New Constructions of Approximately Symmetric Informationally Complete Positive Operator-Valued Measures by Character Sums over Galois Rings

Abstract In quantum information theory, symmetric informationally complete positive operator-valued measures (SIC-POVMs) are related to quantum state tomography, quantum cryptography and foundational studies. It is difficult to construct SIC-POVMs and remains unknown whether SIC-POVMs exist for infinitely dimension. Therefore, researchers have proposed the solution to construct approximately symmetric informationally complete positive operator-valued measures (ASIC-POVMs). This paper proposes a construction of ASIC-POVMs using character sums over Galois rings. The dimension of this ASIC-POVMs is q-1, where q is a prime power. Also, our constructions include partial results about ASIC-POVMs constructed by X. Cao et al (X. Cao, J. Mi and S. Xu: Two constructions of approximately symmetric informationally complete positive operator-valued measures. J. Math. Phys., 58(6), 062201 (2017)). Mathematics Subject Classification (2020) 81P15 · 81P45