Regular structures of lines in complex spaces

We study the properties of regular structures of lines, such as equiangular sets of lines and mutually unbiased bases (MUBs) in a general setting that includes real, complex and quaternionic spaces. We formulate a common generalization of several results in real and complex spaces that also hold in the quaternionic space. A set of lines is called equiangular if the angle between each pair is the same. A set of MUBs is a collection of orthonormal bases such that the angle between vectors from different bases is constant. Regular structures of lines have been studied in several fields such as digital communication, quantum computing, discrete mathematics and analysis. Our new concept of a multipartite equiangular set of lines is a common generalization of equiangular lines and MUBs. We prove a bound on the size of such set of lines, which generalizes the well-known absolute upper bounds. The existence of d + 1 MUBs in Cd is only known for prime power dimensions. We study sets of d + 1 MUBs that are the union of a standard basis and an orbit of the Weyl-Heisenberg group. As an example, we construct such MUBs in prime power dimensions. We also show connections between spherical 2-designs and other structures of lines. Fiducial vectors have been widely used to construct large sets of equiangular lines. A complex vector is fiducial if its orbit under a Weyl-Heisenberg group is an equiangular set of d2 lines. We give a new characterization of fiducial vectors, one that simplifies and significantly reduces the number of equations that must be solved to find a fiducial vector. We consider some possible classes of fiducial vectors and prove several nonexistence results. For example, using our new characterization we prove that the construction of fiducial vectors in small prime dimensions by Appleby (2005) essentially does not generalize. Finally, we give some methods for constructing equiangular sets of lines in complex and quaternionic spaces. We also find numerical fiducial vectors with high precision in Cd , d ≤ 21.

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