New Constructions of Approximately Mutually Unbiased Bases by Character Sums over Galois Rings

Abstract

Mutually unbiased bases (MUBs) are widely used in quantum information processing and play an important role in quantum cryptography, quantum state tomography and communications. It's difficult to construct MUBs and remains unknown whether complete MUBs exist for any non prime power. Therefore, researchers have proposed the solution to construct approximately mutually unbiased bases (AMUBs) by weakening the inner product conditions. This paper constructs q AMUBs of ℂ^q, (q+1) AMUBs of ℂ^q-1 and q AMUBs of ℂ^q-1 by using character sums over Galois rings and finite fields, where q is a power of a prime. The first construction of q AMUBs of ℂ^q is new which illustrates K AMUBs of ℂ^K can be achieved. The second and third constructions in this paper include the partial results about AMUBs constructed by W. Wang et al. in [9].