# Dressed coherent states in finite quantum systems: a cooperative game theory approach

## TL;DR

This paper introduces a novel approach using cooperative game theory to create dressed coherent states in finite quantum systems, enabling better analysis of Hamiltonian dynamics and phase space behavior.

## Contribution

It applies Shapley methodology to renormalize coherent states into density matrices, providing a new formalism for phase space analysis in quantum systems.

## Key findings

- Dressed coherent states resolve the identity and are practically useful.
- Renormalized Q-functions resemble Shapley values, offering new insights.
- The formalism generalizes to sets without resolution of the identity.

## Abstract

A quantum system with variables in Z(d) is considered. Coherent density matrices and coherent projectors of rank n are introduced, and their properties (e.g., the resolution of the identity) are dis- cussed. Cooperative game theory and in particular the Shapley methodology, is used to renormalize coherent states, into a particular type of coherent density matrices (dressed coherent states). The Q-function of a Hermitian operator, is then renormalized into a physical analogue of the Shapley values. Both the Q-function and the Shapley values, are used to study the relocation of a Hamilto- nian in phase space as the coupling constant varies, and its effect on the ground state of the system. The formalism is also generalized for any total set of states, for which we have no resolution of the identity. The dressing formalism leads to density matrices that resolve the identity, and makes them practically useful.