Quantum Tomography under Prior Information

We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify an unknown quantum state which is constrained by prior information? We show that if the prior information restricts the possible states to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the set of all states. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that the minimal number of measurement outcomes (POVM elements) necessary to identify all pure states in a d-dimensional Hilbert space is 4d-3-c(d) alpha(d) for some and alpha(d) being the number of ones appearing in the binary expansion of (d-1).