利用Gauss 和与Jacobi 和构造近似MUB 和SIC-POVM

Abstract

Mutually unbiased bases (MUB) and symmetric informationally complete positive operator-valued measure (SIC-POVM) are both important objects in quantum information theory. While people do not know if there exists a complete MUB for non-prime-power dimension, several versions of approximately MUB have been considered by relaxed the inner product condition. So far there are only finite number of <italic>K</italic> such that SIC- POVMs in Ck have been found. As in the MUB case, several versions of approximately SIC-POVM have been considered by relaxed the inner product condition. In this paper, we use the definitions of approximate MUB and SIC-POVM given by Klappenecker et al. For prime power q, we present simple constructions of <italic>q</italic> approximately MUB (AMUB) for dimension <italic>q</italic>-1, <italic>q</italic>+1 AMUB for dimension <italic>q</italic>-1, which shows the number of orthonormal bases of an AMUB in C^{<italic>k</italic>} can be more than <italic>K</italic>+1, and q AMUB for dimension <italic>q</italic>+1 by Gauss and Jacobi sums. We also present a construction of approximately SIC-POVM (ASIC-POVM) in dimension <italic>q</italic>-1 by Gauss sum.