Generalized Bases with a Resolution of the Identity: A Cooperative Game Theory Approach

Abstract

A quantum system with d-dimensional Hilbert space \(H_d\), is considered. A dresssing mechanism inspired by Shapley’s methodology in cooperative game theory, is used to convert a total set of \(n\ge d\) states (for which we have no resolution of the identity), into a ‘generalized basis’ of n mixed states with a resolution of the identity. Results based on these generalized bases are sensitive to physical changes and robust in the presence of noise. An arbitrary vector is expanded in these generalized bases, in terms of n component vectors. The concept of location index of a Hermitian operator, is introduced. Hermitian operators are studied using the concepts of comonotonic operators and comonotonicity intervals.