On Propagation of Fixed Points of Quantum Operations and Beyond

TL;DR

This paper unifies and extends Noether's theorems for quantum and classical dynamical systems using fixed point propagation of quantum operations, providing new insights and characterizations.

Contribution

It introduces a unified approach to Noether theorems for quantum and classical systems via fixed point analysis of quantum operations and extends existing results with new characterizations.

Findings

Unified framework for Noether theorems in quantum and classical systems

New characterizations of symmetries and conservation laws

Examples and counter-examples illustrating the theory

Abstract

We show that some abstract results on propagation of fixed points for completely positive maps on $C^*$-algebras provide a natural approach to unify recent Noether type theorems on the equivalence of symmetries with conservation laws for dynamical systems of Markov processes, of quantum operations, and of quantum stochastic maps. In addition, we obtain some new Noether type theorems, provide examples and counter-examples, and extend most of the existing results with characterisations in terms of dual infinitesimal generators of the corresponding strongly continuous one-parameter semigroups.