Abstract
We present and generalize the basic ideas underlying recent work aimed at the construction of mutually unbiased bases in finite dimensional Hilbert spaces with the help of group and graph theoretical concepts. In this approach finite groups are used to construct maximal sets of mutually unbiased bases. Thus the prime number restrictions of previous approaches are circumvented and this construction principle sheds new light onto the intricate relation between mutually unbiased bases and characteristic geometrical structures of Hilbert spaces.
Highlights
Unbiased bases of Hilbert spaces, as originally pioneered by Schwinger [1], are of mathematical interest by exhibiting characteristic geometric properties of Hilbert spaces, but they have interesting practical applications in quantum technology
Based on the early work of Charnes and Beth [17] we summarize the basic definitions encompassing the relations between mutually unbiases bases, their basis groups and associated Cayley graphs which are capable of encoding characteristic features of mutually unbiased bases of Hilbert spaces [18,19]
We have discussed and generalized a recently developed group and graph theoretial approach aiming at the construction of large sets of mutually unbiased bases in finite dimensional Hilbert spaces
Summary
Unbiased bases of Hilbert spaces, as originally pioneered by Schwinger [1], are of mathematical interest by exhibiting characteristic geometric properties of Hilbert spaces, but they have interesting practical applications in quantum technology. Besides the possible practical advantages, this method is independent of prime power restrictions of previous techniques and may offer interesting novel conceptual advantages and links to other areas of mathematics The purpose of this manuscript is to present the central ideas of this group and graph theoretical method in a self contained way, and to exhibit new connections between mutually unbiased bases and the symmetries encoded in the related basis groups and basis graphs. The examples of polytopal graphs presented are restricted to low dimensional Hilbert spaces, i.e., d = 2, 3, 4, and do not address the still open questions concerning dimension d = 6 These examples demonstrate interesting new links between mutually unbiased bases and the symmetries of graphs which are not apparent with the more orthodox constructions based on Galois fields in prime power dimensions
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