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

TL;DR

This paper introduces a dressing formalism inspired by cooperative game theory to transform pre-bases into generalized bases with a resolution of the identity, enhancing robustness and sensitivity in quantum state representations.

Contribution

It develops a novel renormalization method using M"obius transforms to create generalized bases from pre-bases, improving physical change detection and noise robustness.

Findings

Generalized bases resolve the identity and are practically useful.

Representations in these bases are robust against noise.

The method detects abrupt physical state changes despite noise.

Abstract

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.