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On Propagation of Fixed Points of Quantum Operations and Beyond

Aurelian Gheondea

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.

On Propagation of Fixed Points of Quantum Operations and Beyond

Abstract

We show that some abstract results on propagation of fixed points for completely positive maps on -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.

Paper Structure

This paper contains 14 sections, 22 theorems, 101 equations.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Propagation of Fixed Points in $C^*$-Algebras
  4. Fixed Points of $w^*$-Continuous One-Parameter Semigroups
  5. An Ergodic Theorem in von Neumann Algebras
  6. Dynamical Systems of Stochastic/Markov Maps: The Noncommutative Case
  7. Discrete Semigroups of Stochastic/Markov Maps.
  8. Continuous One-Parameter Semigroups of Stochastic/Markov Maps.
  9. Dynamics for Markov Processes: The Real Commutative Case
  10. Discrete Stochastic/Markov Semigroups.
  11. Continuous Stochastic/Markov Semigroups.
  12. Constants of Dynamical Quantum Systems
  13. Discrete Quantum Semigroups
  14. Continuous Quantum Semigroups.

Key Result

Proposition 1.1

Let $\Phi_{\mathcal{A}}$ be the unital quantum operation in the Schrödinger picture as in e:pac, its dual $\Phi_{\mathcal{A}}^\sharp$ in the Heisenberg picture, and let $B\in{\mathcal{B}}({\mathcal{H}})$ be a fixed point of $\Phi_{\mathcal{A}}^\sharp$. The following assertions are equivalent:

Theorems & Definitions (38)

  • Proposition 1.1
  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Corollary 2.3
  • proof
  • Theorem 2.4
  • Theorem 2.5
  • proof
  • Theorem 2.6
  • ...and 28 more