Efficient 2-designs from bases exist
We show that in a complex d-dimensional vector space, one can find O(d) bases whose elements form a 2-design. Such vector sets generalize the notion of a maximal collection of mutually unbiased bases (MUBs) MUBs have many applications in quantum information theory (e.g. in state tomography. cloning; or, cryptography) - however it is suspected that maximal sets exist only in prime-power dimensions. Our construction offers an efficient alternative for general dimensions. The findings are based on a framework recently established in [A. Roy and A. Scott, J. Math. Phys. 48, 072110 (2007)] which reduces the construction of such bases to the combinatorial problem of finding certain highly nonlinear functions between abelian groups.
2-designs mutually unbiased bases highly non-linear functions