Quantum state tomography using two-outcome measurements

We investigate the process of quantum state tomography in which the observer may only access measurements on a single qubit. We assume that all measurements are projective and have only two possible outcomes, so that measurements on a system of dimension n are represented by POVMs in which the elements of each POVM are projections onto orthogonal subspaces of dimension n/2. We claim that the optimal measurements for non-adaptive quantum state tomography using two-outcome measurements are described by {\em complex Grassmannian 2-designs}. We also give lower bounds for the size of a Grassmannian 2-design: in a system of dimension n, at least n^2-1 two-outcome measurements must occur in order for the union of the POVM elements to form a 2-design. Finally, we show that this lower bound on the size of a 2-design is tight by constructing 2-designs of minimal size in certain dimensions. In particular, for every n such that there exists a Hadamard matrix of order n, we construct n^2-1 two-outcome measurements which form a complex Grassmannian 2-design in an n-dimensional system.