Trace-2 excluded subsets of special linear groups over finite fields and mutually unbiased maximally entangled bases

Mutually unbiased maximally entangled bases (MUMEBs) in bipartite systems have a close relation with unitary 2-design and have attracted much attention in recent years. In the paper, we construct MUMEBs in C q ⊗ C q with q a power of an odd prime number. For this purpose, we introduce the notation of trace-2 excluded subset of the special linear group S L ( 2 , F q ) over the finite field F q and establish a relation between a trace-2 excluded subset and a set of MUMEBs in C q ⊗ C q . Under this relation, we prove that M ( q , q ) ≥ q 2 - 1 2 by constructing trace-2 excluded subsets in S L ( 2 , F q ) , which highly raises the lower bound of M ( q , q ) given in Liu et al. (Quantum Inf Process 16(6):159, 2017 ) and Xu (Quantum Inf. Process 16(3):65, 2017 ).