Precision bounds in noisy quantum metrology

In an idealistic setting, quantum metrology protocols allow to sense physical parameters with mean squared error that scales as $1/N^2$ with the number of particles involved---substantially surpassing the $1/N$-scaling characteristic to classical statistics. A natural question arises, whether such an impressive enhancement persists when one takes into account the decoherence effects that are unavoidable in any real-life implementation. In this thesis, we resolve a major part of this issue by describing general techniques that allow to quantify the attainable precision in metrological schemes in the presence of uncorrelated noise. We show that the abstract geometrical structure of a quantum channel describing the noisy evolution of a single particle dictates then critical bounds on the ultimate quantum enhancement. Our results prove that an infinitesimal amount of noise is enough to restrict the precision to scale classically in the asymptotic $N$ limit, and thus constrain the maximal improvement to a constant factor. Although for low numbers of particles the decoherence may be ignored, for large $N$ the presence of noise heavily alters the form of both optimal states and measurements attaining the ultimate resolution. However, the established bounds are then typically achievable with use of techniques natural to current experiments. In this work, we thoroughly introduce the necessary concepts and mathematical tools lying behind metrological tasks, including both frequentist and Bayesian estimation theory frameworks. We provide examples of applications of the methods presented to typical qubit noise models, yet we also discuss in detail the phase estimation tasks in Mach-Zehnder interferometry both in the classical and quantum setting---with particular emphasis given to photonic losses while analysing the impact of decoherence.