Construction of ${\epsilon }_d$-ASIC-POVMs via $2$-to-$1$ PN functions and the Li bound

Symmetric informationally complete positive operator-valued measures (SIC-POVMs) in finite dimension $d$ are a particularly attractive case of informationally complete POVMs (IC-POVMs), which consist of $d^2$ subnormalized projectors with equal pairwise fidelity. However, it is difficult to construct SIC-POVMs, and it is not even clear whether there exists an infinite family of SIC-POVMs. To realize some possible applications in quantum information processing, Klappenecker et al. [37] introduced an approximate version of SIC-POVMs called approximately symmetric informationally complete POVMs (ASIC-POVMs). In this paper, we construct a class of ${\epsilon }_d$-ASIC-POVMs in dimension $d=q$ and a class of ${\epsilon }_d$-ASIC-POVMs in dimension $d=q+1$, respectively, where $q$ is a prime power. We prove that all $2$-to-$1$ perfect nonlinear (PN) functions can be used for constructing ${\epsilon }_q$-ASIC-POVMs. We show that the set of vectors corresponding to the ${\epsilon }_q$-ASIC-POVM forms a biangular frame. The construction of ${\epsilon }_{q+1}$-ASIC-POVMs is based on a multiplicative character sum estimate called the Li bound. We show that the set of vectors corresponding to the ${\epsilon }_{q+1}$-ASIC-POVM forms an asymptotically optimal codebook. We characterize “how close” the ${\epsilon }_q$-ASIC-POVMs (resp. ${\epsilon }_{q+1}$-ASIC-POVMs) are from being SIC-POVMs of dimension $q$ (resp. dimension $q+1$). Finally, we explain the significance of constructing ${\epsilon }_d$-ASIC-POVMs.