Construction of 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. [33] introduced an approximate version of SIC-POVMs called approximately symmetric informationally complete POVMs (ASIC-POVMs). In this paper, we present two new constructions of ASIC-POVMs in dimensions $q$ and $q+1$ by $2$-to-$1$ PN functions and the Li bound, respectively, where $q$ is a prime power. In the first construction, we show that all $2$-to-$1$ PN functions can be used for constructing ASIC-POVMs of dimension $q$, which not only generalizes the construction in [33, Theorem 5], but also generalizes the general construction in [11, Theorem III.3]. We show that some $2$-to-$1$ PN functions that do not satisfy the condition in [11, Theorem III.3] can be also utilized for constructing ASIC-POVMs of dimension $q$. We also give a class of biangular frames related to our ASIC-POVMs. The second construction gives a new method to obtain ASIC-POVMs in dimension $q+1$ via a multiplicative character sum estimate called the Li bound.