Characterisation of complex systems using quantum information and sensing techniques

Complex quantum systems require a mix of theoretical models and statis tical methods in order to characterize observed behaviour. In this thesis, we first build a framework that uses recorded data and quantum information tools to rule out incompatible quantum models based on observed dimensional complexity. In the second part, settling on a fixed parametric quantum model but with unknown parameter values, we construct enhanced detection setups for estimation of complex system parameters using non-classical probe states of light. Both of the introduced schemes involve inferring something about the complex system from observed data. We first present a model-independent measure of dynamical complexity based on simulation of quantum dynamics using stroboscopic Markovian dynamics. Tools from classical signal processing enable us to infer the Hilbert space dimension of the complex system evolving under a time-independent Hamiltonian via pulsed in terrogation. We illustrate this using simulated pump-probe spectroscopy data for exciton transport in toy model of coupled dimer with vibrational levels, and light harvesting 2 (LH2) and Fenna-Matthews-Olson (FMO) complexes using data from recent nonlinear ultrafast optical spectroscopy experiments. For the latter we make model-independent inferences commensurate with model-specific ones, including the estimation of the fewest number of parameters needed to fit the experimental data and identifying the spatial extent, i.e., delocalization size, of participating quantum states. Next, we recast spectroscopy using quantum states of light as a parameter estimation problem. Introducing a hierarchy of two-sided equations of motion for arbitrary complex system Hamiltonians and incoming light states, we show that ultimate precision of parameter estimators can be bounded by calculating the quan tum Fisher information (QFI). For the single two-level systems (TLS) probed using single-mode pulses, we show that optimal detection strategies depend on the TLS parameter one is interested in. For the toy model of coupled dimer system, we show that the estimation problem for interstitial coupling is similar to that for the TLS level frequency, and that optimal measurements for complex system Hamilto nian parameter estimation are similar to each other, and di↵erent from coupling Hamiltonian parameters. Using two-mode entangled states in biphoton spectro scopic setup, so that only one of the photons interacts with the complex system, we show that correlated measurement schemes always yield more precise estimators than using uncorrelated local measurements, or single-mode measurements.