Learning from temporal correlations in quantum data

Abstract

Many applications in quantum technologies require precise characterisation and calibration of quantum devices. Oftentimes, one has to estimate dynamical parameters from the outcomes of the quantum systems. In this thesis, we address this problem, focussing specifically on the role of time correlations in the output data. We give novel precision bounds for the estimated parameters, and introduce techniques to saturate such bounds. The achievable precision when estimating a parameter is bounded by the Cramér-Rao bound, in terms of the Fisher information of the outcomes' probability distribution. While estimation is well understood for independent variables, the role of correlations is still largely unexplored. How do the precision bounds for independent random variables generalise to the correlated case? And how can estimation be actually performed, in a computationally feasible manner? We first derive precision bounds for classical stationary processes with a finite memory. We prove an asymptotic expression for the Fisher information in the limit of a large amount of data. Quite counterintuively, correlations can both hamper or improve estimation performances: we give general rules of thumb to determine when they are helpful and when they aren't. We apply our results to temperature estimation on an Ising spin chain, and we find crucially different behaviour between the ferromagnetic and anti-ferromagnetic phases, in line with our heuristics. We then focus on the quantum jump unravelling (QJU) of open quantum system dynamics. These processes describe continuously monitored quantum systems, where time and type of each emission are recorded. We derive analytical expressions for the Fisher information for processes in which all the memory is lost upon every emission, known as renewal processes. Beyond the renewal hypothesis, the problem cannot be treated analytically. We generalise the previously introduced monitoring operator formalism to compute the Fisher information on quantum trajectories. We develop the quantum Gillespie algorithm, for a more efficient simulation on trajectories. We illustrate these numerical methods with case studies from atomic physics and quantum optics. In quantum thermodynamics, one can express the kinetic and thermodynamic uncertainty relations in terms of the Fisher information on trajectories. Exploiting our results on metrology for the QJU, we introduce a tighter version of the kinetic uncertainty relation for quantum systems undergoing continuous monitoring. We consider the problem of learning a quantum channel acting on a system, by only having access to a subsystem, while quantum memory keeps flowing in the unmonitored portion. We term this framework \textit{collisional learning}. As can be expected, not all the features of the channel can be learnt in this scenario, and we characterise the emerging gauge freedoms. We define a new notion of similarity between the estimated channel and the original one, in terms of their action on the monitored subsystem. We develop numerically efficient methods to perform collisional learning, and show that they can effectively learn two-qubits Hamiltonians. Frequently, the output data are not available in raw form, but rather subject to some post-processing. In general, this reduces the Fisher information in the data, and consequently the estimation precision. We derive expressions for the Fisher information contained in the outcomes of two of the most common post-processing strategies, the empirical distribution and the sample average. For quantum jump unravelling outputs, we show what happens if some of the jump channels cannot be monitored, or cannot be distinguished one from each other. This amounts to a reduction of the Fisher information. Quite surprisingly, the renewal nature of the process can change, depending on the post-processing. Throughout the thesis, we discuss hypothesis testing problems on the side of estimation. This is a framework in which, instead of estimating continuous parameters, one chooses the most likely among a discrete sets of models that can have generated the process. We give both analytical and numerical results on the error probabilities. Finally, in a more speculative Outlook chapter, we give some hints at possible directions in which our research could be carried forward.