An operator algebraic approach for generalized Cardano polynomials

Abstract

We develop an operator algebraic framework for generalized Cardano polynomials and show how their structure naturally leads to an operator formulation of Cardano method that is compatible with tools and concepts from quantum information theory. The generalized Cardano polynomials are constructed as a generalization of classical theory of Cardano formula for cubic equation, as well as through the spectral properties of a circular operator W, expressed in terms of n dimensional clock operator. This representation embeds Cardano type identities into the spectral theory of circulant operators. The construction clarifies the algebraic structure and solvability of a family of two parameters odd order polynomials, classically and through operator methods familiar in QIT. As application, spectral calculus on operator algebras is used to construct Cardano generated POVMs.