From imaging to timekeeping. Position, phase and time in quantum technologies

In this thesis, we discuss applications to imaging, positioning, phase and time estimations in quantum mechanics, which encompass technological implications, practical scenarios, and foundational problems. In the first part, we focus on the manipulation of information in terms of images. We discuss imaging in quantum-mechanical terms, by designing a quantum encoder that consists of a multiatom lattice sensor that electromagnetically interacts with the photons diffracted by the object, encoding its two-dimensional shape in the probabilities of a multiqubit quantum state. We introduce a quantum algorithm that can compress such a state, by reducing the number of encoding qubits without altering the overall visual pattern, i.e. a lossy compression. This allows to capture, compress, and send images even with limited quantum resources for storage and communication, providing an exponential speedup, in terms of gates only, over classical downsampling algorithms. Finally, we discuss image classification from a completely different perspective: by processing the information encoded in the spectrum of a single-photon state, without direct image reconstruction. Using the Hong-Ou-Mandel effect, we introduce an interferometric pattern recognition method for binary classification tasks, which shows a constant complexity in the number of computational operations and photons required by each classification instance. Mathematically, the response of the apparatus resembles that of a single artificial neuron, which, however, can classify images bypassing their reconstruction. In the second part, we take a theoretical perspective on the challenges provided by characterizing real multiqubit devices. We summarize the notion of quantum channels as a quantum-mechanical framework for describing noisy processes. In this context, we introduce a tomographic protocol for the reconstruction of multiqubit noise models, which, using a limited number of experimental resources, can partially characterize channels with an exponential speedup over quantum process tomography. We discuss a deconvolution method for mitigating the expectation values of noisy measurements at the output of such channels. In the last part, we consider the estimation of phase and the measurement of time in quantum mechanics. First, we present a novel interferometric protocol, which translates the estimation of an unknown phase into a binary search task. Such a method can work at arbitrary degrees of accuracy, while undergoing the resource scaling set by the Heisenberg limit. Second, we analyse the task of predicting the time at which a particle arrives at a screen, commonly known as the arrival time problem. We compare the theoretical proposals that attempt to solve this issue, also beyond the standard quantum theory. Exploring multiple regimes, we characterize under which conditions such proposals provide different, and experimentally discriminable, predictions, paving the way for future arrival time experiments.