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Decoherence-free algebras in quantum dynamics

Daniele Amato, Paolo Facchi, Arturo Konderak

Abstract

In this Article we analyze the algebraic properties of the asymptotic dynamics of finite-dimensional open quantum systems in the Heisenberg picture. In particular, a natural product (Choi-Effros product) can be defined in the asymptotic regime. Motivated by this structure, we introduce a new space called the Choi-Effros decoherence-free algebra. Interestingly, this space is both a C* -algebra with respect to the composition product, and a B* -algebra with respect to the Choi-Effros product. Moreover, such space admits a direct-sum decomposition revealing a clear relationship with the attractor subspace of the dynamics. In particular, the equality between the attractor subspace and the Choi-Effros decoherence-free algebra is a necessary and sufficient condition for a faithful dynamics. Finally, we show how all the findings do not rely on complete positivity but on the much weaker Schwarz property.

Decoherence-free algebras in quantum dynamics

Abstract

In this Article we analyze the algebraic properties of the asymptotic dynamics of finite-dimensional open quantum systems in the Heisenberg picture. In particular, a natural product (Choi-Effros product) can be defined in the asymptotic regime. Motivated by this structure, we introduce a new space called the Choi-Effros decoherence-free algebra. Interestingly, this space is both a C* -algebra with respect to the composition product, and a B* -algebra with respect to the Choi-Effros product. Moreover, such space admits a direct-sum decomposition revealing a clear relationship with the attractor subspace of the dynamics. In particular, the equality between the attractor subspace and the Choi-Effros decoherence-free algebra is a necessary and sufficient condition for a faithful dynamics. Finally, we show how all the findings do not rely on complete positivity but on the much weaker Schwarz property.
Paper Structure (13 sections, 32 theorems, 131 equations)

This paper contains 13 sections, 32 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Unital completely positive maps
  4. Algebraic structure of the asymptotics
  5. Decoherence-free algebra
  6. Choi-Effros decoherence-free algebra
  7. The qubit case
  8. Examples
  9. Is complete positivity needed?
  10. Choi-Effros product for Schwarz maps
  11. Asymptotics of Schwarz maps
  12. Conclusions
  13. Choi-Effros product and decoherence-free algebra

Key Result

Lemma 2.1

Let $\Phi$ be an idempotent UCP map, namely $\Phi^2=\Phi$. Then

Theorems & Definitions (66)

  • Remark 1
  • Lemma 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Corollary 2.5
  • proof
  • Corollary 2.6
  • proof
  • Definition 2.7
  • ...and 56 more