Mutually unbiased bases: a group and graph theoretical approach

Abstract

In this contribution the main ideas underlying recent work aiming at the construction of mutually unbiased bases in finite dimensional Hilbert spaces are discussed. This approach relies on a systematic use of group and graph theoretical concepts announced by Charnes and Beth (2005 ERATO Conf. on Quantum Information Science) and extended significantly by Charnes (2018 in preparation) recently. A principal feature of this method is its independence of prime number restrictions thus distinguishing it from almost all previous constructions which have relied on finite fields and related concepts of finite geometry. This group and graph theoretical approach offers the possibility to gain new insight into the intricate relation between quantum theoretical complementarity as encoded in mutually unbiased bases and characteristic geometrical structures of the Hilbert space involved.