SIC-POVM - Relation To Spherical T-designs

Relation To Spherical T-designs

A spherical t-design is a set of vectors on the d-dimensional generalized hypersphere, such that the average value of any -order polynomial over is equal to the average of over all normalized vectors . Defining as the t-fold tensor product of the Hilbert spaces, and

as the t-fold tensor product frame operator, it can be shown that a set of normalized vectors with forms a spherical t-design if and only if

It then immediately follows that every SIC-POVM is a 2-design, since

which is precisely the necessary value that satisfies the above theorem.