Bayesian Process Tomography Using Orientation Statistics

Using the fact that completely positive, trace-preserving (CPTP) maps can be represented by matrices of orthonormal columns through the Stinespring representation, we consider a number of Bayesian estimation problems in quantum process tomography that use random matrices with orthonormal columns as prior distributions. The space of matrices with orthonormal columns of a given dimension is known as a Stiefel manifold, and the generalization of angular or directional statistics to random elements on Stiefel manifolds is known as orientation statistics. From the field of orientation statistics we present three main classes of probability distributions: wrapped distributions formed by exponentiating random matrix elements of the Stiefel manifold tangent space, projected distributions formed by taking the QR decomposition of a random matrix, and maximum entropy distributions derived from the statistical theory of exponential families. In particular, this last class allows for the treatment of an average CPTP map as a sufficient statistic. This presentation contains recent results for a number of Bayesian approaches to process tomography, including full Bayesian tomography, maximum a posteriori (MAP) estimation, expected a posteriori (EAP) estimation, and sequential methods.