We investigate the optimal strategies for estimating the Mueller matrix with arbitrary numbers of illumination and analysis states, in the presence of signal-independent additive noise or signal-dependent Poisson shot noise. We demonstrate that the architectures that minimize and equalize the estimation variances for both types of noise sources are based on spherical designs of order 2 or 3, and we provide closed-form expressions of the estimation precision obtained with these optimal measurement strategies. The obtained results are important to design Mueller polarimeters in practice and assess their fundamental limits in terms of estimation precision.
We investigate the optimal strategies for estimating the Mueller matrix with arbitrary numbers of illumination and analysis states, in the presence of signal-independent additive noise or signal-dependent Poisson shot noise. We demonstrate that the architectures that minimize and equalize the estimation variances for both types of noise sources are based on spherical designs of order 2 or 3, and we provide closed-form expressions of the estimation precision obtained with these optimal measurement strategies. The obtained results are important to design Mueller polarimeters in practice and assess their fundamental limits in terms of estimation precision.
Estimation of the Stokes vector is based on projecting the input light on a number N of polarization analysis states. We address the optimization of the distribution of these analysis states on the Poincaré sphere in the presence of signal-dependent Poisson shot noise for an arbitrary value of N. We show that if this distribution forms a spherical 3 design, the Stokes vector is estimated with minimal equally weighted variance and with estimation variances of the last three Stokes parameters equal and independent of the input Stokes vector. We also demonstrate that in the presence of Poisson shot noise, the estimation signal to noise ratio is independent of N, whereas in the presence of signal independent additive noise, it is proportional to 1/N, which means that there is a precision loss in increasing the number of measurements.
We introduce an optimal metagrating design for position metrology under Poisson shot noise. By using proper electric and magnetic resonances inside the metagrating, the position of the metagrating within an interference field can be extracted with high precision in the presence of shot noise, leading to nearly three orders of magnitude improvement compared to the centroid-fitting-based position metrology with the same number of detected photons. Furthermore, we find that the improvement in precision is closely related to the formation of a phase vortex around the optimal design in the parameter space. The link between the metagrating design, the formation of a phase vortex and the fundamental precision limit using classical light gives new insights to all of these fields.
In metrology, Fisher information is an important metric that quantifies the precision that can be achieved in a measurement. For optical measurements using coherent light it has been shown that Fisher information can be expressed simply using the scattering matrix of the system. Fisher information can be maximized over the input modes to achieve maximum information states, which produce optimally precise estimates for a parameter when the system is limited by photon noise. Here, we extend this approach to multiparameter estimation, in which case Fisher information takes the form of a matrix. We consider several scalar functions of the Fisher matrix to optimize the precision in multiple parameters at the same time. We also consider strategies for dealing with nuisance parameters, which can degrade the achievable precision of other parameters but are not of interest to measure. We corroborate our findings numerically using a scattering system of 2D coupled dipoles.
A fundamental challenge in multiparameter quantum estimation arises from the incompatibility of optimal measurements for different parameters, leading to intricate precision trade-offs that obscure the understanding of ultimate quantum limits. Here, we present an approach that precisely quantifies these trade-offs for an arbitrary number of parameters encoded in pure quantum states. Our approach not only derives tight analytical bounds for the trade-offs induced by measurement incompatibility but also provides a systematic methodology to design optimal measurement strategies that saturate these limits. To demonstrate the practical significance of our findings, we apply our framework to quantum radar and obtain a refined Arthurs-Kelly relation that characterizes the ultimate performance for the simultaneous estimation of range and velocity with any given amount of entanglement. This showcases the transformative potential of our findings for a wide range of applications in quantum metrology, sensing, and beyond.
Complete determination of the polarization state of light requires at least four distinct projective measurements of the associated Stokes vector. Stability of state reconstruction, however, hinges on the condition number κ of the corresponding instrument matrix. Optimization of redundant measurement frames with an arbitrary number of analysis states, m, is considered in this Letter in the sense of minimization of κ. The minimum achievable κ is analytically found and shown to be independent of m, except for m=5 where this minimum is unachievable. Distribution of the optimal analysis states over the Poincaré sphere is found to be described by spherical 2 designs, including the Platonic solids as special cases. Higher order polarization properties also play a key role in nonlinear, stochastic, and quantum processes. Optimal measurement schemes for nonlinear measurands of degree t are hence also considered and found to correspond to spherical 2t designs, thereby constituting a generalization of the concept of mutually unbiased bases.
The theory of quantum state estimation is exploited here to investigate the most efficient strategies for this task, especially targeting a complete picture identifying optimal conditions in terms of Fisher information, quantum measurement, and associated estimator. The approach is specified to estimation of the phase of a qubit in a rotation around an arbitrary given axis, equivalent to estimating the phase of an arbitrary single-qubit quantum gate, both in noise-free and then in noisy conditions. In noise-free conditions, we establish the possibility of defining an optimal quantum probe, optimal quantum measurement, and optimal estimator together capable of achieving the ultimate best performance uniformly for any unknown phase. With arbitrary quantum noise, we show that in general the optimal solutions are phase dependent and require adaptive techniques for practical implementation. However, for the important case of the depolarizing noise, we again establish the possibility of a quantum probe, quantum measurement, and estimator uniformly optimal for any unknown phase. In this way, for qubit phase estimation, without and then with quantum noise, we characterize the phase-independent optimal solutions when they generally exist, and also identify the complementary conditions where the optimal solutions are phase dependent and only adaptively implementable.