For any positive real number @math , the @math -frame potential of @math unit vectors @math is defined as @math . In this paper, we focus on the special case @math and establish the unique minimizer of @math for @math . Our results completely solve the minimization problem of @math -frame potential when @math , which confirms a conjecture posed by Chen, Gonzales, Goodman, Kang and Okoudjou.
For any @math , Ehler and Okoudjou provided another bound in @cite_12 : where the equality holds if and only if @math is an equiangular tight frame (ETF) in @math @cite_1 @cite_15 . We take @math as an example. Since there always exist @math unit vectors in @math forming an ETF @cite_3 , then the set of these @math vectors is the minimizer of the @math -frame potential for @math .
@cite_15 @cite_3 @cite_1 @cite_12
2072565885 1665316997 2098598867
We study frames from the viewpoint of coding theory. We introduce a numerical measure of how well a frame reconstructs vectors when some of the frame coefficients of a vector are lost and then attempt to find and classify the frames that are optimal in this setting. Quantum key distribution protocols based on equiangular spherical codes are introducedand their behavior under the intercept resend attack investigated. Such protocols offera greater range of secure noise tolerance and speed options than protocols based ontheir cousins, the mutually-unbiased bases, while also enabling the determination of thechannel noise rate without the need to sacrifice key bits. For fixed number of signalstates in a given dimension, the spherical code protocols offer Alice and Bob more noisetolerance at the price of slower key generation rates. We investigate the optimal configurations of n points on the unit sphere for a class of potential functions. In particular, we characterize these optimal configurations in terms of their approximation properties within frame theory. Furthermore, we consider similar optimal configurations in terms of random distribution of points on the sphere. In this probabilistic setting, we characterize these optimal distributions by means of probabilistic frames. Our work also indicates some connections between statistical shape analysis and frame theory.