Optimal strategies in quantum metrology

In this thesis, we address the problem of optimization in quantum metrology. The usual approach identifies three stages that could possibly be subject to optimization: the preparation of the probing system's state, the measurement, and the data processing. Various numerical and analytic techniques have been developed to tackle these problems, both separately and jointly. A popular approach in the field is the study of precision lower bounds based on the Fisher information. Within this framework, we find bounds on the precision of a quantum sensing procedure that are as tight as possible, and then proceed to optimize them. The assumption is that the solution obtained from such schematization can be useful in experiments. We call this a top-down approach, which starts from the theory and tries to derive practical solutions for the laboratory. If we start our analysis from the experiment instead, we observe that quantum sensors can be manipulated across various parameters, which typically do not exhaust all the theoretical possibilities. In this thesis, we propose a bottom-up approach to optimization in quantum metrology, that consists in identifying the controls of the experiment and in refining the control strategy step-by-step through a training procedure. We are not content with lower bounds, instead  we work with the actual precision computed on the simulations of a model of the sensor. This model should be as close as possible to reality to reproduce all the behaviours observed experimentally. For each sensing platform, pinpointing the optimal controls to enhance the sensor's precision remains a challenging task. While an analytical solution might be out of reach, machine learning offers a promising avenue for many systems of interest, especially given the capabilities of contemporary hardware. Our bottom-up approach is a versatile procedure capable of optimizing a wide range of problems in quantum metrology and estimation by combining model-aware reinforcement learning with Bayesian estimation based on particle filtering. To achieve this, we had to address the challenge of incorporating the many non-differentiable steps of the estimation in the training process, such as measurements and the resampling of the particle filter. Model-aware reinforcement learning is a gradient-based method, where the derivatives of the sensor's precision are obtained through automatic differentiation in the simulation of the experiment. Our approach is suitable for optimizing both non-adaptive and adaptive strategies, using neural networks or other agents. We provide an implementation of this technique in the form of a Python library called qsensoropt, alongside several pre-made applications for relevant physical platforms, namely NV centers and photonic circuits. Leveraging our method, we've achieved results for many examples that surpass the current state-of-the-art in experimental design. In addition to Bayesian estimation, leveraging model-aware reinforcement learning, it is also possible to find optimal controls for the minimization of the Cramér-Rao bound, based on Fisher information. The machine learning approach proposed here is complemented with two theoretical studies. First, we analyse the problem of optimal phase estimation with maximally entangled states, for which we develop an analytical approach based on Lagrangian multipliers. This method allows us to find the best scalings and prefactor rigorously proven for the problem. The second analytical result of the thesis is an application of the asymptotic quantum estimation theory to the quantification of incompatibility in an estimation task. In quantum metrology, in general, different parameters correspond to non-commuting generators. This limits the precision of their simultaneous estimation. This advancement, on one hand, allows us to gain insight into the dynamics of the multiple encoding of parameters on a system; on the other hand, it can be useful for experimental design to choose the probe states that are most capable of encoding multiple parameters, thereby guiding the selection of the input states in the numerical simulations. The role of both these analytical tools is to be complementary to the reinforcement learning approach.