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Maximal orthoplectic fusion frames from mutually unbiased bases and block designs

Authors: Bernhard G. Bodmann and John I. Haas
Journal: Proc. Amer. Math. Soc. 146 (2018), 2601-2616
MSC (2010): Primary 42C15
Published electronically: February 8, 2018
MathSciNet review: 3778161
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Abstract: The construction of optimal line packings in real or complex Euclidean spaces has been shown to be a tantalizingly difficult task, because it includes the problem of finding maximal sets of equiangular lines. In the regime where equiangular lines are not possible, some optimal packings are known, for example, those achieving the orthoplex bound related to maximal sets of mutually unbiased bases. In this paper, we investigate the packing of subspaces instead of lines and determine the implications of maximality in this context. We leverage the existence of real or complex maximal mutually unbiased bases with a combinatorial design strategy in order to find optimal subspace packings that achieve the orthoplex bound. We also show that maximal sets of mutually unbiased bases convert between coordinate projections associated with certain balanced incomplete block designs and Grassmannian 2-designs. Examples of maximal orthoplectic fusion frames already appeared in the works by Shor, Sloane, and by Zauner. They are realized in dimensions that are a power of four in the real case or a power of two in the complex case.

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Additional Information

Bernhard G. Bodmann
Affiliation: Department of Mathematics, 651 Philip G. Hoffman Hall, University of Houston, Houston, Texas 77204-3008

John I. Haas
Affiliation: Department of Mathematics, 219 Mathematical Sciences Building, University of Missouri, Columbia, Missouri 65211

Received by editor(s): September 29, 2016
Received by editor(s) in revised form: September 11, 2017
Published electronically: February 8, 2018
Additional Notes: The first author was supported in part by NSF DMS 1412524.
The second author was supported by NSF ATD 1321779.
Communicated by: Pham Huu Tiep
Article copyright: © Copyright 2018 American Mathematical Society