytically and involves a simple search: finding the largest element in a list. An example is presented which compares what is possible under constraints against the ideal with no constraints, the Quantum Fisher Information.
Quantum metrology comprises a set of techniques and protocols that utilize quantum features for parameter estimation which can in principle outperform any procedure based on classical physics. We formulate the quantum metrology in terms of an optimal control problem and apply Pontryagin's maximum principle to determine the optimal protocol that maximizes the quantum Fisher information for a given evolution time. As the quantum Fisher information involves a derivative with respect to the parameter which one wants to estimate, we devise an augmented dynamical system that explicitly includes gradients of the quantum Fisher information. The necessary conditions derived from Pontryagin's maximum principle are used to quantify the quality of the numerical solution. The proposed formalism is generalized to problems with control constraints, and can also be used to maximize the classical Fisher information for a chosen measurement.
Measurement noise is a major source of noise in quantum metrology.Here, we explore preprocessing protocols that apply quantum controls to the quantum sensor state prior to the final noisy measurement (but after the unknown parameter has been imparted), aiming to maximize the estimation precision.We define the quantum preprocessing-optimized Fisher information, which determines the ultimate precision limit for quantum sensors under measurement noise, and conduct a thorough investigation into optimal preprocessing protocols.First, we formulate the preprocessing optimization problem as a biconvex optimization using the error observable formalism, based on which we prove that unitary controls are optimal for pure states and derive analytical solutions of the optimal controls in several practically relevant cases.Then we prove that for classically mixed states (whose eigenvalues encode the unknown parameter) under commuting-operator measurements, coarse-graining controls are optimal, while unitary controls are suboptimal in certain cases.Finally, we demonstrate that in multiprobe systems where noisy measurements act independently on each probe, the noiseless precision limit can be asymptotically recovered using global controls for a wide range of quantum states and measurements.Applications to noisy Ramsey interferometry and thermometry are presented, as well as explicit circuit constructions of optimal controls.
We study optimal quantum sensing of multiple physical parameters using repeated measurements. In this scenario, the Fisher information framework sets the fundamental limits on sensing performance, yet the optimal states and corresponding measurements that attain these limits remain to be discovered. To address this, we extend the Fisher information approach with a second optimality requirement for a sensor to provide unambiguous estimation of unknown parameters. We propose a systematic method integrating Fisher information and Bayesian approaches to quantum metrology to identify the combination of input states and measurements that satisfies both optimality criteria. Specifically, we frame the optimal sensing problem as an optimization of an asymptotic Bayesian cost function that can be efficiently solved numerically and, in many cases, analytically. We refer to the resulting optimal sensor as a `quantum compass' solution, which serves as a direct multiparameter counterpart to the Greenberger-Horne-Zeilinger state-based interferometer, renowned for achieving the Heisenberg limit in single-parameter metrology. We provide exact quantum compass solutions for paradigmatic multiparameter problem of sensing two and three parameters using an SU(2) sensor. Our metrological cost function opens avenues for quantum variational techniques to design low-depth quantum circuits approaching the optimal sensing performance in the many-repetition scenario. We demonstrate this by constructing simple quantum circuits that achieve the Heisenberg limit for vector field and 3D rotations estimation using a limited set of gates available on a trapped-ion platform. Our work introduces and optimizes sensors for a practical notion of optimality, keeping in mind the ultimate goal of quantum sensors to precisely estimate unknown parameters.
Quantum metrology utilizes quantum resources to enhance the precision of parameter estimation. A central objective is to maximize the Quantum Fisher information (QFI), which determines the ultimate estimation precision based on the Cramér-Rao bound. In this work, we formulate QFI optimization as an optimal control problem and apply Pontryagin’s Minimum Principle (PMP) to derive the optimal control protocol. We show that when the control amplitude is constrained below a certain threshold, the optimal solution reduces to a constant control. This strategy enables the QFI to scale quadratically with the total evolution time T, thereby improving estimation precision with longer probe durations.
As one of the main pillars of quantum technologies, quantum metrology aims to improve measurement precision using techniques from quantum information. The two main strategies to achieve this are the preparation of nonclassical states and the design of optimized measurement observables. We discuss precision limits and optimal strategies in quantum metrology and sensing with a single mode of quantum continuous variables. We focus on the practically most relevant cases of estimating displacements and rotations and provide the sensitivities of the most important classes of states that includes Gaussian states and superpositions of Fock states or coherent states. Fundamental precision limits that are obtained from the quantum Fisher information are compared to the precision of a simple moment-based estimation strategy based on the data obtained from possibly sub-optimal measurement observables, including homodyne, photon number, parity and higher moments. Finally, we summarize some of the main experimental achievements and present emerging platforms for continuous-variable sensing. These results are of particular interest for experiments with quantum light, trapped ions, mechanical oscillators, and microwave resonators.
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.