Statistical theory of controlled quantum dynamics

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We are currently entering a new technological era where harnessing quantum mechanical effects is at the core of applications ranging from computation to cryptography and metrology. The modelling, estimation and identification of quantum open systems are key enabling tools for a broad array of quantum engineering tasks such as state preparation and tomography, quantum feedback control, error correction, and high precision metrology. As quantum devices and measurement techniques become more sophisticated, experimenters have to interpret increasingly complex measurement datasets to obtain accurate information about the system's state and dynamics. Therefore, there is a need for expanding the mathematical foundations of quantum statistics beyond the traditional i.i.d. (independent, identically distributed) framework of state estimation, in order to tackle new inference problems involving correlated states of interacting systems, with realistic modelling of noise, and optimal experimental design. The goal of this proposal is to build a statistical theory of quantum stochastic processes in the framework of quantum input-output (I-O) dynamics. The I-O formalism describes the system of interest (e.g. quantum device, or sensor) as a black-box interacting with the outside world via input and output channels. These can model unwanted effects such as dephasing and leakage, or can be used to monitor and control the system using quantum feedback. The central premise of the proposal is that information about the dynamics is continuously encoded into the output state, and this resource can be exploited to perform tasks such as system identification, estimation and quantum enhanced metrology. The three central objectives are: (1) to develop new mathematical theory pertaining to central limit, concentration for quantum processes, and large deviations, in close relation to the theories of finitely correlated states and hidden Markov processes; (2) to build a general statistical framework for quantum estimation with I-O systems centred around the concept of local asymptotic normality; and (3) to employ these theoretical results for the design of new quantum metrology setups which exploit features such as dynamical phase transitions, including the use of novel methods for statistical learning of time-dependent parameters.