• information geometry
• noisy estimation
• incompatibility
• quantum metrology
• quantum estimation
• qlan
• heisenberg scaling

One of the characterizing feature of quantum mechanics is the incompatibility between observables, understood since the beginning as the Heisenberg uncertainty principle. This is a trade-off relation between the measurements precision of two conjugated physical quantities. In the modern approach of quantum information theory some vagueness about the uncertainty principle has been cleared out, and a whole new theory of incompatibility has been developed. Among today cornerstones of quantum information there is quantum estimation theory, that deals with the precision limits in extraction of parameters characterizing a certain physical state. These parameters may not have an associated observable (temperature, phase shift, noise, ...) and to recover information about them one needs to perform complex data processing of the measured data. When going to the full realm of quantum metrology, entanglement can be exploited to boost the estimation strategy. These developments help in clarifying what can and can't be done within the domain of quantum mechanics. However, far from being just theoretical curiosities they have had and will have great relevance in the fields of applied physics concerning gravitational waves, biology and enhanced imaging, just to cite some applications. As a matter of fact it's hard to overestimate the importance of quantum metrology for the development of quantum technologies: it naturally pops out of fundamental tasks like the calibration of gates for quantum computation, the quantification of the amount of entanglement and/or noise in a given state, the characterization of thermal environments and many others. The objective of this thesis is to explore quantum incompatibility within the domain of quantum metrology. We give a precise quantification of the amount of incompatibility in the estimation of multiple parameters by defining a suitable figure of merit. Its mathematical properties are extensively characterized, and it is computed for some relevant estimations on qubit and qutrit systems; deeply relying on the theory of Quantum Local Asymptotic Normality (QLAN). Then we analyze the metrological process in presence of noise by taking as example the simple scenario of a qubit subject to various disturbances. The computation of the figure of merit points toward the presence of an information-compatibility trade-off, i.e. we can have an improvement in compatibility by dropping some of the information content and vice versa. Further work is dedicated to enforcing full compatibility at the expense of a finite amount of information sacrificed. Then we perform calculations of the noisy figure of merit for the more elaborate scenarios of sequential estimation and multiqubit entangled input, and we expose the interplay between the enhancement in compatibility given by noise and that given by entanglement. Since now we have been talking about incompatibility at the measurement level, but in multiparameter quantum metrology there is also incompatibility at the level of probes, meaning that each parameter has its own optimal input, and in the simultaneous estimation they compete for the emergence of a joint optimal state. We explore this issue by defining a figure of merit for probes incompatibility. Being this work completely theoretical the hope is that it will contribute to advance the edge of quantum technologies and in particular that of today NISQ (Noisy Intermediate Scale Quantum) technologies. It is worth mentioning that in the final appendix we formulate and solve a very harmful loophole in entanglement enhanced metrology (Heisenberg scaling) which could potentially destroy its advantage. We feel that the loophole and its resolution have been overlooked in the literature and were never given a proper treatment.