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Erschienen in:
01.07.2019
Trace-2 excluded subsets of special linear groups over finite fields and mutually unbiased maximally entangled bases
Erschienen in: Quantum Information Processing | Ausgabe 7/2019
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Abstract
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 \(\mathbb {C}^q\otimes \mathbb {C}^q\) 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 \(SL(2,\mathbb {F}_q)\) over the finite field \(\mathbb {F}_q\) and establish a relation between a trace-2 excluded subset and a set of MUMEBs in \(\mathbb {C}^q\otimes \mathbb {C}^q\). Under this relation, we prove that \(M(q,q)\ge \dfrac{q^2-1}{2}\) by constructing trace-2 excluded subsets in \(SL(2,\mathbb {F}_q)\), 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).