2023/10/10 by Meng Cao, Cao, Meng, Xiantao Deng +1
Computer Science · Engineering · Mathematics · #Coding theory and cryptography #FOS: Physical sciences #Mathematical Analysis and Transform Methods #Quantum Physics (quant-ph) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2310.06418
openalex publication_date 2023/10/10 · openalex created_date 2023/10/12 · openalex updated_date 2026/07/28
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 d2 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 εd-ASIC-POVMs in dimension d=q and a class of εd-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 εq-ASIC-POVMs. We show that the set of vectors corresponding to the εq-ASIC-POVM forms a biangular frame. The construction of εq+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 εq+1-ASIC-POVM forms an asymptotically optimal codebook. We characterize "how close" the εq-ASIC-POVMs (resp. εq+1-ASIC-POVMs) are from being SIC-POVMs of dimension q (resp. dimension q+1). Finally, we explain the significance of constructing εd-ASIC-POVMs.