On the Noncommutative Deformation of the Operator Graph Corresponding to the Klein Group

Abstract

We study the noncommutative operator graph ℒ_θdepending on a complex parameter θ recently introduced by M. E. Shirokov to construct channels with positive quantum zero-error capacity having vanishing n-shot capacity. We define a noncommutative group G and an algebra A_θwhich is the quotient of ℂG by a special algebraic relation depending on θ such that the matrix representation ϕ of A_θresults in the algebra ℳ_θgenerated by ℒ_θ. In the case of θ = ±1, the representation ϕ degenerates into a faithful representation of ℂK₄, where K₄is the Klein group. Thus, ℒ_θcan be regarded as a noncommutative deformation of the graph associated with the Klein group. Bibliography: 16 titles.