Maximal orthoplectic fusion frames from mutually unbiased bases and block designs
Open Access
- 8 February 2018
- journal article
- research article
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
- Vol. 146 (6), 2601-2616
- https://doi.org/10.1090/proc/13956
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.Keywords
Funding Information
- National Science Foundation (1412524, NSF ATD 1321779, DMS 1412524)
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