Efficient 2-designs from bases exist

TL;DR

This paper demonstrates the existence of approximately d bases forming 2-designs in any complex d-dimensional space, providing a practical alternative to maximal mutually unbiased bases especially in non-prime-power dimensions.

Contribution

It introduces a novel construction of 2-designs from bases in arbitrary dimensions, extending the applicability beyond prime-power dimensions.

Findings

Constructs O(d) bases forming 2-designs in complex d-dimensional space

Offers an efficient method as an alternative to maximal MUBs in general dimensions

Builds on a framework linking the problem to nonlinear functions between abelian groups

Abstract

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 manifold 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.