Equiangular quantum key distribution in more than two dimensions

We extend the spherical code-based key distribution protocols to qudits with dimensions 4 and 16 by constructing equiangular frames and their companions. We provide methods for equiangular frames in arbitrary dimensions for Alice to use and the companion frames, that have one antipode to eliminate one of the possibilities, made up of qudits with as part of Bob’s code. Non-orthogonal bases that form positive operator-valued measures can be constructed using the tools of frames (overcomplete bases of a Hilbert space), and here we apply them to key distribution that is robust due to the large size of the bases, making it hard to eavesdrop. We demonstrate a method to construct a companion frame for an equiangular tight frame for generated from the discrete Fourier transform, where p is any odd prime. The security analysis is based on the assumption of restricting possible attacks to an intercept/resend scenario, highlighting the advantages of qudit-based over qubit-based protocols.