Renormalization of total sets of states into generalized bases with a resolution of the identity

A total set of states for which we have no resolution of the identity (a `pre-basis'), is considered in a finite dimensional Hilbert space. A dressing formalism renormalizes them into density matrices which resolve the identity, and makes them a `generalized basis', which is practically useful. The dresssing mechanism is inspired by Shapley's methodology in cooperative game theory, and it uses M\"obius transforms. There is non-independence and redundancy in these generalized bases, which is quantified with a Shannon type of entropy. Due to this redundancy, calculations based on generalized bases, are sensitive to physical changes and robust in the presence of noise. For example, the representation of an arbitrary vector in such generalized bases, is robust when noise is inserted in the coefficients. Also in a physical system with ground state which changes abruptly at some value of the coupling constant, the proposed methodology detects such changes, even when noise is added to the parameters in the Hamiltonian of the system.