Quantum conical designs

Complex projective t-designs, particularly sics and full sets of mutually unbiased bases, play an important role in quantum information. We introduce a generalization which we call conical t-designs. They include arbitrary rank symmetric informationally complete measurements (sims) and full sets of arbitrary rank mutually unbiased measurements (mums). They are deeply implicated in the description of entanglement (as we show in a subsequent paper). Viewed in one way a conical two-design is a symmetric decomposition of a separable Werner state (up to a normalization factor). Viewed in another way it is a certain kind of polytope in the Bloch body. In the Bloch body picture sims and full sets of mums form highly symmetric polytopes (a single regular simplex in the one case; the convex hull of a set of orthogonal regular simplices in the other). We give the necessary and sufficient conditions for an arbitrary polytope to be what we call a homogeneous conical two-design. This suggests a way to search for new kinds of projective two-design. (paper)