Quantum Tomographic Reconstruction with Error Bars: a Kalman Filter Approach

We present a novel quantum tomographic reconstruction method based on Bayesian inference via the Kalman filter update equations. The method not only yields the maximum likelihood/optimal Bayesian reconstruction, but also a covariance matrix expressing the measurement uncertainties in a complete way. From this covariance matrix the error bars on any derived quantity can be easily calculated. This is a first step towards the broader goal of devising an omnibus reconstruction method that could be adapted to any tomographic setup with little effort and that treats measurement uncertainties in a statistically well-founded way. In this first part we restrict ourselves to the important subclass of tomography based on measurements with discrete outcomes (as opposed to continuous ones), and we also ignore any measurement imperfections (dark counts, less than unit detector efficiency, etc.), which will be treated in a follow-up paper. We illustrate our general theory on real tomography experiments of quantum optical information processing elements.