  Complex projective t-designs, particularly SICs and full sets of MUBs, 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 2-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 2-design. This suggests a way to search for new
kinds of projective 2-design.
