Quantum Tomographic Reconstruction with Error Bars: a Kalman Filter Approach

          @misc{ pirsa_PIRSA:08080033,
            doi = {10.48660/08080033},
            url = {https://pirsa.org/08080033},
            author = {Audenaert, Koenraad},
            keywords = {},
            language = {en},
            title = {Quantum Tomographic Reconstruction with Error Bars: a Kalman Filter Approach},
            publisher = {Perimeter Institute},
            year = {2008},
            month = {aug},
            note = {PIRSA:08080033 see, \url{https://pirsa.org}}
          }
          

Abstract

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. 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 further work. We illustrate the general theory on two real tomography experiments of quantum optical information processing elements.