Construction of εd-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 ε d -ASIC-POVMs in dimension d = q and a class of ε 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 ε q -ASIC-POVMs. We show that the set of vectors corresponding to the ε q -ASIC-POVM forms a biangular frame. The construction of ε 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 ε q + 1 -ASIC-POVM forms an asymptotically optimal codebook. We characterize "how close" the ε q -ASIC-POVMs (resp. ε q + 1 -ASIC-POVMs) are from being SIC-POVMs of dimension q (resp. dimension q + 1). Finally, we explain the significance of constructing ε d -ASIC-POVMs. [ABSTRACT FROM AUTHOR]